j. differential geometry

65 (2003) 341-467

ENUMERATION OF GENUS-TWO CURVES WITH A FIXED COMPLEX STRUCTURE IN ℙ^{2} AND ℙ^{3}

ALEKSEY ZINGER

Abstract

The main concrete result of this paper is enumeration of genus-two curves with complex structure fixed in ℙ^{2} and ℙ^{3}. Along the way, rational curves with certain simple singularities are counted as well. While the methods described can be used to count positive-genus curves in some other cases, the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces.

1. Introduction

1.1 Background and results

Let (Σ, j_{Σ}) be a nonsingular Riemann surface of genus g ≥ 2, and let d, n be positive integers with d ≥ 1 and n ≥ 2. Denote by H_{Σ,d}(ℙ^{n}) the set of simple holomorphic maps from Σ to ℙ^{n} of degree d. Let μ = (μ_{1}, …, μ_{N}) be an N-tuple of proper complex submanifolds of ℙ^{n} such that

∑_{l=1}^{l=N} codim_{ℂ}μ_{l} = d(n + 1) − n(g − 1) + N. (1.1)

The author was partially supported by NSF Graduate Research Fellowship and NSF grant DMS-9803166.

Received 01/31/2002.

If these submanifolds are in general position, the cardinality of the set

(1.2) H_{Σ,d}(μ) = {(y_{1}, …, y_{N}; u) : u ∈ H_{Σ,d}(ℙ^{n});

y_{l} ∈ Σ, u(y_{l}) ∈ μ_{l} ∀l = 1, …, N}

is finite, and its cardinality depends only on the homology classes of μ_{1}, …, μ_{N}. The group Aut(Σ) of holomorphic automorphisms of Σ acts freely on H_{Σ,d}(μ). For this reason, algebraic geometers prefer to consider the ratio of the cardinality of the set H_{Σ,d}(μ) and the order of the group Aut(Σ). For a dense open subset of complex structures on Σ, the cardinality of the set H_{Σ,d}(μ) has the same order. The same is true of the set Aut(Σ). If j_{Σ} lies in this open subset, we denote the above ratio by n_{g,d}(μ). This number is precisely the number of irreducible, nodal degree-d genus-g curves in ℙ^{n} with a fixed generic complex structure on the normalization and passing through the constraints μ_{1}, …, μ_{N}.

For g = 0, 1, one can define the numbers n_{g,d}(μ) for constraints of appropriate total codimension by counting the number of equivalence classes under the action of the now infinite group Aut(Σ) on the set H_{Σ,d}(μ) defined as in (1.2) above. It is shown in [10] that

n_{0,d}(μ) = RT_{0,d}(μ_{1}, μ_{2}, μ_{3}; μ_{4}, …, μ_{N}),

where RT_{0,d}(·; ·) denotes the symplectic invariant of ℙ^{n} as defined in [10]. For g = 1, in [4] the difference

RT_{g,d}(μ_{1}; μ_{2}, …, μ_{N}) − 2n_{g,d}(μ)

is expressed as an intersection number on a blowup of the space of degree-d (N +1)-marked rational curves passing through the constraints μ_{1}, …, μ_{N}. This number is shown to be computable, and explicit formulas are given in the n = 2, 3 cases. On the other hand, the symplectic invariant is easily computable from the two composition laws of [10]. A completely different approach for the n = 2, g = 1 case is given in [8]. Using this algebraic approach, [5] express n_{2,d} in the n = 2 case in terms of the numbers n_{0,d′} with d′ ≤ d.

In this paper, we extend the approach of [4] to compute the difference

RT_{2,d}(·; μ_{1}, …, μ_{N}) − 2n_{2,d}(μ)

in the n = 2, 3 cases. The reason for the factor of two above is that the automorphism group of a generic genus-two Riemann surface has order two. The following two theorems are the main results of this paper. The two tables list some low-degree genus-two numbers. Evidence in support of the two formulas is described in Subsection 5.8, where more low-degree numbers for ℙ^{3} are also given.

Theorem 1.1. Let n_{2,d} denote the number of genus-two degree-d curves that pass through 3d − 2 points in general position in ℙ^{2} and have a fixed generic complex structure. With n_{d} = n_{0,d},

n_{2,d} = 3(d^{2} − 1)n_{d}

+ \frac{1}{2} ∑_{d_{1}+d_{2}=d}(d_{1}^{2}d_{2}^{2} + 28 − 16\frac{9d_{1}d_{2} − 1}{3d − 2}) \binom{3d − 2}{3d_{1} − 1}d_{1}d_{2}n_{d_{1}}n_{d_{2}}.

\begin{matrix}d & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ n_{2,d} & 0 & 0 & 0 & 14,400 & 6,350,400 & 3,931,128,000 & 3,718,909,209,600 \end{matrix}

Theorem 1.2. If d is a positive integer and μ is a tuple of p points and q lines in general position in ℙ^{3} with 2p + q = 4d − 3,

2n_{2,d}(μ) = RT_{2,d}(·; μ) − CR(μ),

where CR(μ) is the sum of the intersection numbers of explicit tautological classes in the space of stable rational maps into ℙ^{3}.

degree  4  5  6
(p,q) (3,7) (2,9) (1,11) (8,1)  (0,17)  (10,1)
n_{2,d}(μ) 14,400 307,200 4,748,160 9,600 7,494,574,433,280 1,301,760

A formula for CR(μ) is given in Theorem 5.28. Intersection numbers of tautological classes are shown to be computable in [9]. In fact, we give a method of computing these numbers along the lines of that in [4], which is slightly different from the method of [9]; see Subsection 5.7.

The numbers we obtain in the n = 2 case are different from the numbers given in [5]. However, our numbers can be recovered via the approach of [5]. In particular,

n_{2,d} = 6 (n_{2,d}^{KQR} + τ_{d}) ,

where τ_{d} is the number of degree-d tacnodal rational curves passing through (3d − 2) points in general position in ℙ^{2}. The factor of six is a minor omission on the authors’ part. The contribution of 6τ_{d} arises from a three-component stratum [5] ruled out by their Remark 3.12, which is stated without a proof. Details can be found in [14].

This paper combines the topological tools of Section 3 with the explicit analytic structure theorems of [13]. Together these give a general framework that will hopefully provide a way of computing positive-genus enumerative invariants from the symplectic ones in any homogeneous Kahler manifold. In fact, the methods of this paper should apply, with very little change, at least up to genus seven in ℙ^{2}, to the g = 3 case in ℙ^{3}, and to the g = 2 case in ℙ^{4}. Genus-three plane fixed-complex-structure curves have been enumerated; see [15].

Along the way, we enumerate cuspidal rational curves in ℙ^{2} and twocomponent rational curves connected at a tacnode in ℙ^{3}; see Lemmas 5.4 and 5.5. The formula of Lemma 5.4 is not new. However, the methods of this paper can be used to count rational curves with singularities of “local nature.” By “local nature,” we mean that a description of the singularities can be given that involves at most one point of each component of the normalization of the curve. For example, a tacnode on a one-component curve is not of “local nature,” but a tacnode at the node common to two irreducible components of a curve is. So is a cusp of any arbitrary pre-specified form. Unlike many approaches in algebraic geometry, our methods are not limited to ℙ^{2} and apply just as well to arbitrary-dimensional projective spaces. In fact, the machinery itself can be used on other homogeneous manifolds to express counts of singular rational curves in terms of intersections of tautological classes on moduli spaces of rational maps. However, there is no general method of computing these intersections for homogeneous manifolds other than the projective spaces.

The author is grateful to T. Mrowka for pointing out the paper [4] and many useful discussions, and G. Tian for first introducing him to Gromov-Witten invariants. The author also thanks R. Vakil for sharing some of his expertise in enumerative algebraic geometry, and A. J. de Jong and J. Starr for help with understanding [5].

1.2 Summary

If ν ∈ Γ(Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n}), let M_{Σ,ν,d} denote the set of all smooth maps u from Σ to ℙ^{n} of degree d such that \overline{∂}u|_{z} = ν|_{(z,u(z))} for all z ∈ Σ. If μ is as above, put

M_{Σ,ν,d}(μ) = {(y_{1}, …, y_{N}; u) : u ∈ M_{Σ,ν,d};

y_{l} ∈ Σ, u(y_{l}) ∈ μ_{l} ∀l = 1, …, N}.

For a generic ν, M_{Σ,ν,d} is a smooth finite-dimensional oriented manifold, and M_{Σ,ν,d}(μ) is a zero-dimensional finite submanifold of M_{Σ,ν,d} × Σ^{N}, whose cardinality (with sign) depends only the homology classes of μ_{1}, …, μ_{N}; see [10]. The symplectic invariant RT_{g,d}(; μ) is the signed cardinality of the set M_{Σ,ν,d}(μ).

If ∥ν_{i}∥_{C^{0}} ⟶ 0 and (\underline{y}_{i}; u_{i}) ∈ M_{Σ,ν_{i},d}(μ), then a subsequence of \{(\underline{y}_{i}; u_{i})\}_{i=1}^{∞} must converge in the stable-map topology to one of the following:

(1) an element of H_{Σ,d}(μ);

(2) (Σ_{T} , \underline{y}, u), where Σ_{T} is a bubble tree of S^{2}’s attached to Σ with marked points y_{1}, …, y_{N}, and u : Σ_{T} ⟶ ℙ^{n} is a holomorphic map such that u(y_{l}) ∈ μ_{l} for l = 1, …, N, and:

(2a) u|_{Σ} is simple and the tree contains at least one S^{2}; (2b) u|_{Σ} is multiply-covered; (2c) u|_{Σ} is constant and the tree contains at least one S^{2}.

By Proposition 6.6, the case (2a) does not occur if the constraints are in general position. Furthermore, if g = 2, (2b) cannot occur either if n = 2, 3 or if n = 4 and d ≠ 2. It is well-known that n_{2,2}(μ) = 0, and thus the case n = 4 and g = d = 2 presents no interest. Our approach will be to take t very small and to count the number of elements of M_{Σ,tν,d}(μ) that lie near the maps of type (2c). The rest of the elements of M_{Σ,tν,d}(μ) must lie near the space H_{Σ,d}(μ). By Proposition 3.30 in [13] and Corollary 6.5, there is a one-to-one correspondence between the elements of H_{Σ,d}(μ) and the nearby elements of M_{Σ,tν,d}(μ), at least if d ≥ 3. If d = 1, 2, H_{Σ,d}(μ) = ∅; see the proof of Proposition 6.6. Thus, we are able to compute the cardinality of H_{Σ,d}(μ) by computing the total number of elements of M_{Σ,tν,d}(μ) that lie near the maps of type (2c).

In Subsection 1.3, we summarize our notation for spaces of bubble maps and vector bundle over them. For details, the reader is referred to [13]. In Section 2, we describe an obstruction-bundle setup and state Theorem 2.7, which relates the elements of M_{Σ,d,tν}(μ) lying near the maps of type (2c) to the zero set of a map between two bundles. We also describe the local structure of certain spaces of stable rational maps. These spaces are very familiar in algebraic geometry, but for our computations in Section 5 we need the analytic estimate of Theorem 2.8.

In Section 3, we introduce a category of mostly smooth (ms) objects and maps and present the topological tools used in Section 4. We view moduli spaces of rational maps as ms-manifolds, rather than as stacks. This approach allows to study the behavior of certain bundle sections over these topological spaces using the analytic estimate of Theorem 2.8.

In Section 4, we use the topological tools of Subsection 3.1 to show that the number of zeros of the maps of Theorem 2.7 is the same as the number of zeros of explicit affine maps between vector bundles over cartesian products of spaces of rational maps with Σ^{k}. The results of this simplification are summarized in Subsection 4.9. In Section 5, we relate the zeros of these affine maps to the intersection numbers of spaces of stable rational maps into ℙ^{n}. We use Theorem 2.8 and Section 3 for local excess-intersection type of computations. We conclude with the very explicit formula of Theorem 1.1 in the n = 2 case and a somewhat less explicit one of Theorem 5.28 in the n = 3 case.

1.3 Notation

In this subsection, we give a brief description of the most important notation used in this paper. See Section 2 in [13] for more details.

Let q_{N}, q_{S} : ℂ ⟶ S^{2} ⊂ ℝ^{3} be the stereographic projections mapping the origin in ℂ to the north and south poles, respectively. Explicitly,

(1.3)  q_{N}(z) = (\frac{2z}{1 + |z|^{2}} , \frac{1 − |z|^{2}}{1 + |z|^{2}} ) ∈ ℂ × ℝ,
q_{S}(z) = (\frac{2z}{1 + |z|^{2}} , \frac{−1 + |z|^{2}}{1 + |z|^{2}} ) .

We denote the south pole of S^{2}, i.e., the point (0, 0, −1) ∈ ℝ^{3}, by ∞. Let

e_{∞} = (0, 0, 1) = dq_{S}|_{0}(\frac{∂}{∂s}) ∈ T_{∞}S^{2}, (1.4)

where we write z = s + it ∈ ℂ. We identify ℂ with S^{2} − {∞} via the map q_{N}. If N is any nonnegative integer, let [N] = {1, …, N}.

Definition 1.3.

(1) A finite partially ordered set I is a linearly ordered set if for all i_{1}, i_{2}, h ∈ I such that i_{1}, i_{2} < h, either i_{1} ≤ i_{2} or i_{2} ≤ i_{1}.

(2) A linearly ordered set I is a rooted tree if I has a unique minimal element, i.e., there exists ˆ0 ∈ I such that ˆ0 ≤ i for all i ∈ I.

If I is a linearly ordered set, let Î be the subset of the non-minimal elements of I. For every h ∈ Î, denote by ι_{h} ∈ I the largest element of I which is smaller than h. We call ι : Î ⟶ I the attaching map of I. Suppose I = ⨆_{k∈K} I_{k} is the splitting of I into rooted trees such that k is

the minimal element of I_{k}. If ˆ1 ∉ I, we define the linearly ordered set I ⊔_{k} ˆ1 to be the set I ⊔ {ˆ1} with all partial-order relations of I along with the relations

k < ˆ1, ˆ1 < h if h ∈ Î_{k}.

If I is a rooted tree, we write I ⊔ ˆ1 for I ⊔_{k} ˆ1.

If S = Σ or S = S^{2} and M is a finite set, a ℙ^{n}-valued bubble map with M-marked points is a tuple

b = (S, M, I; x, (j, y), u),

where I is a linearly ordered set, and

x : Î ⟶ S ∪ S^{2}, j : M ⟶ I, y : M ⟶ S ∪ S^{2}, and
u : I ⟶ C^{∞}(S; ℙ^{n}) ∪ C^{∞}(S^{2}; ℙ^{n})

are maps such that

x_{h} ∈ \begin{cases}S^{2} − \{∞\}, & if ι_{h} ∈ Î; \\ S, & if ι_{h} ∉ Î, \end{cases} y_{l} ∈ \begin{cases}S^{2} − \{∞\}, & if j_{l} ∈ Î; \\ S, & if j_{l} ∉ Î, \end{cases}

u_{i} ∈ \begin{cases}C^{∞}(S^{2}; ℙ^{n}), & if i ∈ Î; \\ C^{∞}(S; ℙ^{n}), & if i ∉ Î, \end{cases}

and u_{h}(∞) = u_{ι_{h}}(x_{h}) for all h ∈ Î. We associate such a tuple with Riemann surface

Σ_{b} = (⨆_{i∈I} Σ_{b,i}) / ∼, where Σ_{b,i} = \begin{cases}\{i\} × S^{2}, & if i ∈ Î; \\ \{i\} × S, & if i ∉ Î, \end{cases}
and (h, ∞) ∼ (ι_{h}, x_{h}) ∀h ∈ Î,

with marked points (j_{l}, y_{l}) ∈ Σ_{b,j_{l}}, and continuous map u_{b} : Σ_{b} ⟶ ℙ^{n}, given by u_{b}|_{Σ_{b,i}} = u_{i} for all i ∈ I. We require that all the singular points of Σ_{b}, i.e., (ι_{h}, x_{h}) ∈ Σ_{b,ι_{h}} for h ∈ Î, and all the marked points be distinct. In addition, if Σ_{b,i} = S^{2} and u_{i∗}[S^{2}] = 0 ∈ H_{2}(ℙ^{n}; ℤ), then Σ_{b,i} must contain at least two singular and/or marked points of Σ_{b} other than (i, ∞). Two bubble maps b and b′ are equivalent if there exists a homeomorphism ϕ : Σ_{b} ⟶ Σ_{b′} such that u_{b} = u_{b′} ∘ ϕ, ϕ(j_{l}, y_{l}) = (j′_{l}, y′_{l}) for all l ∈ M, ϕ|_{Σ_{b,i}} is holomorphic for all i ∈ I, and ϕ|_{Σ_{b,i}} = Id if S = Σ and i ∈ I − Î.

The general structure of bubble maps is described by tuples T = (S, M, I; j, \underline{d}), with d_{i} ∈ ℤ describing the degree of the map u_{b} on Σ_{b,i}. We call such tuples bubble types. Bubble type T is simple if I is a rooted tree; T is is basic if Î = ∅; T is semiprimitive if ι_{h} ∉ Î for all h ∈ Î. We call semiprimitive bubble type T primitive if j_{l} ∈ Î for all j_{l} ∈ M. The above equivalence relation on the set of bubble maps induces an equivalence relation on the set of bubble types. For each h, i ∈ I, let

H_{i}T = {h ∈ Î: ι_{h} = i}, M_{i}T = {l ∈ M : j_{l} = i},

χ_{T} h = \begin{cases}0, & if d_{i} = 0 ∀i ≤ h; \\ 1, & if d_{h} ≠ 0, but d_{i} = 0 ∀i < h; \\ 2, & otherwise. \end{cases}

Let H_{T} denote the space of all holomorphic bubble maps with structure T .

The automorphism group of every bubble type T we encounter in Sections 4 and 5 is trivial. Thus, every bubble type discussed below is presumed to be automorphism-free.

If S = Σ, we denote by M_{T} the set of equivalence classes of bubble maps in H_{T}. Then there exists M_{T}^{(0)} ⊂ H_{T} such that M_{T} is the quotient of M_{T}^{(0)} by an (S^{1})Î-action. Corresponding to this action, we obtain |Î| line orbi-bundles {L_{h}T ⟶ M_{T} : h ∈ Î}. The bundle of gluing parameters in this case is

F T = ⨁_{h∈Î} F_{h}T , where F_{h,[b]}T = \begin{cases}L_{h,[b]}T ⊗ L_{ι_{h},[b]}^{∗}T , & if ι_{h} ∈ Î; \\ L_{h,[b]}T ⊗ T_{x_{h}}Σ, & if ι_{h} ∉ Î. \end{cases}

Let F^{∅}T = {υ = (υ_{h})_{h∈Î} ∈ F T : υ_{h} ≠ 0 ∀h ∈ Î}. Each line orbibundle F_{h}T ⟶ M_{T} is the quotient of a line bundle F_{h}^{(0)}T ⟶ M_{T}^{(0)} by a G_{T} ≡ (S^{1})Î-action. We denote by F^{(∅)}T the preimage of F^{∅}T in

F^{(0)}T ≡ ⨁_{h∈Î}F_{h}^{(0)}T . The bundles F^{∅}T , F^{(∅)}T , and F_{h}^{(0)}T are defined

even if the automorphism group of T is nontrivial.

For each bubble type T = (S^{2}, M, I; j, \underline{d}), let

U_{T} = {[b] : b = (S^{2}, M, I; x, (j, y), u) ∈ H_{T} ,

u_{i_{1}}(∞) = u_{i_{2}}(∞) ∀i_{1}, i_{2} ∈ I − Î}.

Similarly to the S = Σ case above, U_{T} is the quotient of a subset B_{T} of H_{T} by a G̃_{T} ≡ (S^{1})^{I}-action. Denote by U_{T}^{(0)} the quotient of B_{T} by G_{T} ≡ (S^{1})Î ⊂ G̃_{T}. Then U_{T} is the quotient of U_{T}^{(0)} by the residual

G_{T}^{∗} ≡ (S^{1})^{I−Î} ⊂ G̃_{T}

action. Corresponding to these quotients, we obtain line orbi-bundles {L_{h}T ⟶ U_{T}^{(0)} : h ∈ Î} and {L_{i}T ⟶ U_{T} : i ∈ I}. Let

F T =⨁_{h∈Î} F_{h}T ⟶U_{T}^{(0)}, where F_{h,[b]}T =\begin{cases}L_{h,[b]}T ⊗ L_{ι_{h},[b]}^{∗}T , & if ι_{h} ∈ Î; \\ L_{h,[b]}T , & if ι_{h} ∉ Î; \end{cases}

FT = ⨁_{h∈Î} F_{h}T ⟶ U_{T} , where F_{h,[b]}T = L_{h,[b]}T ⊗ L_{ι_{h},[b]}^{∗}T .

The orbi-bundles F_{h}T and F_{i}T are quotients of line bundles over B_{T} similarly to the S = Σ case.

The stable-map topology on the space of equivalence classes of bubble maps induces a partial ordering on the set of bubble types and their equivalence classes such that the spaces

\overline{M}_{T} = ⋃_{T′≤T} M_{T′}, \overline{U}_{T}^{(0)} = ⋃_{T′≤T} U_{T′}^{(0)}, and \overline{U}_{T} = ⋃_{T′≤T} U_{T′}

are compact and Hausdorff. The G_{T}^{∗} -action on U_{T}^{(0)} extends to an action on \overline{U}_{T}^{(0)}, and thus line orbi-bundles L_{i}T ⟶ U_{T} with i ∈ I − Î extend over \overline{U}_{T}. The evaluation maps

ev_{l} : H_{T} ⟶ ℙ^{n}, ev_{l}((S, M, I; x, (j, y), u)) = u_{j_{l}}(y_{l}),

descend to all the quotients and induce continuous maps on \overline{M}_{T}, \overline{U}_{T}, and \overline{U}_{T}^{(0)}. If μ = μ_{M} is an M-tuple of submanifolds of ℙ^{n}, let

M_{T} (μ) = {b ∈ M_{T} : ev_{l}(b) ∈ μ_{l} ∀l ∈ M}

and define spaces U_{T} (μ), \overline{U}_{T} (μ), etc. in a similar way. If S = S^{2}, we define another evaluation map,

ev : B_{T} ⟶ ℙ^{n} by ev((S^{2}, M, I; x, (j, y), u)) = uˆ0(∞),

where ˆ0 is any minimal element of I. This map descends to U_{T}^{(0)} and U_{T}. If μ = μ_{{ˆ0}⊔M} is a tuple of constraints, let

U_{T} (μˆ0; μ_{M}) = {b ∈ U_{T} (μ_{M}) : ev(b) ∈ μˆ0}

and define U_{T}^{(0)}(μˆ0; μ_{M}), etc. similarly. If S = Σ, T is a simple bubble type, and dˆ0 = 0, define

ev : H_{T} ⟶ ℙ^{n} by ev((Σ, M, I; x, (j, y), u)) = uˆ0(Σ).

This map is well-defined, since uˆ0 is a degree-zero holomorphic map and thus is constant.

If T is any bubble type, let ⟨T ⟩ be the basic bubble such that T ≤ ⟨T ⟩. If T is a simple bubble type, let \overline{T} be the bubble type obtained from T by dropping the minimal element ˆ0 from the indexing set I and the subset Mˆ0T from M. Note that if T is primitive, \overline{T} is basic.

Finally, if X is any space, F ⟶ X a normed vector bundle, and δ : X ⟶ ℝ is any function, let

F_{δ} = {(b, v) ∈ F : |v|_{b} < δ(b)}.

Similarly, if Ω is a subset of F, let Ω_{δ} = F_{δ} ∩ Ω. If υ = (b, v) ∈ F, denote by b_{υ} the image of υ under the bundle projection map, i.e., b in this case.

2. Analysis

2.1 The basic setup

In this section, we focus on bubble types T = (S, M, I; j, \underline{d}) such that either S = S^{2} or dˆ0 = 0. In the first case, we describe a small neighborhood of U_{T} (μ) in \overline{U}_{⟨T ⟩}(μ) and the behavior of sections of certain bundles over \overline{U}_{⟨T ⟩}(μ) near U_{T} (μ); see Theorem 2.8. This theorem is deduced from Theorem 3.33 in [13]. If T is a simple bubble type, S = Σ, and dˆ0 = 0, we describe the elements of M_{Σ,tν,d}(μ) lying near M_{T} (μ) as the zero set of a map defined on an open subset of the bundle F T ; see Theorem 2.7. The map takes values in a bundle over M_{T} (μ), which is the analogue of Taubes’s obstruction bundle of [11] in this setting. Theorem 2.7 is a consequence of Theorem 3.29 in [13], which requires us to make two major choices. This is done in the next two subsections.

If T = (S, M, I; j, \underline{d}) and S = S^{2}, by Corollaries 6.3 and 6.5, T is a (ℙ^{n}, J)-regular bubble type in the sense of Definition 3.1 in [13]. This regularity property implies that:

(R1) H_{T} is a smooth manifold.

(R2) For any b = (S, M, I; x, (j, y), u) ∈ H_{T}, a neighborhood of b in H_{T}, is modeled on

ker (D_{b} : Γ(b) ⟶ Γ^{0,1}(b)) ⊕ ⨁_{h∈I} T_{x_{h}}Σ_{b,ι_{h}} ⊕ ⨁_{l∈M} T_{y_{l}}Σ_{b,j_{l}}.

(R3) D_{b} : Γ(b) ⟶ Γ^{0,1}(b) is surjective for all b ∈ H_{T}.

Here Γ^{0,1}(b) denotes the space of u_{b}^{∗}T ℙ^{n}-valued (0, 1)-forms on the components of Σ_{b}, while Γ(b) is the set of vector fields ξ on the components of Σ_{b} that agree at the nodes and such that ξ(i_{1}, ∞) = ξ(i_{2}, ∞) for all i_{1}, i_{2} ∈ I − Î. The operator D_{b} is the linearization of the \overline{∂}-operator with respect to a connection in T ℙ^{n}. Along H_{Σ}, it is independent of the choice of the connection. On the other hand, if T is a simple bubble type, S = Σ, and dˆ0 = 0, by the same two corollaries, T is a (ℙ^{n}, J)- semiregular bubble type in the sense of Definition 3.2 in [13]. This means that (R1) and (R2) are satisfied, with Γ(b) defined as above but omitting the last condition. Property (R3) is not satisfied, and in fact by the two corollaries,

coker D_{b} ≈ H_{Σ}^{0,1} ⊗ T_{ev(b)}ℙ^{n} ∀b ∈ H_{T} ,

where H_{Σ}^{0,1} is the space of harmonic (0, 1)-forms on Σ. This cokernel bundle descends to a bundle Γ_{−}^{0,1} ⟶ M_{T}, which will be our obstruction bundle.

If S = Σ, for the gluing construction in [13], we choose a smooth family {g_{b,ˆ0} : b ∈ H_{T} } of metrics on Σ such that for all

b = (Σ, M, I; x, (j, y), u) ∈ H_{T} ,

the metric g_{b,ˆ0} is flat on a neighborhood of x_{h} in Σ for all h ∈ Î such that ι_{h} = ˆ0. This family of metrics, in fact, depends only on the sets {x_{h} : ι_{h} = ˆ0}. Along with the standard metric on S^{2}, the metric g_{b,ˆ0} induces a Riemannian metric g_{b} = (g_{b,i})_{i∈I} on Σ_{b} = ⋃_{i∈I} Σ_{b,i}. If

S = S^{2}, we take g_{b,i} to be the standard metric on Σ_{b,i} = S^{2} for all i ∈ I. With notation as above, if x_{h}, z ∈ Σ_{b,ˆ0} = Σ, let r_{b,h}(z) = d_{g_{b,ˆ0}}(x_{h}, z). If x_{h}, z ∈ Σ_{b,i} = S^{2} and z ≠ ∞, let r_{b,h}(z) = |z − x_{h}|.

For each υ = (b, v_{h})_{h∈Î} ∈ F^{(0)}T sufficiently small, in [13] we then define a complex curve Σ_{υ}, smooth maps q_{υ} : Σ_{υ} ⟶ Σ_{b} and q_{υ,i} : Σ_{υ,ˆ0} ⟶ Σ_{b} for i ∈ I, and Riemannian metric g_{υ} on Σ on Σ_{υ} such that:

(G1) The linearly ordered set corresponding to Σ_{υ} is

I(υ) ≡ I − {h ∈ Î: v_{h} ≠ 0}.

(G2) The map q_{υ}|_{Σ_{υ,ˆ0}} factors through each of the maps q_{υ,i}.

(G3) q_{υ} : (Σ_{υ}, g_{υ}) ⟶ (Σ_{b}, g_{b}) is an isometry (and thus holomorphic) outside of the annuli

A_{υ,h}^{+} = q_{υ,ι_{h}}^{−1} (\{z ∈ Σ_{b,ι_{h}} : |v_{h}|^{\frac{1}{2}} ≤ r_{b,h}(z) ≤ 2|v_{h}|^{\frac{1}{2}} \}); (2.1)

A_{υ,h}^{−} = q_{υ,ι_{h}}^{−1} (\{z ∈ Σ_{b,ι_{h}} : \frac{1}{2}|v_{h}|^{\frac{1}{2}} ≤ r_{b,h}(z) ≤ |v_{h}|^{\frac{1}{2}} \}) .

(G4) q_{υ,ι_{h}} : (A_{υ,h}^{±}, g_{υ}) ⟶ (q_{υ,ι_{h}}(A_{υ,h}^{±}), g_{b}) is an isometry.

The map q_{υ} collapses disjoint circles on Σ_{υ} and identifies the resulting surfaces with S^{2} in a manner encoded by υ. Alternatively, (Σ_{υ}, g_{υ}) can be viewed as the surface obtained by smoothing (some of) the nodes of Σ_{b}. The maps q_{υ} and q_{υ,i} are constructed explicitly by fixing a smooth function β : ℝ ⟶ [0, 1] such that

(2.2) β(t) = \begin{cases}0, & if t ≤ 1; \\ 1, & if t ≥ 2, \end{cases} and β′(t) > 0 if t ∈ (1, 2).

If r > 0, let β_{r} ∈ C^{∞}(ℝ; ℝ) be given by β_{r}(t) = β(r^{− \frac{1}{2}} t). Note that

supp(β_{r}) = [r^{\frac{1}{2}} , 2r^{\frac{1}{2}} ], ∥β′_{r}∥_{C^{0}} ≤ C_{β}r^{− \frac{1}{2}} , and ∥β′′_{r}∥_{C^{0}} ≤ C_{β}r^{−1}. (2.3)

These cutoff functions will not appear in the main statements of this paper, but they do show up in the proofs of Lemma 2.1, Theorem 2.8, and Proposition 4.4. Having constructed the maps q_{υ}, we let b(υ) = (Σ_{υ}, u_{υ}) = (Σ_{υ}, u_{b} ∘ q_{υ}). The marked points on Σ_{υ} are the preimages of the marked points of Σ_{b} under the map q_{υ}.

We also need to choose a smooth family {g_{ℙ^{n},b} : b ∈ M_{T}^{(0)}} of metrics

on ℙ^{n} invariant under the equivalence relation on M_{T}^{(0)} if S = Σ and on B_{T} if S = S^{2}. While taking g_{ℙ^{n},b} to be the standard metric on ℙ^{n} may be the canonical choice, for computational reasons it is more convenient to take g_{ℙ^{n},b} = g_{ℙ^{n},ev(b)}, where {g_{ℙ^{n},q} : q ∈ ℙ^{n}} is the family of metrics of Lemma 2.1.

Lemma 2.1. There exist r_{ℙ^{n}} > 0 and a smooth family of Kahler metrics {g_{ℙ^{n},q} : q ∈ ℙ^{n}} on ℙ^{n} with the following property. If B_{q}(q′, r) ⊂ ℙ^{n} denotes the g_{ℙ^{n},q}-geodesic ball about q′ of radius r, the triple (B_{q}(q, r_{ℙ^{n}}), J, g_{ℙ^{n},q}) is isomorphic to a ball in ℂ^{n} for all q ∈ ℙ^{n}.

Proof. On the open set U_{0} = {[X_{0} : ⋯: X_{n}] ∈ ℙ^{n} : X_{0} ≠ 0}, the Fubini-Study symplectic form is given by

ω_{ℙ^{n}} = \frac{i}{2π} ∂\overline{∂} ln(1 + f_{0}), where f_{0}([X_{0} : ⋯: X_{n}]) = ∑_{k∈[n]} |X_{k}/X_{0}|^{2}; (2.4)

see [3, p. 31]. Let q = [1 : 0 : ⋯: 0]. Set

ω_{ℙ^{n},q,ϵ} = \frac{i}{2π} ∂\overline{∂}\{f_{0} + (β_{ϵ^{2}} ∘ f_{0})( ln(1 + f_{0}) − f_{0})\}. (2.5)

Note that ω_{ℙ^{n},q,ϵ} agrees with ω_{ℙ^{n}} outside of the set {f_{0} ≤ 2ϵ} and with the standard symplectic form ω_{ℂ^{n}} on {f_{0} ≤ ϵ}. Here we view ω_{ℂ^{n}} as a form on U_{0} via the coordinates z_{0,k} = X_{k}/X_{0}, k ∈ [n]. In particular, ω_{ℙ^{n},q,ϵ} is globally defined, and the corresponding Riemannian metric on {f_{0} ≤ ϵ} is flat. Furthermore,

(2.6)  ω_{ℙ^{n},q,ϵ} = {(1 − β_{ϵ^{2}} ∘ f_{0})ω_{ℂ^{n}} + (β_{ϵ^{2}} ∘ f_{0})ω_{ℙ^{n}}}

+ \frac{i}{2π} \{(∂(β_{ϵ^{2}} ∘ f_{0}))(\overline{∂}f̃_{0})

− (\overline{∂}(β_{ϵ^{2}} ∘ f_{0}))(∂f̃_{0}) + (∂\overline{∂}(β_{ϵ^{2}} ∘ f_{0}))f̃_{0}\},

where f̃_{0} = ln(1 + f_{0}) − f_{0}. On the set \{f_{0} ≤ 2ϵ\} with ϵ ≤ \frac{1}{2},

∥f̃_{0}∥_{C^{0}} ≤ Cϵ^{2} and ∥df̃_{0}∥_{C^{0}} ≤ Cϵ^{\frac{3}{2}} , (2.7) where ∥df̃_{0}∥_{C^{0}} denotes the C^{0}-norm with respect to the standard metric on ℂ^{n}. Furthermore, by (2.3),

(2.8)  ∥d(β_{ϵ^{2}} ∘ f_{0})∥_{C^{0}} ≤ Cϵ^{−1}ϵ^{\frac{1}{2}} ,
∥∇^{2}(β_{ϵ^{2}} ∘ f_{0})∥_{C^{0}} ≤ C(ϵ^{−2}ϵ^{\frac{1}{2}} ϵ^{\frac{1}{2}} + ϵ^{−1}),

where again all the norms are computed with respect to the standard metric on ℂ^{n}. Equations (2.7) and (2.8) imply that the term on the second line of (2.6) tends to 0 as ϵ goes to 0. Thus by (2.6), we can choose ϵ > 0 such that ω_{ℙ^{n},q} ≡ ω_{ℙ^{n},q,ϵ} is a symplectic form on all of ℙ^{n}. Note that ω_{ℙ^{n},q} is invariant under the action of the stabilizer of q in SU_{n+1}, which is the subgroup

Stab_{p}(SU_{n+1}) = \{( \begin{matrix}\overline{det(h)} & 0 \\ 0 & h \end{matrix} ) : h ∈ U_{n}\} ⊂ SU_{n+1}.

We can define a smooth family of symplectic Kahler forms on ℙ^{n} by

ω_{ℙ^{n},g·q} = g^{∗}ω_{ℙ^{n},q}, g ∈ SU_{n+1}.

The above invariance property of ω_{ℙ^{n},q} insures that ω_{ℙ^{n},g·q} depends only on g · q. We can now take g_{ℙ^{n},g·q} to be the metric corresponding to the symplectic form ω_{ℙ^{n},g·q} and the standard complex structure J on ℙ^{n}.

We denote by exp_{b} and Π_{b,X} for X ∈ T ℙ^{n} the g_{ℙ^{n},b}-exponential map and g_{ℙ^{n},b}-parallel transport along the g_{ℙ^{n},b}-geodesic for X, respectively. If υ ∈ F^{(0)}T , let

g_{ℙ^{n},υ} = g_{ℙ^{n},b_{υ}}, exp_{υ} = exp_{b_{υ}}, Π_{υ,X} = Π_{b_{υ},X}.

If υ ∈ F^{(0)} is sufficiently small, we define L^{2}-norms inner-products on

Γ(υ) ≡ Γ(b(υ)) and Γ^{0,1}(υ) ≡ Γ^{0,1}(b(υ))

via the metrics g_{ℙ^{n},υ} and g_{υ} in the usual way. Denote by D_{υ} the linearization of the \overline{∂}-operator with respect to the metric g_{ℙ^{n},υ} on ℙ^{n} and by D_{υ}^{∗} its formal adjoint with respect to the above (L^{2}, υ) inner-product. We fix p > 2 and denote by ∥ · ∥_{υ,p,1} and ∥ · ∥_{υ,p} the modified Sobolev (L_{1}^{p}, g_{ℙ^{n},υ}, g_{υ}) and (L^{p}, g_{ℙ^{n},υ}, g_{υ}) norms of [6] on Γ(υ) and Γ^{0,1}(υ), respectively. Let L_{1}^{p}(υ) and L^{p}(υ) be the corresponding completions. A description of the modified Sobolev norms in the notation of this paper can be found in [13]. They are needed only for certain technical aspects of this paper.

2.2 Obstruction bundle

In this subsection, in the case S = Σ, we choose an obstruction bundle over F^{(∅)}T_{δ} in the sense of Definition 3.13 in [13] with δ ∈ C^{∞}(M_{T}; ℝ^{+}) sufficiently small.

Let δ_{T} ∈ C^{∞}(M_{T}; ℝ^{+}) be such that

4δ_{T} (b)∥du_{i}∥_{b,C^{0}} < r_{ℙ^{n}} ∀b = (Σ, M, I; x, (j, y), u) ∈ M_{T} , i ∈ I.

We assume that the above function δ is such that 8δ^{\frac{1}{2}} < δ_{T}. If υ ∈ F^{(∅)}T_{δ} and Xψ ∈ T_{ev(b_{υ})}ℙ^{n} ⊗ H_{Σ}^{0,1}, define R_{υ}Xψ ∈ Γ^{0,1}(u_{υ}) as follows. If z ∈ Σ_{υ} = Σ is such that q_{υ}(z) ∈ Σ_{b_{υ},h} for some h ∈ Î with χ_{T} h = 1 and |q_{S}^{−1}(q_{υ}(z))| ≤ 2δ_{T} (b_{υ}), by our assumption on δ_{T}, we can define \overline{u}_{υ}(z) ∈ T_{ev(b_{υ})}ℙ^{n} by

exp_{υ,ev(b_{υ})} \overline{u}_{υ}(z) = u_{υ}(z), |\overline{u}_{υ}(z)| < r_{ℙ^{n}}.

Given z ∈ Σ, let h_{z} ∈ I be such that q_{υ}(z) ∈ Σ_{b_{υ},h_{z}}. If w ∈ T_{z}Σ, put

R_{υ}Xψ|_{z}w = \begin{cases}0, & if χ_{T} h_{z} = 2; \\ β(δ_{T} (b_{υ})|q_{υ}z|)(ψ|_{z}w)Π_{υ,\overline{u}_{υ}(z)}X, & if χ_{T} h_{z} = 1; \\ (ψ|_{z}w)X, & if χ_{T} h_{z} = 0. \end{cases}

Let Γ_{−}^{0,1}(υ) be the image of T_{ev(b_{υ})}ℙ^{n} ⊗ H_{Σ}^{0,1} under the map R_{υ}. Denote by π_{υ,−}^{0,1} the (L^{2}, υ)-orthogonal projection of L^{p}(υ) onto Γ_{−}^{0,1}(υ).

The spaces Γ_{−}^{0,1}(υ) form our obstruction bundle over F^{(∅)}T . We need to show that these spaces satisfy the requirements of Definition 3.13 in [13]. First, the rate of change of π_{υ,−}^{0,1} with respect to changes in υ should be controlled by a function of b_{υ} only. The proof of this fact is similar to the proof of the second statement of (5) of Lemma 3.6 in [13]. The next lemma implies that the remaining conditions are also satisfied. For any h ∈ Î, put

|υ|_{h} = ∏_{i∈Î,i≤h} |v_{i}|.

Lemma 2.2. For any υ ∈ F^{(∅)}T_{δ} and Xψ ∈ T_{ev(b_{υ})}ℙ^{n} ⊗ H_{Σ}^{0,1}, D_{υ}^{∗}R_{υ}Xψ vanishes outside of the annuli

Ã_{υ,h} ≡ q_{υ}^{−1}({(h, z) ∈ Σ_{b_{υ},h} : δ_{T} (b_{υ}) ≤ |q_{S}^{−1}(z)| ≤ 2δ_{T} (b_{υ})})

with h ∈ Î such that χ_{T} h = 1. Furthermore, there exists C ∈ C^{∞}(M_{T}; ℝ^{+}) such that:

(1)  ∥D_{υ}^{∗}R_{υ}Xψ∥_{υ,C^{0}} ≤ C(b_{υ}) ( ∑_{χ_{T} h=1} |υ|_{h}) |X|_{υ}∥ψ∥_{2};

(2) (1 − C(b_{υ})^{−1}|υ|^{\frac{2}{p̃}})∥Xψ∥_{υ,p̃} ≤ ∥R_{υ}Xψ∥_{υ,p̃}
≤ (1 + C(b_{υ})^{−1}|υ|^{\frac{2}{p̃}})∥Xψ∥_{υ,p̃},

where p̃ = 2, p.

Proof. The first statement and estimate (2) are immediate from the definition of R_{υ}Xψ and of the norms; see [13]. Let (s, t) be the conformal coordinates on Ã_{υ,h} given by q_{υ}(s, t) = s + it ∈ ℂ. Write g_{υ} = θ^{−2}(s, t)(ds^{2} + dt^{2}). Then

θ = \frac{1}{2} (1 + s^{2} + t^{2}) . (2.9)

Put

(2.10)  ξ(s, t) = {R_{υ}Xψ}_{(s,t)}∂_{s}
= β(δ_{T} (b_{υ})\sqrt{s^{2} + t^{2}})(ψ|_{(s,t)}∂_{s})Π_{υ,\overline{u}_{υ}(s,t)}X.

Then by [7, p. 29],

D_{υ}^{∗}R_{υ}Xψ|_{z} = θ^{2} (−\frac{D}{ds}ξ + J \frac{D}{dt}ξ) , (2.11)

where \frac{D}{ds} and \frac{D}{dt} denote covariant differentiation with respect to the metric g_{ℙ^{n},υ} on ℙ^{n}. Since this metric is flat on the support of ξ and ψ ∈ H_{Σ}^{0,1}, Equations (2.9)–(2.11) give

(2.12) D_{υ}^{∗}R_{υ}Xψ|_{z} = \frac{(1 + s^{2} + t^{2})^{2}}{4} \{β′|_{δ_{T} (b_{υ})\sqrt{s^{2}+t^{2}}}δ_{T} (b_{υ})\frac{−s + it}{\sqrt{s^{2} + t^{2}}} \}
· (ψ|_{(s,t)}∂_{s})Π_{υ,\overline{u}_{υ}(s,t)}X.

Since the right-hand-side of (2.12) vanishes unless δ_{T} (b_{υ})^{−1} ≤ \sqrt{s^{2} + t^{2}} ≤ 2δ_{T} (b_{υ})^{−1}, it follows that

|D_{υ}^{∗}R_{υ}Xψ|_{υ,z} ≤ C(b_{υ})|ψ|_{(s,t)}∂_{s}||X|_{υ} ≤ C′(b_{υ})|υ|_{h}∥ψ∥_{2}|X| (2.13)

Claim (1) follows from (2.13).

Let R̃_{υ} : H_{Σ}^{0,1} ⊗ T_{ev(b_{υ})}ℙ^{n} ⟶ Γ_{−}(υ) be the adjoint of R_{υ}^{−1}, i.e.,

⟨⟨R̃_{υ}Xψ, R_{υ}X′ψ′⟩⟩_{υ,2} = ⟨⟨Xψ, X′ψ′⟩⟩_{b_{υ},2} = ⟨X, X′⟩_{b_{υ}}⟨ψ, ψ′⟩_{2} (2.14)

for all X, X′ ∈ T_{ev(b_{υ})}ℙ^{n} and ψ, ψ′ ∈ H_{Σ}^{0,1}. By Lemma 2.2, ∥R̃_{υ} −R_{υ}∥_{2} ≤ C(b_{υ})|υ|.

2.3 Tangent-bundle model

We now describe our choice for a tangent-bundle model, which is the subject of Definition 3.11 in [13].

For any υ ∈ F^{(0)}T sufficiently small and ξ ∈ Γ(b_{υ}), define R_{υ}ξ ∈ L_{1}^{p}(υ) by {R_{υ}ξ}(z) = ξ(q_{υ}(z)). Let Γ_{−}(υ) be the image of ker(D_{b_{υ}}) under the map R_{υ}. Denote by Γ_{+}(υ) its (L^{2}, υ)-orthogonal complement in L_{1}^{p}(υ). Let π_{υ,±} be the (L^{2}, υ)-orthogonal projection onto Γ_{±}(υ).

If x ∈ Σ, let H_{Σ}^{−}(x) = {ψ ∈ H_{Σ}^{0,1} : ψ|_{x} = 0}. This is a codimensionone subspace of H_{Σ}^{0,1} for all x ∈ Σ; see [3]. Denote by H_{Σ}^{+}(x) its L^{2}-orthogonal complement. The space H_{Σ}^{+}(x) is independent of the choice of a Kahler metric on (Σ, j_{Σ}). For any h ∈ Î, we put x̃_{h}(υ) = q_{υ,ι_{h}}^{−1}(ι_{h}, x_{h}). Fix h^{∗} ∈ Î such that χ_{T} h^{∗} = 1. Let

\overline{Γ}_{−}(υ) = D_{υ}^{∗}R_{υ}(H_{Σ}^{+}(x_{h^{∗}}(υ)) ⊗ T_{ev(b_{υ})}ℙ^{n}).

Denote by \overline{Γ}_{+}(υ) the (L^{2}, υ)-orthogonal complement of \overline{Γ}_{−}(υ) in L_{1}^{p}(υ) and by \overline{π}_{υ,±} the (L^{2}, g_{υ})-orthogonal projections onto \overline{Γ}_{±}(υ). Let Γ̃_{+}(υ) be the image of Γ_{+}(υ) under \overline{π}_{υ,+} and let Γ̃_{−}(υ) be the (L^{2}, υ)-orthogonal complement of Γ̃_{+}(υ) in L_{1}^{p}(u_{υ}).

The spaces Γ̃_{−}(υ) will be our tangent-space model. We need to check that the requirements of Definition 3.11 in [13] are satisfied. Let

{h ∈ Î: χ_{T} (h) = 1} = {h_{1} = h^{∗}, h_{2}, …, h_{m}}.

If z ∈ Σ_{b,h_{r}} is such that |q_{S}^{−1}(z)| ≤ 2δ(b), define \overline{u}_{h_{r}}(z) ∈ T_{ev(b)}ℙ^{n} by

exp_{b,ev(b)} \overline{u}_{h_{r}}(z) = u_{h_{r}}(z), |\overline{u}_{h_{r}}(z)|_{b} < r_{ℙ^{n}}.

If X ∈ T_{ev(b)}ℙ^{n}, define R_{b,h_{r}}X ∈ Γ(u_{h_{r}}) by

R_{b,h_{r}}X(z) = \begin{cases}0, & if |z| ≥ 2δ_{T} (b)^{−1}; \\ β′|_{δ_{T} (b)|z|} \frac{(1+|z|^{2})^{2}z}{|z|} Π_{b,\overline{u}_{hr} (z)}X, & otherwise. \end{cases}

Since R_{b,h_{r}}X vanishes at all the nodes of Σ_{b} by assumption on δ_{T}, we can extend R_{b,h_{r}}X by zero to an element of Γ(b). If c = c_{[m]} ∈ ℂ^{[m]} is different from zero, let

\overline{Γ}_{−}(b; c) = \{ ∑_{r∈[m]} c_{r}R_{b,h_{r}}X : X ∈ T_{ev(b)}ℙ^{n}\} .

Denote by \overline{Γ}_{+}(b; c) the (L^{2}, b)-orthogonal complement of \overline{Γ}_{−}(b; c) in Γ(b). Let \overline{π}_{(b;c),±} be the corresponding (L^{2}, b)-orthogonal projection maps. Let Γ̃_{+}(b; c) = \overline{π}_{(b,c),+}(Γ_{+}(b)) and let Γ̃_{−}(b; c) be its (b, L^{2})- orthogonal complement.

Lemma 2.3. There exist δ, C ∈ C^{∞}(M_{T}^{(0)}; ℝ^{+}) such that for all υ ∈ F^{(∅)}T_{δ} and ξ ∈ \overline{Γ}_{−}(υ),

∥ξ∥_{υ,p,1} ≤ C(b_{υ})∥ξ∥_{υ,2}.

In addition, dim_{ℂ} \overline{Γ}_{−}(υ) = dim_{ℂ} \overline{Γ}_{−}(b_{υ}; c) = n for any nonzero c ∈ ℂ^{m}. Furthermore, if υ_{k} ⟶ b ∈ M_{T}^{(0)} and ξ_{k} ∈ \overline{Γ}_{−}(υ) is such that ∥ξ_{k}∥_{υ_{k},2} = 1, then there exists a nonzero c ∈ ℂ^{m} and ξ ∈ \overline{Γ}_{−}(b; c) with ∥ξ∥_{b,2} = 1 such that a subsequence of {ξ_{k}} C^{0}-converges to ξ.

Remark. The last statement means that a subsequence of {ξ_{k}} C^{0}-converges to ξ on compact subsets of Σ_{b}^{∗} and the norms ∥ξ_{k}∥_{υ_{k},p,1} are uniformly bounded; see Definition 3.9 in [13].

Proof. (1) Let ψ be a generator of H_{Σ,+}^{0,1}(x̃_{h_{1}}(υ)). If X ∈ T_{ev(b_{υ})}ℙ^{n} and r ∈ [m], define R_{υ,h_{r}}X ∈ Γ(u_{υ}) as follows. If q_{υ}(z) ∈ Σ_{b_{υ},h_{r}}, let

R_{υ,h_{r}}X(z) = ( ∑_{r∈[m]} |ψ_{x̃_{r}(υ)}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s}|)^{−1} \frac{(1 + |q_{υ}z|^{2})^{2}q_{υ}z}{|q_{υ}z|}
· β′|_{δ_{T} (b_{υ})|q_{υ}z|}(ψ_{z}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s})Π_{b_{υ},\overline{u}_{υ}(z)}X.

Note that the sum is not zero, since ψ|_{x̃_{h_{1}}(υ)} ≠ 0. If q_{υ}(z) ∉ Σ_{b_{υ},h_{r}}, we let R_{υ,h_{r}}X(z) = 0. Since the modified Sobolev norms are equivalent to the standard ones away from the thin necks of (Σ_{υ}, g_{υ}),

(2.15) ∥R_{υ,h_{r}}X∥_{υ,p,1} ≤ C(b_{υ}) ( ∑_{r∈[m]} |ψ_{x̃_{r}(υ)}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s}|)^{−1}
· |ψ_{z}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s}||X|_{υ}
≤ C′(b_{υ})∥R_{υ,h_{r}}X∥_{υ,2}.

By the proof of Lemma 2.2, if ξ ∈ \overline{Γ}_{−}(υ),

ξ = R_{υ}X ≡ ∑_{r∈[m]} R_{υ,h_{r}}X,

for some X ∈ T_{ev(b_{υ})}ℙ^{n}. Thus, the first two statements of the lemma follow from (2.15).

(2) If υ_{k} ⟶ b and ξ_{k} = R_{υ}X_{k} ∈ \overline{Γ}_{−}(υ_{k}) is such that ∥ξ_{k}∥_{υ_{k},2} = 1, then it is immediate from (1) that a subsequence of ξ_{k} C^{0}-converges to

∑_{r∈[m]} c_{r}R_{b,h_{r}}X, where

X = lim_{k⟶∞} X_{k}, (2.16)

c_{r} = lim_{k⟶∞} ( ∑_{r∈[m]} |ψ_{x̃_{r}(υ)}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s}|)^{−1} (ψ_{x̃_{r}(υ)}d(q_{υ,h_{r}}^{−1} ∘ q_{N})∂_{s}).

The two limits in (2.16) exist after passing to a subsequence of the original sequence. This proves the last statement of the lemma.

Lemma 2.4. There exist δ, C ∈ C^{∞}(M_{T}^{(0)}; ℝ^{+}) such that for all υ ∈ F^{(∅)}T_{δ} and ξ ∈ Γ̃_{−}(υ),

∥ξ∥_{υ,p,1} ≤ C(b_{υ})∥ξ∥_{υ,2}.

Proof. Let Γ_{−+}(υ) be the (υ, L^{2})-orthogonal complement of π_{υ,−}(\overline{Γ}_{−}(υ)) in Γ_{−}(υ). Then

Γ̃_{−}(υ) = Γ_{−+}(υ) ⊕ \overline{Γ}_{−}(υ).

Since this decomposition is (L^{2}, υ)-orthogonal, we can assume that either ξ ∈ Γ_{−+}(υ) or ξ ∈ \overline{Γ}_{−}(υ). In the first case, the statement is obvious, since Γ_{−+}(υ) ⊂ Γ_{−}(υ). The second case is proved in Lemma 2.3.

Corollary 2.5. Suppose υ_{k} ∈ F^{(0)}T_{δ} and υ_{k} ⟶ b ∈ M_{T}^{(0)}. If {ξ_{υ_{k},l}} is an (L^{2}, υ)-orthonormal basis for Γ̃_{−}(υ_{k}), then there exists a nonzero c ∈ ℂ^{m} and an (L^{2}, b)-orthonormal basis {ξ_{b,l}} for Γ̃_{−}(b; c) such that after passing to a subsequence ξ_{υ_{k},l} C^{0}-converges to ξ_{b,l} for all l.

Proof. If ξ_{k,l} ∈ \overline{Γ}_{−}(υ_{k}), by Lemma 2.3 a subsequence of \{ξ_{k,l}\} C^{0}- converges to an element of ξ_{l} ∈ \overline{Γ}_{−}(b; c) for some nonzero c ∈ ℂ^{n} dependent on the sequence {υ_{k}}. Furthermore, orthonormal pairs of such elements C^{0}-converge to an orthonormal pair in \overline{Γ}_{−}(b). If ξ_{k,l} ∈ \overline{Γ}_{−+}(υ_{k}) ⊂ Γ_{−}(υ_{k}), then by definition of Γ_{−}(υ_{k}), a subsequence of \{ξ_{k,l}\} C^{0}-converge to an element ξ_{l} ∈ Γ_{−}(b), which must be orthogonal to \overline{Γ}_{−}(b; c); see Lemma 3.10 in [13]. Thus, a subsequence of \{\{ξ_{k,l}\}\} C^{0}- converges to an orthonormal set of vectors in Γ̃_{−}(b), which implies that dim_{ℂ} Γ̃_{−}(b; c) ≥ dim_{ℂ} Γ̃_{−}(υ_{k}). However,

dim_{ℂ} Γ̃_{−}(b; c) = dim_{ℂ} Γ_{−+}(b; c) + dim_{ℂ} \overline{Γ}_{−}(b; c)

= dim_{ℂ} Γ_{−}(b) + (dim_{ℂ} \overline{Γ}_{−}(b; c) − dim_{ℂ} π_{b,−}\overline{Γ}_{−}(b; c));

dim_{ℂ} Γ̃_{−}(υ_{k}) = dim_{ℂ} Γ_{−+}(υ_{k}) + dim_{ℂ} \overline{Γ}_{−}(υ_{k})

= dim_{ℂ} Γ_{−}(υ_{k}) + (dim_{ℂ} \overline{Γ}_{−}(υ_{k}) − dim_{ℂ} π_{υ_{k},−}\overline{Γ}_{−}(υ_{k})) ,

where Γ_{−+}(b; c) denotes the (L^{2}, b)-complement of π_{b,−}\overline{Γ}_{−}(b; c) in Γ_{−}(b). Since Γ_{−}(υ_{k}) and Γ_{−}(b) have the same dimension, in order to conclude the proof, it is sufficient to show that

π_{b,−} : \overline{Γ}_{−}(b; c) ⟶ Γ_{−}(b; c)

is an isomorphism; see Lemma 2.6.

Lemma 2.6. There exists C ∈ C^{∞}(M_{T}^{(0)}; ℝ^{+}) such that for all b ∈ M_{T}^{(0)}, nonzero c ∈ ℂ^{m}, and ξ ∈ \overline{Γ}_{−}(b; c)

∥ξ∥_{b,2} ≤ C(b_{ω})∥π_{b,−}ξ∥_{b,2}.

Proof. Suppose X ∈ T_{ev(b)}ℙ^{n}. We define R̃_{b,h_{r}}X ∈ Γ^{0,1}(u_{h_{r}}), outside of ∞ ∈ Σ_{b,h_{r}}, by

R̃_{b,h_{r}}X|_{x} = 4β(δ_{T} (b)|q_{N}^{−1}(x)|)(q_{N}^{−1∗}d\overline{z})Π_{b,\overline{u}_{hr} (x)}X,

where d\overline{z} is the usual (0, 1)-form on ℂ. By the same computation as in the proof of Lemma 2.2, R_{b,h_{r}}X = D_{b,u_{hr}}^{∗} R̃_{b,h_{r}}X. Thus, if ξ = ξÎ ∈ ker D_{b} and 2δ < δ_{T} (b), by integration by parts,

⟨⟨ξ, R_{b,h_{r}}X⟩⟩_{b} = ⟨⟨ξ_{h_{r}}, D_{b,u_{hr}}^{∗} R̃_{b,h_{r}}X⟩⟩_{b} (2.17)

= 2iδ_{T} (b)^{−1} ∫_{|q_{N}^{−1}(x)|=δ^{−1}} ⟨ξ_{h_{r}}(x), Π_{b,\overline{u}_{hr} (x)}X⟩_{b}q_{N}^{−1∗}dz,

since D_{b,u_{hr}} ξ_{h_{r}} = 0. Using the change of variables with x = q_{N}(w^{−1}), we obtain

∫_{|q_{N}^{−1}(x)|=δ^{−1}} ⟨ξ_{h_{r}}(x), Π_{b,\overline{u}_{hr} (x)}X⟩_{b}q_{N}^{−1∗}dz (2.18)

= − ∫_{|w|=δ} ⟨ξ_{h_{r}}|_{q_{N} (w^{−1})}, Π_{b,\overline{u}_{hr} (q_{N} (w^{−1}))}X⟩_{b} \frac{dw}{w^{2}}

= −2πi\frac{d}{dw} ⟨ξ_{h^{∗}}|_{q_{N} (w^{−1})}, Π_{b,\overline{u}_{hr} (q_{N} (w^{−1}))}X⟩_{b}|_{w=0}

= −2πi\frac{d}{d\overline{z}} ⟨ξ_{h_{r}}|_{q_{S}(z)}, Π_{b,\overline{u}_{hr} (q_{S}(z))}X⟩_{b}|_{z=0}

= −2πi⟨\frac{D}{ds}(ξ_{h_{r}} ∘ q_{S})|_{z=0}, X⟩,

since D_{b,u_{hr}} ξ_{h_{r}} = 0. It follows from (2.17) and (2.18) that for any ξ = ξ_{[M]} ∈ ker(D_{b}),

⟨⟨ξ, ∑_{r∈[m]} c_{r}R_{b,h_{r}}X⟩⟩_{b} = 4πδ_{T} (b)^{−1} ∑_{r∈[m]} c_{r}⟨\frac{D}{ds}(ξ_{h_{r}} ∘ q_{S})|_{z=0}, X⟩. (2.19)

Along with Corollary 6.3, Equations (2.19) gives

∥π_{b,−} ∑_{r∈[m]} c_{r}R_{b,h_{r}}X∥_{b,2} (2.20)

≥ C(b)|c_{r^{∗}}| sup_{ξ_{[M]}∈ker(D_{b}),∥ξ_{[M]}∥=1} ⟨\frac{D}{ds}(ξ_{h_{r^{∗}}} ∘ q_{S})|_{z=0}, X⟩_{b}

≥ C′(b)|c_{r^{∗}}||X| ≥ C′′(b)∥ ∑_{r∈[m]} c_{r}R_{b,h_{r}}X∥_{b,2},

where r^{∗} ∈ [m] is such that |c_{r^{∗}}| = sup_{r} |c_{r}|. Since the right-hand side of (2.20) must be a continuous function of b, the claim follows.

The statement of Corollary 2.5 is precisely Condition (1) of Definition 3.11 in [13]. The other two conditions require that the rate of change of the (L^{2}, υ)-orthogonal projection onto Γ̃_{−}(υ) be controlled by a function of b_{υ} only. This is a consequence of the convergence described in the Corollary 2.5, i.e., we can use the same argument as described in the remark following Lemma 3.6 in [13], but with Γ_{−}(b) replaced by the appropriate space Γ_{−}(b; c) (depending on υ).

2.4 Structure theorem, S = Σ

If T = (Σ, [N], I; j, \underline{d}) is a simple bubble type and μ is an N-tuple of complex submanifolds of ℙ^{n} such that the evaluation map,

ev_{[N]} ≡ ev_{1} × ⋯ × ev_{N} : M_{T} ⟶ (ℙ^{n})^{N},

is transversal to μ_{1} ×⋯×μ_{N}, M_{T} (μ) is a complex submanifold of M_{T}. Let N^{μ}T be its normal bundle. If S is a complex submanifold of M, denote its normal bundle by N S and an identification of small neighborhoods of S in N S and in M_{T} by ϕ_{S}. For any complex vector bundle V ⟶ M_{T}, we denote by Φ_{S} an identification of ϕ_{S}^{∗}V and π_{N S}^{∗}V such that its restriction to the fibers over S is the identity. We assume that Φ_{S} preserves F^{∅}T ⊂ F T . Let

F^{∅}S = {(b, n⃗, υ) ∈ N S ⊕ F S : (b, υ) ∈ F^{∅}T }.

If ev_{[N]}|_{S} is transversal to μ_{1} × ⋯ × μ_{N}, S(μ) ≡ S ∩ M_{T} (μ) is a complex submanifold of S with normal bundle N^{μ}T . Let ϕ_{S}^{μ} and Φ_{S}^{μ} be the analogues of ϕ_{S} and Φ_{S} for the bundle N^{μ}T ⟶ S(μ). We assume the bundle N^{μ}T is normed. We call the pair (Φ_{S}, Φ_{S}^{μ}) a regularization of S(μ) if it satisfies a certain minor compatibility condition. For the purposes of this paper, it suffices to say that once Φ_{S} is chosen, it is a condition on Φ_{S}^{μ}|_{F T}; see Subsection 3.8 in [13] for details. However, the exact nature of Φ_{S}^{μ}|_{F T} is irrelevant for our computational purposes. Finally, we denote by \overline{C}_{(d;N)}^{∞}(Σ; μ) the space of all bubble maps (Σ, [N], I; x, (j, y), u) such that ∑_{i∈I} u_{i∗}[Σ_{b,i}] = dλ, where λ ∈ H_{2}(ℙ^{n}; Z) is the class of a line, and u_{j_{l}}(y_{l}) ∈ μ_{l} for all l ∈ [N].

Theorem 2.7. Suppose d is a positive integer, T = (Σ, [N], I; j, \underline{d}) is a simple bubble type with dˆ0 = 0 and ∑_{i∈I} d_{i} = d, S ⊂ M_{T} is a complex

submanifold, and

ν ∈ Γ^{0,1}(Σ × ℙ^{n}; Λ_{J,j}^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n})

is a generic section. Let μ be an N-tuple of complex submanifolds of ℙ^{n} in general position of total codimension

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N,

and (Φ_{S}, Φ_{S}^{μ}) a regularization of S(μ). Then for every precompact open subset K of S(μ), there exist a neighborhood U_{K} of K in \overline{C}_{(d;N)}^{∞}(Σ; μ) and δ, ϵ, C > 0 with the following property. For every t ∈ (0, ϵ), there exist a section

φ_{S,tν}^{μ} ∈ Γ(F^{∅}S_{δ}|_{K}; π_{F S}^{∗}N^{μ}S), with |φ_{S,tν}^{μ}(υ)|_{b_{υ}} ≤ C(t + |υ|^{\frac{1}{p}} ),

and a sign-preserving bijection between M_{Σ,tν,d}(μ) ∩ U_{K} and the zero set of the section ψ_{S,tν}^{μ} defined by

ψ_{S,tν}^{μ} ∈ Γ (F^{∅}S_{δ}|_{K}; π_{F S}^{∗} (H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n})) ,
Π_{b_{υ},ϕ_{S}^{μ}φ_{S,tν}^{μ}(υ)}ψ_{S,tν}^{μ}(υ) = ψ_{S,tν}(Φ_{S}^{μ}(φ_{S,tν}^{μ}(υ)));
ψ_{S,tν} ∈ Γ (F^{∅}S_{δ}|_{S∩U_{K}}; π_{F S}^{∗} (H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n})) ,
Π_{b_{υ},ϕ_{S}(υ)}ψ_{S,tν}(υ) = ψ_{T ,tν}(Φ_{S}(υ));
ψ_{T ,tν} ∈ Γ (F^{∅}T_{δ}|_{M_{T} ∩U_{K}}; π_{F T}^{∗} (H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n})) ,
R̃_{υ}ψ_{T ,tν}(υ) = π_{υ,−}^{0,1}(tν_{υ,t} − \overline{∂}u_{υ} − D_{υ}ξ_{υ,tν}),

where Π_{b,b′} denotes the g_{ℙ^{n},b}-parallel transport along the g_{ℙ^{n},b}-geodesics from ev(b) to ev(b′) whenever d_{ℙ^{n}}(ev(b), ev(b′)) < r_{ℙ^{n}}, ξ_{υ,tν} ∈ Γ̃_{+}(υ),

∥ν_{υ,t} − ν∥_{υ,2} ≤ C(t + |υ|^{\frac{1}{p}} ), and ∥ξ_{υ,tν}∥_{υ,p,1} ≤ C(t + |υ|^{\frac{1}{p}} ).

Proof. This theorem follows immediately from Theorem 3.29 in [13] applied to the obstruction bundle setup of Subsections 2.2 and 2.3. The only refinement is that we drop the term η̃_{υ,tν} from the definition of ψ_{T ,tν}. This is because it vanishes on the support of the (0, 1)-forms in Γ_{−}^{0,1}(υ), provided δ is sufficiently small. Thus, π_{υ,−}^{0,1}η̃_{υ,tν} = 0.

2.5 Structure theorem, S = S^{2}

In this subsection, we define sections D_{⟨T ⟩,k}^{(m)}, where k ∈ I − Î, of the bundle L_{k}^{∗}T^{⊗m} ⊗ev^{∗}T ℙ^{n} over \overline{U}_{⟨T ⟩}(μ), and describe their behavior with respect to the gluing maps near each space U_{T} (μ). In Section 4, the number of elements of M_{Σ,tν,d}(μ) lying near each space M_{T} (μ) will be expressed as the number of zeros of affine maps between certain bundles. These affine maps will involve the sections D_{\overline{T} ,k}^{(m)}. Their behavior near various boundary strata is the foundation for the local computations of Section 5.

If b = (S^{2}, M, I; x, (j, y), u) ∈ B_{T}, m ≥ 1, and k ∈ I, let

D_{T ,k}^{(m)}b = \frac{2}{(m − 1)!} \frac{D^{m−1}}{ds^{m−1}} \frac{d}{ds}(u_{k} ∘ q_{S})|_{(s,t)=0},

where the covariant derivatives are taken with respect to the metric g_{ℙ^{n},b} and s + it ∈ ℂ. If T^{∗} is a basic bubble type, the maps D_{T ,k}^{(m)} with

T < T^{∗} and k ∈ I − Î induce a continuous section of ev^{∗}T ℙ^{n} over \overline{U}_{T^{∗}}^{(0)} and a continuous section of the bundle L_{k}^{∗}T^{∗⊗m} ⊗ ev^{∗}T ℙ^{n} over \overline{U}_{T^{∗}}, described by

D_{T^{∗},k}^{(m)}[b, c_{k}] = c_{k}^{m}D_{T ,k}^{(m)}b, if b ∈ U_{T}^{(0)}, c_{k} ∈ ℂ.

We will often write D_{T ,k} instead of D_{T ,k}^{(1)}. If T is simple, we will abbre-

viate D_{T ,k}^{(m)} as D^{(m)}. If T = (Σ, [N], I; j, \underline{d}) is a simple bubble type and

k ∈ Î, let D_{T ,k}^{(m)} denote the section D_{\overline{T} ,k}^{(m)}.

Theorem 2.8. If T^{∗} = (S^{2}, M, I^{∗}; j, \underline{d}^{∗}) is a basic bubble type and μ is an M-tuple of constraints in general position, the spaces \overline{U}_{T^{∗}}^{(0)}(μ) and \overline{U}_{T^{∗}}(μ) are oriented topological orbifolds. If T < T^{∗}, there exist G_{T^{∗}}-invariant functions δ, C ∈ C^{∞}(U_{T}^{(0)}(μ); ℝ^{+}) and G_{T^{∗}}-equivariant continuous map

γ_{T}^{μ} : F T_{δ}|_{U_{T}^{(0)}(μ)} ⟶ \overline{U}_{T^{∗}}^{(0)}(μ),

which is an orientation-preserving homeomorphism onto an open neighborhood of U_{T}^{(0)}(μ) in \overline{U}_{T^{∗}}^{(0)}(μ) and is identity on U_{T}^{(0)}(μ). This map is smooth on F^{∅}T_{δ}. Furthermore, for any

υ = [(b, v_{h})_{h∈Î}] = [(S^{2}, M, I; x, (j, y), u), (v_{h})_{v∈Î}] ∈ F T_{δ}|_{U_{T}^{(0)}(μ)},

|Π_{b_{υ},ev(γ_{T}^{μ} (υ))}^{−1}(D_{T^{∗},k}γ_{T}^{μ} (υ)) − 2 ∑_{h∈I_{k},χ_{T} h=1} ( ∏_{i∈Î,i≤h} v_{i}) (du_{h}|_{∞}e_{∞})|

≤ C(b_{υ})|υ|^{\frac{1}{p}} ∑_{h∈I_{k},χ_{T} h=1} ( ∏_{i∈Î,i≤h} |v_{i}|) ,

where I_{k} ⊂ I is the rooted tree containing k.

Remark. This theorem states that there exists an identification γ_{T}^{μ} : FT_{δ} ⟶ \overline{U}_{T^{∗}}(μ) of neighborhoods of U_{T} (μ) in FT and in \overline{U}_{T^{∗}}(μ).

Furthermore, with appropriate identifications,

|D_{T^{∗},k}γ_{T}^{μ} (υ) − α_{T} (ρ_{T} (υ))| ≤ C(b_{υ})|υ|^{\frac{1}{p}} |ρ_{T} (υ)|, where (2.21)

ρ_{T} (υ) = (b, (ṽ_{h})_{χ_{T} h=1}) ∈ F̃T ≡ ⨁_{χ_{T} h=1} L_{h}T ⊗ L∗ ; ι̃_{h}T ;

ṽ_{h} = ∏_{i∈Î,i≤h} v_{i}; ι̃_{h} = min{i ∈ I : i < h};

α_{T} (b, (ṽ_{h})_{χ_{T} h=1}) = ∑_{h∈I_{k},χ_{T} h=1} D_{T ,h}ṽ_{h}.

This estimate is used frequently in Section 5. Note that if T is a semiprimitive bubble type, the bundle FT is defined over \overline{U}_{T} (μ). However, FT is not the normal bundle of \overline{U}_{T} (μ) in \overline{U}_{⟨T ⟩}(μ) unless Mˆ0T ⊔ Hˆ0T is a two-element set; see [9]. The theorem implies only that the restrictions of the normal bundle of \overline{U}_{T} (μ) in \overline{U}_{⟨T ⟩}(μ) and of FT to U_{T} (μ) are isomorphic.

Proof. (1) All statements of this theorem, except for the analytic estimate, follow immediately from Theorem 3.33 in [13]. We deduce the analytic estimate from (2) of Theorem 3.33. Let

γ_{T}^{μ} (υ) = (S^{2}, M, I(υ); x(υ), (j(υ), y(υ)), ũ_{υ}).

By Theorem 3.33, there exist a holomorphic bubble map

b′ = [S^{2}, M, I; x′, (j, y′), u′]

such that d_{C^{k}}(b, b′) ≤ C(b_{υ})|υ|^{\frac{1}{p}} and with appropriate identifications, ũ_{υ} = exp_{b′,u_{b′}∘q_{υ}} ξ for some ξ ∈ Γ(u_{b′} ∘ q_{υ}) with ∥ξ∥_{b,C^{0}} ≤ C(b_{υ})|υ|^{\frac{1}{p}}. Thus, for the purposes of proving the analytic estimate, we can assume that u_{υ} = exp_{b,u_{b}∘q_{υ}} ξ_{υ} for ξ ∈ Γ(u_{b} ∘ q_{υ}) with ∥ξ_{υ}∥_{b,C^{0}} ≤ C(b_{υ})|υ|^{\frac{1}{p}}, i.e., it is enough to prove the estimate for the map γ̃_{T} as defined in [13] with T a simple bubble type. If d_{k} ≠ 0, the claim is immediate from the usual Sobolev and elliptic estimates near (k, ∞). Thus, we assume that dˆ0 = 0. For future use, we obtain equations describing the behavior of D^{(m)}γ̃_{T} (υ) for all m ≥ 1.

(2) We identify B_{g_{ℙ^{n},b}}(ev(b), \frac{1}{2}r_{ℙ^{n}}) with an open subset of ℂ^{n} via the g_{ℙ^{n},ev(b)}-parallel transport along the geodesics from ev(b). We assume that δ ∈ C^{∞}(U_{T}^{(0)}; ℝ^{+}) satisfies

C(b)δ(b)^{\frac{1}{2p}} + δ(b)^{\frac{1}{2}} (∑_{i∈M} ∥du_{i}∥_{b,C^{0}}) < \frac{1}{2}r_{ℙ^{n}}.

Let q : B_{1}(0; ℂ) ⟶ S^{2} be the local stretching map as in Subsection 2.2 of [13] with v = 1, defined with respect to the standard metric on ℂ. Let f_{υ} = u_{υ} ∘q and f̃_{υ} = ũ_{υ} ∘q. We denote the usual complex coordinate on ℂ by z. For any z ∈ B_{1}(0; ℂ), let i_{υ}(z) be such that q_{υ}(q(z)) ∈ Σ_{b,i_{υ}(z)}. If X ∈ T_{ev(b)}ℙ^{n} and m ≥ 1, define R_{υ}Xψ^{(m)} ∈ Γ^{0,1}(f̃_{υ}) by

R_{υ}Xψ^{(m)}|_{z} = \begin{cases}X\overline{z}^{m−1}d\overline{z}, & if χ_{T} i_{υ}(z) = 0; \\ β(δ(b_{υ})|q_{υ}(q(z))|)X\overline{z}^{m−1}d\overline{z}, & if χ_{T} i_{υ}(z) = 1; \\ 0, & if χ_{T} i_{υ}(z) = 2. \end{cases}

Note that if χ_{T} i_{υ}(z) = 0, or χ_{T} i_{υ}(z) = 1 and β(δ(b_{υ})|q_{υ}(q(z))|) ≠ 0, f̃_{υ}(z) lies in B_{g_{ℙ^{n},b}}(ev(b), \frac{1}{2}r_{ℙ^{n}}). Thus, R_{υ}Xψ^{(m)} is well-defined. We now compute ⟨⟨\overline{∂}f̃_{υ}, R_{υ}Xψ^{(m)}⟩⟩ in two ways and compare the results. First, note that the map f̃_{υ} is holomorphic outside of the annulus

Aˆ0(υ) ≡ B_{1}(0; ℂ) − B_{\frac{1}{2}} (0; ℂ).

Thus, by the same computation as in the proof of Lemma 4.3, we see that

⟨⟨\overline{∂}f̃_{υ}, R_{υ}Xψ^{(m)}⟩⟩ = −\frac{π}{m}⟨D^{(m)}γ̃_{T} (υ), X⟩. (2.22)

(3) Since f̃_{υ} = exp_{ev(b),f_{υ}}(ξ_{υ} ∘ q) and f_{υ} is constant on Aˆ0(υ),

2i⟨⟨\overline{∂}f̃_{υ}, R_{υ}Xψ^{(m)}⟩⟩ = ∫_{Aˆ0(υ)} ⟨\frac{\overline{∂}}{∂\overline{z}} (ξ_{υ} ∘ q), X⟩ z^{m−1}d\overline{z} ∧ dz (2.23)

Denote by A+ ˆ0(υ) and A− ; ˆ0(υ) the outer and inner boundary of Aˆ0(υ), respectively. For every h ∈ Î with χ_{T} h = 1, let

A_{h}(υ) = q_{υ,ι_{h}}^{−1}(\{z ∈ Σ_{b_{υ},ι_{h}} : 4δ(b_{υ})^{−1}|v_{h}| ≤ |ϕ_{b,h}^{−1}z| ≤ |v_{h}|^{\frac{1}{2}} \}) ⊂ Σ_{b_{υ},ˆ0}.

Denote by A_{h}^{±}(υ) the outer and inner boundary of A_{h}(υ). Let w be the complex coordinate on ℂ ⊂ Σ_{b_{υ},ˆ0} = S^{2}. Note that q is holomorphic inside of A− ; ˆ0(υ) and outside of q^{−1}(A_{h}^{−}(υ)). Furthermore, since u_{b} and ũ_{υ} are both holomorphic, on the image of this set under q

\frac{\overline{∂}}{∂\overline{w}} ξ_{υ} = −\frac{\overline{∂}}{∂\overline{w}} u_{υ}.

The last quantity vanishes outside of the annuli A_{h}(υ). Thus by integration by parts,

∫_{Aˆ0(υ)} ⟨\frac{\overline{∂}}{∂\overline{z}} (ξ_{υ} ∘ q), X⟩ z^{m−1}d\overline{z} ∧ dz (2.24)

= ∑_{χ_{T} h=1} ( ∫_{q^{−1}(A_{h}(υ))} ⟨(\frac{\overline{∂}u_{υ}}{∂\overline{w}} ) \overline{(\frac{∂q}{∂z} )}, X⟩ z^{m−1}d\overline{z} ∧ dz

+ ∫_{q^{−1}(A_{h}^{−}(υ))} ⟨ξ_{υ} ∘ q, X⟩z^{m−1}dz)

= ∑_{χ_{T} h=1} ( ∫_{A_{h}(υ)} ⟨\frac{\overline{∂}u_{υ}}{∂\overline{w}} , X⟩ g d\overline{w} ∧ dw + ∫_{A_{h}^{−}(υ)} ⟨ξ_{υ}, X⟩g dw),

where g(w) = w^{m−1}. Since ξ_{υ} ∘ q is constant on A+ ˆ0(υ), the second boundary term is zero. Note that the radius of A_{h}^{−}(υ) in ℂ ⊂ S^{2} is bounded by C(b_{υ})|ṽ_{h}|. Furthermore, |g| ≤ C_{m}(b_{υ}) on A_{h}^{−}(υ). It follows that

| ∫_{A_{h}^{−}(υ)} ⟨ξ_{υ}, X⟩gdw| ≤ C_{m}(b_{υ})|υ|^{\frac{1}{p}} |ṽ_{h}|. (2.25)

On the other hand, by the same computation as in the proof of Lemma 4.3,

∫_{A_{h}(υ)}⟨\frac{\overline{∂}u_{υ}}{∂\overline{w}} , X⟩ g d\overline{w} ∧ dw = −2i ∑_{m′=1}^{m′=m} \frac{πa_{m′,h}(υ)}{m′} \binom{m − 1}{m′ − 1}v_{h}^{m′}(D_{T ,h}^{(m′)}b). (2.26)

Combining Equations (2.22)–(2.26), we see that

|⟨D^{(m)}γ̃_{T} (υ), X⟩ − 2m ∑_{χ_{T} h=1} ṽ_{h}(du_{h}|_{∞}e_{∞})| ≤ C(b_{υ})|υ|^{\frac{1}{p}} ( ∑_{χ_{T} h=1} |ṽ_{h}|) . (2.27)

3. Topology

3.1 Maps between vector bundles

In Section 4, we express the number of zeros of the maps ψ_{T ,tν}^{μ} (and ψ_{S,tν}^{μ} for certain submanifolds S of M_{T}) of Theorem 2.7 in terms of the number of zeros of affine maps between the same vector bundles. The topological justification for this reduction is discussed in this subsection. Subsections 3.2 and 3.3 are used in the explicit computations of Section 5. For simplicity, we state all the results for smooth vector bundles over smooth manifolds, but similar statements apply in the orbifold category. However, in the cases of g = 2, n = 2, 3, 4, the spaces involved are actually manifolds.

Let I denote the unit interval [0, 1]. If Z is a compact oriented zerodimensional manifold, we denote the signed cardinality of Z by ^{±}|Z|. All vector bundles we encounter in this subsection will be assumed to be smooth, complex, and normed.

Definition 3.1. Suppose M is a smooth manifold and F, O ⟶ M are vector bundles.

(1) If F = ⨁_{i=1}^{i=k} F_{i}, bundle map α : F ⟶ O is a polynomial of degree

d_{[k]} if for each i ∈ [k] there exists

p_{i} ∈ Γ(M; F_{i}^{∗⊗d_{i}} ⊗ O) for i ∈ [k] s.t.

α(υ) = ∑_{i=1}^{i=k} p_{i}(υ_{i}^{d_{i}}) ∀υ = (υ_{i})_{i∈[k]} ∈ ⨁_{i=1}^{i=k} F_{i}.

(2) If α : F ⟶ O is a polynomial, the rank of α is the number

rk α ≡ max{rk_{b}α : b ∈ M}, where rk_{b}α = dim_{ℂ} (Im α_{b}).

Polynomial α : F ⟶ O is of constant rank if rk_{b}α = rk α for all b ∈ M; α is nondegenerate if rk_{b}α = rk F for all b ∈ M.

(3) If Ω is an open subset of I × F, O is a vector bundle, and

{ϕ_{t}} = {ϕ_{t} : {υ ∈ F : (t, υ) ∈ Ω} ⟶ O}

is a family of smooth bundle maps, bundle map α : F ⟶ O is a dominant term for {ϕ_{t}} if there exists ε ∈ C^{0}(I × F; ℝ) such that

|ϕ_{t}(υ) − α(υ)| ≤ ε(t, υ)(t + |α(υ)|)

∀(t, υ) ∈ Ω and lim_{(t,υ)⟶0} ε(t, υ) = 0.

Dominant term α : F ⟶ O of {ϕ_{t}} is the resolvent of {ϕ_{t}} if α is a polynomial of constant rank.

In (2) above, by dim_{ℂ}(Im α_{b}) we mean the dimension of the image of α_{b} as an analytic subvariety of the fiber O_{b}. Note that if \overline{Ω} ⊂ I × F contains a neighborhood of {0} × M, the resolvent of {ϕ_{t}} is unique (if it exists).

Lemma 3.2. Suppose M is a smooth manifold,

(1) F ≡ F^{−} ⊕ F^{+} ⟶ M and O ≡ O^{−} ⊕ O^{+} ⟶ M are vector bundles;

(2) Ω is an open subset of I ×F and {ϕ_{t} : {υ ∈ F : (t, υ) ∈ Ω} ⟶ O} is a family of smooth maps;

(3) α : F ⟶ O is a dominant term for {ϕ_{t}} s.t. α(F^{+}) ⊂ O^{+}, α^{−} ≡ π^{−} ∘α|_{F^{−}} is a constant-rank polynomial, where π^{−} : O^{−} ⊕O^{+} ⟶ O^{−} is the projection map, and (dim M + 2rk α^{−}) < 2rk O^{−};

(4) \overline{ν} = (\overline{ν}^{−}, \overline{ν}^{+}) ∈ Γ(M; O^{−} ⊕ O^{+}) is generic with respect to α^{−}.

Then for every compact subset K of M, there exist δ_{K} > 0 and a neighborhood U_{F} (K) of K in F such that the map

ψ_{t} : \{υ ∈ F : (t, υ) ∈ Ω\} ⟶ O, ψ_{t}(υ) = t\overline{ν}_{υ} + ϕ_{t}(υ),

has no zeros on {υ ∈ U_{F} (K) : (t, υ) ∈ Ω} for all t ∈ (0, δ_{K}).

Proof. (1) Suppose υ̃ ∈ Ω_{δ_{K}}|_{K} and ψ_{t}(υ̃) = 0. Then by our assumptions on ϕ_{t},

|α(υ̃)| ≤ C_{K}(t + \overline{ε}_{K}(δ_{K})|α(υ̃)|),

where C_{K} > 0 depends only on K (and \overline{ν}) and \overline{ε}_{K} is a continuous function vanishing at zero. Thus, if δ_{K} > 0 is sufficiently small,

|α(υ̃)| ≤ 2C_{K}t ∀t < δ_{K}, υ̃ ∈ F_{δ_{K}}|_{K} s.t. ψ_{t}(υ̃) = 0. (3.1)

(2) Let F^{−} = ⨁_{i=1}^{i=k} F_{i} ⟶ M be the bundles and p_{i} ∈ Γ(M; F_{i}^{∗⊗d_{i}} ⊗

O^{−}) the sections as in (1) of Definition 3.1 corresponding to α^{−}. Define

φ_{t} ∈ Γ(M; End(F^{−})) by φ_{t}(υ_{i}) = t^{−1/d_{i}}υ_{i} if υ_{i} ∈ F_{i}.

Then by our assumption on ϕ_{t} and Equation (3.1),

|\overline{ν}^{−} + α^{−}(φ_{t}(υ^{−}))| ≤ C̃_{K} \overline{ε}_{K}(δ_{K}) ∀t < δ_{K}, υ̃ ∈ F_{δ_{K}}|_{K} s.t. ψ_{t}(υ̃) = 0, (3.2)

where C̃_{K} is determined by K. Since α^{−} has constant rank, the image of α^{−} is closed and is the total space of a bundle of affine analytic varieties of complex dimension rk α^{−} < rk O^{−} − \frac{1}{2} dim M. Thus, by assumption (4) of the lemma, \overline{ν}^{−} does not intersect the image of α^{−}, and there exists ϵ_{K} > 0 such that

|\overline{ν}^{−} + α^{−}(υ^{−})| ≥ ϵ_{K} ∀υ ∈ F^{−}|_{K}. (3.3)

If ϵ_{K} > C̃_{K} \overline{ε}_{K}(δ_{K}), by (3.2) and (3.3), π^{−} ∘ ψ_{t} (and thus ψ_{t}) has no zeros on F_{δ_{K}}|_{K}.

We will call family {ϕ_{t} : {υ ∈ F : (t, υ) ∈ Ω} ⟶ O} of smooth maps hollow if it admits a dominant term α that satisfies hypothesis (3) of Lemma 3.2.

Definition 3.3. Suppose M is a smooth manifold and F ⟶ M is a vector bundle.

(1) Subset Y of F is small if Y contains no fiber of F and there exists a smooth manifold Z of dimension (dim F − 1) and a smooth map f : Z ⟶ F such that the image of f is closed in F and contains Y.

(2) If F, F̃ ⟶ M are smooth complex vector bundles, ρ ∈ Γ(M; F^{∗⊗d} ⊗ F̃) induces a d̃-to-1 cover F ⟶ F̃ if the map

F_{b} ⟶ F̃_{b}, υ ⟶ ρ(υ) ≡ ρ(υ^{d}),

is d̃-to-1 on a dense open subset of every fiber F_{b} of F.

Lemma 3.4. Suppose M is a smooth manifold, F = ⨁_{i=1}^{i=k} F_{i} and O

are vector bundles over M, and

α = ∑_{i=1}^{i=k} p_{i} : F ⟶ O, where p_{i} ∈ Γ(M; F_{i}^{∗⊗d_{i}} ⊗ O),

is a nondegenerate polynomial. Then there exists a small subset Y_{α}

of F = ⨁_{i=1}^{i=k} F_{i}, which is invariant under scalar multiplication in each

component separately, with the following property. If K is a compact subset of O − α(Y_{α}), there exists C_{K} > 0 such that

|υ| ≤ C_{K}|α(υ)| ∀ υ ∈ F s.t. α(υ) ∈ K.

Proof. (1) Let Y_{α} ⊂ F be the closed subset on which the differential of the fiberwise map υ ⟶ α(υ) does not have full rank, i.e., its rank is less than rk F. Since α is nondegenerate, Y_{α} contains no fiber of F. By our assumptions on α,

D(α|_{F_{b}})|_{υ} = (D(p_{1}|_{F_{1,b}})|_{υ_{1}}, …, D(p_{k}|_{F_{k,b}})|_{υ_{k}}) : F_{1} ⊕ ⋯ ⊕ F_{k} ⟶ O,

∀ b ∈ M, υ = υ_{[k]} ∈ ⨁_{i=1}^{i=k} F_{i}.

Since p_{i}|_{F_{i,b}} is a homogeneous polynomial of degree d_{i}, its derivative is a homogeneous polynomial of degree (d_{i} − 1). Thus, Y_{α} is preserved under scalar multiplication in each component separately. It also clearly satisfies the second condition of (1) of Definition 3.3.

(2) On F −Y_{α}, α is a covering map onto its image with the number of leaves bounded by some number N_{α}. Thus, if K is any compact subset of O − α(Y_{α}), α^{−1}(K) is a compact subset of F. Therefore, there exists C_{K} such that

|υ| ≤ C_{K}|α(υ)| ∀υ ∈ F s.t. α(υ) ∈ K.

Note that if 0 ∉ α(Y_{α}), then α is a linear injection on every fiber, and the above inequality holds on all of F.

Lemma 3.5. Suppose M is a smooth manifold,

(1) F = ⨁_{i=1}^{i=k} F_{i} and O are vector bundles over M with rk F + \frac{1}{2} dim M=

rk O;

(2) Y is a small subset of F = ⨁_{i=1}^{i=k} F_{i}, which is invariant under the

scalar multiplication in each component separately;

(3) Ω is an open subset of I × F such that Ω ∪ ({0} × X) is a neighborhood of {0} × X in I × (F − (Y − X)); (4) {ϕ_{t} : {υ ∈ F : (t, υ) ∈ Ω} ⟶ O} is a family of smooth maps;

(5) nondegenerate polynomial α : F ⟶ O is the resolvent of {ϕ_{t}};

(6) \overline{ν} ∈ Γ(M; O) is generic with respect to (Y, α), and the map

F ⟶ O, υ ⟶ \overline{ν}_{υ} + α(υ), (3.4)

has a finite number of (transverse) zeros.

If ψ_{t} is transversal to zero for all t, there exists a compact subset K_{α,\overline{ν}} of M with the following property. If K is a precompact open subset of M containing K_{α,\overline{ν}}, there exist δ_{K}, ϵ_{K} > 0 such that for all t ∈ (0, ϵ_{K}),

±|\{υ ∈ F_{δ_{K}}|_{K} : (t, υ) ∈ Ω, ψ_{t}(υ) = 0\}| =^{±} |\{υ ∈ F : \overline{ν}_{υ} + α(υ) = 0\}|,

where ψ_{t}(υ) = t\overline{ν}_{υ} + ϕ_{t}(υ) as before. Furthermore, all the zeros of ψ_{t}|_{F_{δ_{K}} |K} lie over K_{α,\overline{ν}}.

Proof. (1) Since the map in (3.4) has a finite number of zeros, all of them lie in the interior of F_{C_{α,\overline{ν}}}|_{K_{α,\overline{ν}}} for some compact subset K_{α,\overline{ν}} of M and number C_{α,\overline{ν}} > 0. Suppose K ⊂ M is a precompact open subset containing K_{α,\overline{ν}}, δ_{K} > 0 is such that F_{δ_{K}}|_{K−Y} ⊂ Ω, and υ̃ ∈ Ω_{δ_{K}}|_{K} is such that ψ_{t}(υ̃) = 0. By the same argument as in the proof of Lemma 3.2, if δ_{K} > 0 is sufficiently small,

(3.5) |α(υ̃)| ≤ C_{K}t and |t\overline{ν}υ̃ + α(υ̃)| ≤ \overline{ε}_{K}(δ_{K})t ∀t < δ_{K}, υ̃ ∈ F_{δ_{K}}|_{K} s.t. ψ_{t}(υ̃) = 0,

where C_{K} and \overline{ε}_{K} = \overline{ε}_{K}(δ_{K}) depend only on K, and \overline{ε}_{K}(δ_{K}) tends to zero with δ_{K}. Let ϕ_{t} : F ⟶ F be the map defined in (2) of the proof of Lemma 3.2, with F^{−} replaced by F. By (3.5),

(3.6) α(ϕ_{t}(υ̃)) ∈ K\overline{ν}(K; C_{K}, \overline{ε}_{K}(δ_{K})) ≡ \{ϖ ∈ O_{C_{K}} : |\overline{ν}_{ϖ} + ϖ| ≤ \overline{ε}_{K}(δ_{K})\} ∀t < δ_{K}, υ̃ ∈ F_{δ_{K}}|_{K} s.t. ψ_{t}(υ̃) = 0.

(2) If \overline{ν} is generic, the map in (3.4) does not vanish on Y_{α}, where Y_{α} is as in Lemma 3.4. Since α(Y_{α}) is a closed subset of O, there exists ϵ_{K} > 0 such that

|\overline{ν}_{υ} + α(υ)| > ϵ_{K} ∀ υ ∈ Y_{α}|_{K}.

Thus, if \overline{ϵ}_{K}(δ_{K}) < ϵ_{K}, K\overline{ν}(K; C_{K}, \overline{ε}_{K}(δ_{K})) is a compact subset of O disjoint from α(Y_{α}). Then by (3.6) and Lemma 3.4,

|ϕ_{t}(υ̃)| ≤ C_{K}^{∗} ∀t < δ_{K}, υ̃ ∈ F_{δ_{K}}|_{K} s.t. ψ_{t}(υ̃) = 0, (3.7)

where C_{K}^{∗} depends only on K.

(3) There is a one-to-one sign-preserving correspondence between the zeros of ψ_{t} on Ω_{δ_{K}}|_{K} and the zeros of

ψ̃_{t} : Ω_{δ_{K}}(K, t) ≡ {υ ∈ F : (t, ϕ_{t}^{−1}(υ)) ∈ Ω_{δ_{K}}|_{K}} ⟶ O,

ψ̃_{t}(υ) = t^{−1}ψ_{t}(ϕ_{t}^{−1}(υ)).

By (3.7), all the zeros of ψ̃_{t} on Ω_{δ_{K}}(K, t) are in fact contained in F_{C_{K}^{∗}}|_{K}. We can assume that C_{K}^{∗} > C_{α,\overline{ν}}. By our assumptions on ϕ_{t},

|ψ̃_{t}(υ) − (\overline{ν}_{υ} + α(υ))| ≤ C_{K} \overline{ϵ}_{K}(δ_{K}) ∀υ ∈ Ω_{δ_{K}}(K, t) ∩ F_{C_{K}^{∗}}|_{K}, (3.8)

where C_{K} > 0 depends only on K. We define a cobordism between the zeros of ψ̃_{t} and the zeros of \overline{ν} + α on Ω_{δ_{K}}(K, t) ∩ F_{C_{K}^{∗}}|_{K} by

Ψ : I × Ω_{δ_{K}}(K, t) ∩ F_{C_{K}^{∗}}|_{K} ⟶ O,
Ψ_{τ} (υ) = τψ̃_{t}(υ) + (1 − τ)(\overline{ν}_{υ} + α(υ)) + η_{τ} (υ),

where η : I × Ω_{δ_{K}}(K, t) ⟶ O is any smooth function with very small C^{0}-norm such that η_{0} = η_{1} = 0 and Ψ is transversal to zero. It remains to see that Ψ^{−1}(0) is compact. Suppose Ψ_{τ_{r}}(υ_{r}) = 0 and (τ_{r}, υ_{r}) converges (τ̃, υ̃) ∈ I×F_{2C_{K}^{∗}}|\overline{K}; we need to show that υ̃∈Ω_{δ_{K}}(K, t) ∩ F_{C_{K}^{∗}}|_{K}. By Equation (3.8),

(3.9) |\overline{ν}_{υ_{r}} + α(υ_{r})| ≤ C_{K} \overline{ϵ}_{K}(δ_{K}) + ∥η∥_{C^{0}} ∀r ⟹ |\overline{ν}υ̃ + α(υ̃)| ≤ C_{K} \overline{ϵ}_{K}(δ_{K}) + ∥η∥_{C^{0}}.

On the other hand, since \overline{ν} is generic, the map in (3.4) does not vanish on Y. Furthermore, all the zeros of this map are contained in the interior of F_{C_{α,\overline{ν}}}|_{K_{α,\overline{ν}}}. Thus, by compactness,

ϵ̃_{K} ≡ inf \{|\overline{ν}_{υ} + α(υ)| : υ ∈ (Y ∩ F_{2C_{K}^{∗}}) ∪ (F_{C_{K}^{∗}}|_{K} − F_{C_{α,\overline{ν}}}|_{K_{α,\overline{ν}}})\} > 0, (3.10)

where ϵ̃_{K} depends only on K. If ϵ̃_{K} > C_{K} \overline{ϵ}_{K}(δ_{K}) + ∥η∥_{C^{0}}, by (3.9) and (3.10),

υ̃ ∈ F_{C_{α,\overline{ν}}}|_{K_{α,\overline{ν}}} ⊂ F_{C_{K}^{∗}}|_{K} − Y ⊂ Ω_{δ_{K}}(K, t).

The last inclusion follows from the very first assumption on δ_{K} above. We conclude that Ψ^{−1}(0) is compact.

Corollary 3.6. Suppose M is a smooth oriented manifold,

(1) F ≡ F^{−} ⊕ F^{+}, F̃^{−}, and O ≡ O^{−} ⊕ O^{+} are vector bundles over M with

rk F^{−} = rk F̃^{−} = rk O^{−} − \frac{1}{2} dim M and rk F^{+} = rk O^{+};

(2) ρ ∈ Γ(M; F^{−∗⊗k} ⊗ F̃^{−}) induces a d̃-to-1 cover F ⟶ F̃, and α^{−} ∈ Γ(M; F̃^{−∗} ⊗ O^{−});

(3) α : F ⟶ O is a nondegenerate polynomial such that α^{+} ≡ α|_{F^{+}} : F^{+} ⟶ O^{+} is linear and π^{−} ∘ α = α^{−} ∘ ρ;

(4) Y is a small subset of F, which is invariant under the scalar multiplication in each component separately;

(5) Ω is an open subset of I × F such that Ω ∪ X is a neighborhood of {0} × X in I × (F − (Y − X));

(6) {ϕ_{t} : {υ ∈ F : (t, υ) ∈ Ω} ⟶ O} is a family of smooth maps with resolvent α;

(7) \overline{ν} = (\overline{ν}^{−}, \overline{ν}^{+}) ∈ Γ(M; O^{−}⊕O^{+}) is generic with respect to (α^{+}, α^{−}, ρ, Y), and the map

F̃^{−} ⟶ O^{−}, ϖ ⟶ \overline{ν}_{ϖ}^{−} + α^{−}(ϖ), (3.11)

has a finite number of (transverse) zeros.

If ψ_{t} is transversal to zero for all t, there exists a compact subset K_{α,\overline{ν}} of M with the following property: If K is a precompact open subset of M containing K_{α,\overline{ν}}, there exist δ_{K}, ϵ_{K} > 0 such that for all t ∈ (0, ϵ_{K}),

±|{υ ∈ F_{δ_{K}}|_{K} : (t, υ) ∈ Ω, ψ_{t}(υ) = 0}|
= d̃·^{^{±}|}\{ϖ ∈ F̃^{−} : \overline{ν}_{ϖ}^{−} + α^{−}(ϖ) = 0\}|,

where ψ_{t}(υ) = t\overline{ν}_{υ} + ϕ_{t}(υ). Furthermore, all the zeros of ψ_{t}|_{F_{δ_{K}} |K} lie over K_{α,\overline{ν}}.

Proof. Let K_{α,\overline{ν}} and δ_{K} > 0 be as in Lemma 3.5. Then if K is a precompact open subset of M, for all t ∈ (0, ϵ_{K}) the signed number of zeros of ψ_{t} on Ω_{δ_{K}}|_{K} is the same as the signed number of solutions of

F |_{K} ⟶ O, \begin{cases}\overline{ν}_{υ}^{−} + α^{−}(ρ(υ^{−})) = 0 ∈ O^{−}; \\ \overline{ν}_{υ}^{+} + α^{+}(υ^{+}) + π^{+}(α(υ^{−})) = 0 ∈ O^{+}. \end{cases} (3.12)

For every solution of the first equation, there is a unique solution of the second equation. Since α^{+} is complex-linear on the fibers, the signed number of solutions of (3.12) is the same as the signed number of solutions of the first equation. Since the first equation has no solutions on Y_{α^{−}} if \overline{ν} is generic and ρ is d̃-to-1 outside of Y_{α^{−}}, ρ induces a d̃-to-1 sign-preserving map from the set of zeros of (3.11) to the set of solutions of the first equation.

3.2 Contributions to the Euler class

If \overline{M} is a smooth oriented compact n-manifold and V ⟶ \overline{M} is an oriented vector bundle of rank n, the euler class of V is the number of zeros of any section s : \overline{M} ⟶ V which is transverse to the zero set. In this subsection, under slightly more topological assumptions on \overline{M} and V, we discuss a relationship between subsets of the zero set of a non-transverse section and the euler class of V.

Definition 3.7.

(1) Compact oriented topological manifold \overline{M} = M_{n} ⊔ ⨆_{i=0}^{i=n−2} M_{i} of

dimension n is mostly smooth, or ms, if:

(1a) Each M_{i} is a smooth manifold of dimension i, and M ≡ M_{n} is a dense open subset of \overline{M}.

(1b) For each i ∈ [n − 2], \overline{M}_{i} − M_{i} ⊂ ⋃_{j=0}^{j−2} M_{j};

(2) If \overline{Z} = Z⊔⨆ Z_{j} and \overline{M} = M⊔⨆ M_{i} are ms-manifolds, continuous map π : Z ⟶ M is an ms-map if for each j there exists i such that π : Z_{j} ⟶ M_{i} is a smooth map.

(3) If \overline{M} is an ms-manifold, topological vector bundle V ⟶ \overline{M} is an ms-bundle if V |_{M_{i}} is a smooth vector bundle for i = n and all i ∈ [n − 2].

(4) If V ⟶ \overline{M} is an ms-bundle, continuous section s : M ⟶ V is an ms-section if s|_{M_{i}} is C^{2}-smooth for i = n and all i ∈ [n − 2].

The dense open submanifold M of \overline{M} will be called the smooth base of \overline{M}. Note that if E ⟶ \overline{M} is an ms-bundle, then the (complex) projectivization ℙE of E is an ms-manifold. Furthermore, the projection map π_{E} : ℙE ⟶ \overline{M} is an ms-map, and the tautological line bundle γ_{E} ⟶ ℙE is an ms-bundle.

If V ⟶ \overline{M} is an ms-bundle, we denote the space of ms-sections of V by Γ(\overline{M}; V). Using (4) of Definition 3.7, we define an ms-polynomial map between two ms-bundles analogously to (1) of Definition 3.1. We topologize Γ(\overline{M}; V) as follows. If s_{k}, s ∈ Γ(\overline{M}; V), the sequence \{s_{k}\} converges to s if s_{k} converges to s in the C^{0}-norm on all of \overline{M} and in the C^{2}-norm on compact subsets of M_{i} for i = n and all i ∈ [n − 2]. The C^{0}-norm is defined with respect to the norm on V ⟶ \overline{M}. In order to define the C^{2}-norm on compact subsets of M_{i}, we fix a connection in each smooth bundle in V ⟶ M_{i}.

Definition 3.8. Let \overline{M} be an ms-manifold as in Definition 3.7.

(1) If Z ⊂ M_{i} is a smooth oriented submanifold, a normal-bundle model for Z is a tuple (F, Y, ϑ), where:

(1a) F ⟶ Z is a smooth complex normed vector bundle and Y is a small subset of F.

(1b) For some δ ∈ C^{∞}(Z; ℝ^{+}), ϑ : F_{δ} − (Y − Z) ⟶ \overline{M} is a continuous map such that:

(1b-i) ϑ : F_{δ} − (Y − Z) ⟶ \overline{M} is a homeomorphism onto an

open neighborhood of Z in M ∪ Z; (1b-ii) ϑ|_{Z} is the identity map, and ϑ : F_{δ} − (Y − Z) ⟶ M

is an orientation preserving diffeomorphism on an open subset of M.

(2) A closure of a normal-bundle model (F, Y, ϑ) is a tuple (\overline{Z}, F̃, π), where:

(2a) \overline{Z} is an ms-manifold with smooth base Z;

(2b) π : \overline{Z} ⟶ \overline{M} is an ms-map such that π|_{Z} is the identity;

(2c) F̃ ⟶ \overline{Z} is an ms-bundle such that F̃|_{Z} = F.

If Z is a smooth submanifold of M, an identification of the normal bundle N Z of Z in M with a neighborhood of Z in M induces a normal bundle model for Z. Definition 3.8 extends this standard construction to the ms-category.

Definition 3.9. Suppose E, O ⟶ \overline{M} are ms-bundles and α : E ⟶ O is an ms-polynomial.

(1) Subset Z of M is α-regular if there exist a normal bundle model (F, Y, ϑ) for Z, constant-rank polynomial p : F ⊕E ⟶ O over Z, smooth bundle isomorphisms ϑ_{E}: ϑ^{∗}E ⟶ π_{F}^{∗} E and ϑ_{O} : ϑ^{∗}O ⟶ π_{F}^{∗} O covering the identity on F_{δ} − (Y − Z), and ε ∈ C(F; ℝ) such that:

(1a) ϑ_{E} and ϑ_{O} are smooth on F_{δ} − Y − Z and restrict to the

identity over Z; (1b) lim_{w⟶0} ε(w) = 0; (1c) |ϑ_{O}α(ϑ_{E}^{−1}(w, υ)) − p(w, υ)| ≤ ε(w)|p(w, υ)| for all w ∈ F_{δ} −

(Y − X), υ ∈ E.

(2) α is a regular polynomial if \overline{M} is a union of finitely many α-regular subsets.

Lemma 3.10. Suppose E, O ⟶ \overline{M} are ms-bundles, such that rk E + \frac{1}{2} dim \overline{M} = rk O, and α : E ⟶ O is a regular polynomial, such that α is nondegenerate on M. Let ν ∈ Γ(\overline{M}; O) be an ms-section such that the map

ψ_{α,ν} : E ⟶ O, ψ_{α,ν}(υ) = ν_{υ} + α(υ),

does not vanish on E|_{\overline{M}−M} and is transversal to the zero set in O|_{M}. Then ψ_{α,ν}^{−1}(0) is finite, and N(α) ≡^{±} |ψ_{α,ν}^{−1}(0)| is independent of the choice of ν as above.

Proof. (1) We first show that for every x ∈ \overline{M} − M there exists a neighborhood U of x in \overline{M} such that ψ_{α,ν} does not vanish on E|_{U}. By (2) of Definition 3.9, there exists an α-regular subset Z of \overline{M} containing x. Let (F, Y, ϑ), δ, p, ϑ_{E}, ϑ_{O}, and ε be as in (1) of Definition 3.9. It can be assumed that δ is such that

ε(w) < \frac{1}{2} and |ν_{ϑ(w)}| ≤ 2|ν_{w}| ≡ 2|ν_{b_{w}} | ∀w ∈ F_{δ} − (Y − Z).

Then, if ψ_{α,ν}(ϑ_{E}^{−1}(w, υ)) = 0 for some (w, υ) ∈ F ⊕ E with w ∈ F_{δ} − (Y − Z), |α(w, υ)| ≤ 4|ν_{w}| by (1c) of Definition 3.9. Thus, if {(w_{k}, υ_{k})} ⊂ F ⊕E is such that ψ_{α,ν}(ϑ_{E}^{−1}(w_{k}, υ_{k}) = 0 and w_{k} ⟶ x ∈ F, a subsequence of {α(w_{k}, υ_{k})} converges to an element ϖ ∈ O_{x}. Since α is a polynomial map of constant rank, there exists (0, υ) ∈ F ⊕ E such that α(0, υ) = ϖ. Since α(0, υ) = p(0, υ), it follows that ψ_{α,ν}(υ) = 0 contrary to the assumption.

(2) By (1), there exists a compact subset K_{α,ν} of M such that ψ_{α,ν}^{−1}(0) ⊂ E|_{K_{α,ν}}. Since ψ_{α,ν} is transversal to zero, ν(M) ∩ α(Y_{α}) = ∅, where Y_{α} ⊂ E|_{M} is as in Lemma 3.4. It follows that ψ_{α,ν}^{−1}(0) is a finite subset of E|_{M}.

(3) The final claim of the lemma is obtained by constructing a cobordism between ψ_{α,ν} and ψ_{α,ν′}. More precisely, we take a smooth family {ν_{τ} : τ ∈ I} of ms-sections of O such that ν_{0} = ν, ν_{1} = ν′, ψ_{α,ν_{τ}}^{−1} (0) ⊂ E|_{M}, and the section

Ψ_{α} : I × E ⟶ O, Ψ_{α}(τ, υ) = ψ_{α,ν_{τ}} (υ),

is transversal to the zero set in O. Such a family can always be chosen, since \overline{M} − M has codimension two in \overline{M}. Then, by the same argument as in (1) and (2), Ψ_{α}^{−1}(0) is a smooth compact oriented submanifold of E|_{M} with boundary ψ_{α,ν_{1}}^{−1}(0) − ψ_{α,ν_{0}}^{−1}(0).

Definition 3.11. Suppose \overline{M} is an ms-manifold of dimension 2n, V ⟶ \overline{M} is an ms-bundle of rank n, s ∈ Γ(\overline{M}; V), and Z ⊂ M_{i}∩s^{−1}(0).

(1) Z is s-hollow if there exist a normal bundle model (F, Y, ϑ) for Z and a bundle isomorphism ϑ_{V} : ϑ^{∗}V ⟶ π_{F}^{∗} V, covering the identity on F_{δ} − (Y − Z), such that:

(1a) ϑ_{V} |_{F_{δ}−Y −Z} is smooth and ϑ_{V} |_{Z} is the identity;

(1b) ϕ_{0} ≡ ϑ_{V} ∘ ϑ^{∗}s : F_{δ} − (Y − Z) ⟶ V is hollow.

(2) Z is s-regular if there exist a normal bundle model (F, Y, ϑ) for Z with closure (\overline{Z}, F̃, π), regular polynomial α : F̃ ⟶ π^{∗}V, and a bundle isomorphism ϑ_{V} : ϑ^{∗}V ⟶ π_{F}^{∗} V covering the identity on F_{δ} − (Y − Z), such that:

(2a) ϑ_{V} |_{F_{δ}−Y −Z} is smooth and ϑ_{V} |_{Z} is the identity;

(2b) α|_{Z} is nondegenerate and is the resolvent for ϕ_{0} ≡ ϑ_{V} ∘ ϑ^{∗}s : F_{δ} − (Y − Z) ⟶ V, and Y is preserved under scalar multiplication in each of the components of F for the splitting corresponding to α as in (1) of Definition 3.1.

Lemma 3.12. If (\overline{M},V, s) and (Z, F,Y, ϑ) are as in Definition 3.11, there exist a number C_{Z}(s) ∈ ℤ, which equals zero if Z is s-hollow, and a dense open subset Γ_{Z}(s) ⊂ Γ(\overline{M}; V) with the following properties. For every ν ∈ Γ_{Z}(s):

(1) There exists ϵ_{ν} > 0 such that for all t ∈ (0, ϵ_{ν}), all the zeros of tν + s are contained in M and (tν + s)|_{M} is transversal to the zero set in V.

(2) There exist a compact subset K_{ν} ⊂ Z, open neighborhood U_{ν}(K) of K in \overline{M} for each compact subset K ⊂ Z, and ϵ_{ν}(U) ∈ (0, ϵ_{ν}) for each open subset U of \overline{M} such that

±|{b ∈ U : tν(b) + s(b) = 0}| = C_{Z}(s) if t ∈ (0, ϵ_{ν}(U)), K_{ν} ⊂ K ⊂ U ⊂ U_{ν}(K).

Proof. It is clear that we can choose a dense open subset Γ′_{Z}(s) ⊂ Γ(\overline{M}; V) such that every ν ∈ Γ′_{Z}(s) satisfies requirement (1) of the lemma. If Z is s-hollow, we also need that \overline{ν} ≡ ν|_{Z} is generic with respect to the corresponding polynomial α^{−} in the sense of the proof of Lemma 3.2. We can then take K_{ν} = ∅. If Z is s-regular, let \overline{ν} = π^{∗}ν ∈ Γ(\overline{Z}; π^{∗}V). By Lemma 3.10, the second part of (6) of Lemma 3.5 is satisfied, as long as tν + s is transversal to the zero set on each smooth strata. The other requirements on \overline{ν} in Lemma 3.5 are finitely many transversality properties. We then take

C_{Z}(s) = ^{±}|\{υ ∈ F : \overline{ν}_{υ} + α(υ) = 0\}|.

By Lemma 3.10, this number is well-defined.

The total number of zeros of a section tν + s satisfying condition (1) of Lemma 3.12 is precisely the euler class e(V) of the bundle V ⟶ \overline{M}. Thus, due to (2) of Lemma 3.12, we call C_{Z}(s) the scontribution (or simply contribution) of Z to e(V). If Z is any subset of \overline{M} such that Z ∩ s^{−1}(0) satisfies the requirements of Definition 3.11, let C_{Z}(s) = C_{Z∩s^{−1}(0)}(s). In addition, if Z is a closed subset of \overline{M} such that s^{−1}(0) − Z is also closed, we can easily define C_{Z}(s) by Lemma 3.12.

Corollary 3.13. Let V ⟶ \overline{M} be an ms-bundle of rank n over an ms-manifold of dimension 2n. Suppose U is an open subset of M and s ∈ Γ(\overline{M}; V) is such that s|_{U} is transversal to the zero set.

(1) If s^{−1}(0)∩U is a finite set, ^{±}|s^{−1}(0)∩U| = ⟨e(V), [\overline{M}]⟩−C_{\overline{M}−U}(s).

(2) If \overline{M} − U = ⨆_{i=1}^{i=k} Z_{i}, where each Z_{i} is s-hollow or s-regular, then

s^{−1}(0) ∩ U is finite, and

±|s^{−1}(0)∩U| = ⟨e(V), [\overline{M}]⟩−C_{\overline{M}−U}(s) = ⟨e(V), [\overline{M}]⟩−∑_{i=1}^{i=k} C_{Z_{i}}(s).

If Z_{i} is s-hollow, C_{Z_{i}}(s) = 0. If Z_{i} is s-regular and α_{i} : F̃_{i} ⟶ V is the corresponding polynomial,

C_{Z_{i}}(s) = ^{±}|\{υ ∈ F̃_{i} : \overline{ν}_{υ} + α_{i}(υ) = 0\}| ≡ N(α_{i}),

where \overline{ν} ∈ Γ(\overline{Z}_{i}; V) is a generic section. Finally, if α_{i} ∈ Γ(\overline{Z}_{i}; F̃_{i}^{∗⊗k} ⊗ π^{∗}V) has constant rank over \overline{Z}_{i} and factors through a k̃-to-1 cover ρ_{i} : F̃_{i} ⟶ F̃_{i}^{⊗k},

C_{Z_{i}}(s) = k̃⟨e(π^{∗}V/α_{i}(F̃_{i})), [\overline{Z}_{i}]⟩.

All statements of this corollary have already been proved. A splitting of the zero set as in (2) of Corollary 3.13 always exists in the complexanalytic category. It should be possible to generalize the constructions of this subsection to an arbitrary compact oriented topological manifold. However, Lemma 3.10 will no longer be valid, and another approach will be needed to deal with the zeros of ψ_{α,ν} that tend to infinity. For the cases that we encounter in Section 5, the version of s-regularity of Definition 3.11 suffices.

3.3 Zeros of polynomial maps

We now present a procedure for computing the number of zeros of a polynomial map between two complex vector bundles over a compact oriented manifold. All the polynomials we encounter in Section 5 are of degree-one. Thus, we focus on the degree-one case, but discuss the general case at the end for the sake of completeness.

Suppose \overline{M} is an ms-manifold, E, O ⟶ \overline{M} are ms-bundles such that rk E + \frac{1}{2} dim \overline{M} = rk O, and α ∈ Γ(\overline{M}; E^{∗} ⊗ O) is an ms-section. Let \overline{ν} ∈ Γ(\overline{M}; O) be such that \overline{ν} has no zeros, the map

ψ_{α,\overline{ν}} ≡ \overline{ν} + α : E ⟶ O

is transversal to the zero set in O on E|_{M}, and all its zeros are contained in E|_{M}. The first step in our procedure of determining the number of zeros of ψ_{α,\overline{ν}} reduces this issue to the case E is a line bundle. Let ℙE be the projectivization of E (over ℂ) and let γ_{E} ⟶ ℙE be the tautological line bundle. Then α induces an ms-section α_{E} ∈ Γ(ℙE; γ_{E}^{∗} ⊗ π_{E}^{∗}O), where π_{E} : ℙE ⟶ \overline{M} is the bundle projection map. The number of zeros of ψ_{α,\overline{ν}} is the same as the number of zeros of the induced map

ψ_{α,\overline{ν}}^{E} ≡ π_{E}^{∗}\overline{ν} + α_{E} : γ_{E} ⟶ π_{E}^{∗}O.

Thus, we can always reduce the computation to the case E is a line bundle.

The second step describes the number of zeros of ψ_{α,\overline{ν}} topologically in the case E is a line bundle. Since \overline{ν} has no zeros, it spans a trivial subbundle ℂ\overline{ν} of O. Let O^{⊥} be the quotient of O by this trivial subbundle. Denote the ℂ\overline{ν}- and O^{⊥}-components of α by α^{t} and α^{⊥}, respectively. Then the zeros of ψ_{α,\overline{ν}} are described by

\begin{cases}\overline{ν}_{b} + α_{b}^{t}(v) = 0 ∈ ℂ\overline{ν}; \\ α_{b}^{⊥}(v) = 0 ∈ O^{⊥}; \end{cases} b ∈ \overline{M}, v ∈ E_{b}. (3.13)

Since \overline{ν} does not vanish, all solutions of the first equations (3.13) are nonzero. The solution of the second equation with nonzero v is (E − \overline{M})|_{α^{⊥−1}(0)}. Furthermore, if b ∈ α^{⊥−1}(0) and α(b) ≠ 0, α^{t} : E ⟶ (ℂ\overline{ν})_{b} is an isomorphism. Thus, for every b ∈ α^{⊥−1}(0) − α^{−1}(0), there exists a unique v ∈ E_{b} solving the first equation in (3.13), and the sign of (b, v) as a zero of ψ_{α,\overline{ν}} agrees with the sign of b as a zero of α^{⊥}. On the other hand, (3.13) has no solutions on E|_{α^{−1}(0)}. It follows that the number of zeros of ψ_{α,\overline{ν}} is the number of zeros of α^{⊥} on \overline{M} − α^{−1}(0), i.e.,

±|ψ^{−1}(0)| = ⟨e(E^{∗} ⊗ O^{⊥}), [\overline{M}]⟩ − C_{α^{−1}(0)}(α^{⊥}); (3.14)

see Corollary 3.13.

As discussed in the previous subsection, computing C_{Z}(s) in reasonably good cases reduces to counting the number of zeros of polynomial maps between vector bundles over ms-manifolds, but with the rank of the target bundle one less than the rank of the bundle O we started with. Thus, this process will eventually terminate. The lemma below summarizes the last two paragraphs. Let λ_{E} = c_{1}(γ_{E}^{∗}).

Lemma 3.14. Suppose \overline{M} is an ms-manifold and E, O ⟶ \overline{M} are ms-bundles such that

rk E + \frac{1}{2} dim \overline{M} = rk O.

If α ∈ Γ(\overline{M}; E^{∗} ⊗ O) and \overline{ν} ∈ Γ(\overline{M}; O) are such that α is regular, \overline{ν} has no zeros, the map

ψ_{α,\overline{ν}} ≡ \overline{ν} + α : E ⟶ O

is transversal to the zero set on E|_{M}, and all its zeros are contained in E|_{M}, then ψ_{α,\overline{ν}}^{−1}(0) is a finite set, ^{±}|ψ_{α,\overline{ν}}^{−1}(0)| depends only on α, and

N(α) ≡ ^{±}|ψ_{α,\overline{ν}}^{−1}(0)| = ⟨c(O)c(E)^{−1}, [\overline{M})]⟩ − C_{α_{E}^{−1}(0)}(α_{E}^{⊥}).

Proof. Let n = rk E, m= rk O, and λ_{E} = c_{1}(γ_{E}^{∗}). From Lemma 3.10, Equation (3.14), and the construction above, we obtain the first two claims of the lemma along with

(3.15)  N(α) = ∑_{k=0}^{k=m−1} ⟨c_{k}(O^{⊥})λ_{E}^{m−1−k}, [ℙE]⟩ − C_{α_{E}^{−1}(0)}(α_{E}^{⊥})

= ∑_{k=0}^{k=m−1} ⟨c_{k}(O)λ_{E}^{m−1−k}, [ℙE]⟩ − C_{α_{E}^{−1}(0)}(α_{E}^{⊥}).

On the other hand,

λ_{E}^{n} + ∑_{k=1}^{k=n} c_{k}(E)λ_{E}^{n−k} = 0 ∈ H^{2n}(ℙE) and (3.16)

⟨μλ_{E}^{n−1}, [ℙE]⟩ = ⟨μ, [\overline{M}]⟩ ∀μ ∈ H^{2m−2n}(\overline{M});

see [2] for example. The last statement of the lemma follows from (3.15) and (3.16).

Remark. If α : E ⟶ O is a polynomial, and not just a linear map, the first step in computing the number of zeros of the map ψ_{α,\overline{ν}} = \overline{ν} + α would be to reduce to the case α is a linear map via a projectivization construction similar to the one in the second paragraph of this subsection. For example, suppose α = p_{1} + p_{2}, where p_{i} ∈ Γ(\overline{M}; E_{i}^{∗⊗d_{i}} ⊗ O) and E = E_{1} ⊕ E_{2}. Then the number of zeros of ψ_{α,\overline{ν}} is the same as the number of zeros of

ψ_{α,\overline{ν}}^{E_{1}} ≡ π_{E_{1}}^{∗}\overline{ν} + p_{1,E_{1}} + π_{E_{1}}^{∗}p_{2} : γ_{E_{1}} ⊕ π_{E_{1}}^{∗}E_{2} ⟶ π_{E_{1}}^{∗}O

over ℙE_{1}, where p_{1,E_{1}} ∈ Γ(ℙE_{1}; γ_{E_{1}}^{∗⊗d_{1}}) is the section induced by p_{1}. If \overline{ν} is generic, this number is d_{1}-times the number of zeros of the map

ψ̃_{α,\overline{ν}}^{E_{1}} ≡ π_{E_{1}}^{∗}\overline{ν} + p_{1,E_{1}} + π_{E_{1}}^{∗}p_{2} : γ_{E_{1}}^{⊗d_{1}} ⊕ π_{E_{1}}^{∗}E_{2} ⟶ π_{E_{1}}^{∗}O.

Note that p_{1,E_{1}} is linear on γ_{E_{1}}^{⊗d_{1}}. Taking the projection of π_{E_{1}}^{∗}E_{2} over ℙE_{1} and repeating the above procedure, we obtain an affine map

ψ_{α,\overline{ν}}^{E_{1},E_{2}} : π_{E_{2}}^{∗}γ_{E_{1}}^{⊗d_{1}} ⊕ γ_{π_{E_{1}}^{∗}E_{2}}^{⊗d_{2}} ⟶ π_{π_{E_{1}}^{∗}E_{2}}^{∗}π_{E_{1}}^{∗}O.

4. Resolvents for {ψ_{T ,tν}^{μ}} and {ψ_{S,tν}^{μ}}

4.1 A power series expansion for π_{υ,−}^{0,1}\overline{∂}u_{υ}

Throughout this section, we assume that T = (Σ, [N], I; j, \underline{d}) is a simple bubble type, with dˆ0 = 0 and ∑_{i∈I} d_{i} = d, and μ is an N-tuple of constraints in general position of total codimension

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Our goal is to extract leading-order terms from the bundle map ψ_{T ,tν}^{μ} of Theorem 2.7 and to describe the zero set of ψ_{T ,tν}^{μ} as the union of the zero sets of affine maps between finite-rank vector bundles. The main topological tool is Subsection 3.1.

Nearly all of this subsection is devoted to obtaining the power series expansion for π_{υ,−}^{0,1}\overline{∂}u_{υ} of Proposition 4.4. However, we first state an estimate for π_{υ,−}^{0,1}ν_{υ,t}, which is immediate from Theorem 2.7.

Let {ψ_{j}} denote an orthonormal basis for H_{Σ}^{0,1}. Given q ∈ ℙ^{n} and an orthonormal basis {X_{i}} for T_{q}ℙ^{n}, put

\overline{ν}_{q} = ∑_{i=1,j=1}^{i=n,j=g} ( ∫_{z∈Σ} ⟨ν(z, q), X_{i}ψ_{j}⟩_{z})X_{i}ψ_{j} ≡ π_{H_{Σ}^{0,1}}ν(·, q) ∈ H_{Σ}^{0,1} ⊗ T_{q}ℙ^{n}.

Note that \overline{ν} is well-defined.

Lemma 4.1. There exist δ, C ∈ C^{∞}(M_{T}^{(0)}; ℝ^{+}) such that for all υ ∈ F^{(∅)}T_{δ} and t ∈ (0, δ(b_{υ})),

∥π_{υ,−}^{0,1}ν_{υ,t} − R̃_{υ}\overline{ν}_{ev(b_{υ})}∥_{υ,2} ≤ C(b_{υ})(t + |υ|^{\frac{1}{p}} ).

Suppose υ = ((Σ, [N], I; x, (j, y), u), (v_{h})_{h∈Î}) ∈ F^{(∅)}T is such that q_{υ} is defined. For any h ∈ Î, let h̃(T ) = min{i ∈ Î : i ≤ h}. By the basic gluing construction of Subsection 2.2 in [13],

ṽ_{h} = dϕ_{b_{υ},h̃(T )}|_{x̃_{h}(υ)}(dq_{υ,ι_{h}}^{−1}|_{x̃_{h}(υ)}dϕ_{b_{υ},h}^{−1}|_{0}v_{h}) = ∏_{i∈Î,i≤h} v_{i} ∈ T_{x_{h̃(T )}}Σ,

where ϕ_{b_{υ},h} is a holomorphic identification of neighborhoods of x_{h} in Σ_{b_{υ},ι_{h}} and in F_{b_{υ},h}^{(0)} ≡ T_{x_{h}}Σ_{b_{υ},h}. If Σ_{b_{υ},h} = S^{2}, we also identify T_{x_{h}}Σ_{b_{υ},h} with ℂ with the map q_{N}.

Lemma 4.2. For all υ ∈ F^{(∅)}T such that q_{υ} is defined, \overline{∂}u_{υ} vanishes outside of the annuli A_{υ,h}^{−} with χ_{T} h = 1 and A_{υ,h}^{±} with χ_{T} h = 2. Furthermore, there exists δ ∈ C^{∞}(M_{T}; ℝ^{+}) such that for all υ ∈ F^{(∅)}T_{δ} and h ∈ Î with χ_{T} h = 1, on Ã_{υ,h}^{−}− ≡ \{z ∈ F_{h,b_{υ}}^{(0)} : \frac{1}{2}|v_{h}|^{\frac{1}{2}} ≤ |z|_{b_{υ}} ≤ |v_{h}|^{\frac{1}{2}} \},

Π_{b_{υ},\overline{u}_{υ}(z)}^{−1}\overline{∂}(u_{υ} ∘ q_{υ,ι_{h}}^{−1}) ∘ dϕ_{b_{υ},h}^{−1}|_{z}

= −|v_{h}|^{− \frac{1}{2}} ( ∑_{m≥1} (1 − β_{|v_{h}|}(2|z|))^{(m−1)}D_{T ,h}^{(m)}([b_{υ}, (\frac{v_{h}}{z} )]))\overline{∂}β|_{2|v_{h}|^{− \frac{1}{2}} z},

where \overline{u}_{υ}(z) ∈ T_{ev(b_{υ})}ℙ^{n} is given by

exp_{b_{υ},ev(b_{υ})} \overline{u}_{υ}(z) = u_{υ}(q_{υ,ι_{h}}^{−1}ϕ_{b_{υ},h}^{−1}(z))
= u_{h}(q_{h,(x_{h},v_{h})}ϕ_{b_{υ},h}^{−1}(z)), |\overline{u}_{υ}(z)|_{b_{υ}} < r_{ℙ^{n}}.

This sum converges uniformly on Ã_{υ,h}^{−}.

Remark. By construction, q_{υ} = q_{υ,(x_{h},v_{h})} ∘ q_{υ,ι_{h}} on A_{υ,h}^{−}, and on q_{υ,ι_{h}}(A_{υ,h}^{−})

q_{υ,(x_{h},v_{h})}(z) = (h, q_{S}p_{h,(x_{h},v_{h})}(z)), where
p_{h,(x_{h},v_{h})}(z) = (1 − β_{|v_{h}|}(2|ϕ_{b_{υ},h}z|))\overline{(\frac{v_{h}}{ϕ_{b_{υ},h}z} )}.

Proof. The first claim follows from (G3); see Subsection 2.1. If y ∈ Σ_{b_{υ},h} and |q_{S}^{−1}(y)| ≤ 2δ_{T} (b_{υ}), define \overline{u}_{h}(y) ∈ T_{ev(b_{υ})}ℙ^{n} by

exp_{b_{υ},ev(b_{υ})} \overline{u}_{h}(y) = u_{h}(y), |\overline{u}_{h}(y)|_{b_{υ}} < r_{ℙ^{n}}.

By construction, u_{υ} ∘ q_{υ,ι_{h}}^{−1} = u_{h} ∘ q_{υ} ∘ q_{υ,ι_{h}}^{−1} on q_{υ,ι_{h}}(A_{υ,h}^{−}). Since Π_{b_{υ},\overline{u}_{υ}}^{−1} ∘ du_{h} is ℂ-linear on q_{υ}(A_{υ,h}^{−}), for any z ∈ Ã_{υ,h}^{−}

Π_{b_{υ},\overline{u}_{υ}(·)}^{−1}\overline{∂}(u_{υ} ∘ q_{υ,ι_{h}}^{−1}) ∘ dϕ_{b_{υ},h}^{−1}|_{z} (4.1)

= Π_{b_{υ},\overline{u}_{υ}(·)}^{−1}du_{h} ∘ \overline{∂}(q_{υ} ∘ q_{υ,ι_{h}}^{−1}) ∘ dϕ_{b_{υ},h}^{−1}|_{z}

= −2|v_{h}|^{− \frac{1}{2}} (\frac{v_{h}}{z} ) Π_{b_{υ},\overline{u}_{υ}(·)}^{−1}(du_{h} ∘ dq_{S})|_{p_{h,(x_{h},v_{h})}ϕ_{υ,h}^{−1}(z)} ∘ ∂β|_{2|v_{h}|^{− \frac{1}{2}} z};

see Lemma 2.2 in [13]. Since g_{ℙ^{n},b_{υ}} is flat on u_{υ}(A_{υ,h}^{−}) by our choice of metrics,

Π_{b_{υ},\overline{u}_{υ}}^{−1}(du_{h} ∘ dq_{S}) = d(\overline{u}_{h} ∘ q_{S}) (4.2)

on q_{S}^{−1}q_{υ}(A_{υ,h}^{−}). Since \overline{u}_{h} ∘ q_{S} is antiholomorphic and the metric g_{ℙ^{n},b_{υ}} is flat near ev(b_{υ}),

(4.3) d(\overline{u}_{h} ∘ q_{S})|_{x} (\frac{∂}{∂s}) = d(\overline{u}_{h} ∘ q_{S})|_{x} (\frac{∂}{∂\overline{y}} )

= ∑_{m≥1} \frac{\overline{x}^{m−1}}{(m − 1)!} \frac{d^{m}}{d\overline{y}^{m}} (\overline{u}_{h} ∘ q_{S})|_{(s,t)=0}

= ∑_{m≥1} \frac{\overline{x}^{m−1}}{(m − 1)!} \frac{D^{m−1}}{ds^{m−1}} \frac{d}{ds}(u_{h} ∘ q_{S})|_{(s,t)=0},

for any x ∈ q_{υ}(A_{υ,h}^{−}), where y = s + it ∈ ℂ is the complex coordinate. The second claim follows from Equations (4.1)–(4.3). For the last claim, note that the sum converges uniformly on Ã_{υ,h}^{−} as long as q_{υ}(A_{υ,h}^{−}) is contained in the ball of convergence for the power series expansion for \overline{u}_{h} at 0.

If ψ ∈ H_{Σ}^{0,1}, b ∈ M_{T}^{(0)}, m ≥ 1, and the metric g_{b,ˆ0} is flat near x, we

define D_{b,x}^{(m)}ψ ∈ T_{x}^{0,1}Σ^{⊗m} as follows. If (s, t) are conformal coordinates centered at x such that s^{2} + t^{2} is the square of the g_{b,ˆ0}-distance to x, let

\{D_{b,x}^{(m)}ψ\} (\frac{∂}{∂s}) ≡ \{D_{b,x}^{(m)}ψ\}(\underbrace{\frac{∂}{∂s}, …, \frac{∂}{∂s}}_{m})

= \frac{π}{m!}\{\frac{D^{m−1}}{ds^{m−1}} ψ_{j}|_{(s,t)=0}\} (\frac{∂}{∂s}) ,

where the covariant derivatives are taken with respect to the metric g_{b,ˆ0}. Since ψ_{j} ∈ H_{Σ}^{0,1}, ψ_{j} = f(ds − idt) for some anti-holomorphic function f. Since g_{b,ˆ0} is flat near x, it follows that D_{b,x}^{(m)}ψ ∈ T_{x}^{0,1}Σ^{⊗m}. If {ψ_{j}} is an

orthonormal basis for H_{Σ}^{0,1}, let s_{b,x}^{(m)} ∈ T_{x}^{∗}Σ^{⊗m} ⊗ H_{Σ}^{0,1} be given by

s_{b,x}^{(m)}(v) ≡ s_{b,x}^{(m)}(\underbrace{v, …, v}_{m}) = ∑_{j∈[g]} \overline{\{D_{b,x}^{(m)}ψ_{j}\}(v)}ψ_{j}.

The section s_{b,x}^{(m)} is always independent of the choice of a basis for H_{Σ}^{0,1},

but is dependent on the choice of the metric g_{b,ˆ0} if m > 1. However, s_{b,x}^{(1)} depends only on (Σ, j); we denote this section by s_{Σ,x}. By [3, p. 246], s_{Σ,x} does not vanish and thus spans a subbundle of Σ × H_{Σ}^{0,1} ⟶ Σ. We denote this subbundle by H_{Σ}^{+} and its orthogonal complement by H_{Σ}^{−}. A slightly different description of these bundles is given in Subsection 2.3. Let

π^{+}, π^{−} ∈ Γ(Σ; (Σ × H_{Σ}^{0,1})^{∗} ⊗ H_{Σ}^{±})

be the corresponding orthogonal projection maps. Denote by s_{b,x}^{(m,±)} the

composition π_{x}^{±} ∘ s_{b,x}^{(m,±)}.

Lemma 4.3. There exists δ ∈ C^{∞}(M_{T}; ℝ^{+}) such that for all υ ∈ F^{(∅)}T_{δ}, X ∈ T_{ev(b_{υ})}ℙ^{n}, and ψ ∈ H_{Σ}^{0,1},

⟨⟨π_{υ,−}^{0,1}\overline{∂}u_{υ}, R_{υ}Xψ⟩⟩_{υ,2}

= − ∑_{m≥1} ∑_{χ_{T} h=1} ⟨D_{T ,h}^{(m)}b_{υ}, X⟩\overline{(\{D_{b_{υ},x̃_{h}(υ)}^{(m)}ψ\}((dϕ_{b_{υ},x_{h̃(T )}}|_{x̃_{h}(υ)})^{−1}ṽ_{h}))}.

Furthermore, the sum is absolutely convergent.

Proof. Since ⟨\overline{∂}u_{υ}, R_{υ}Xψ⟩ = 0 outside of the annuli A_{υ,h}^{−} with χ_{T} h = 1,

⟨⟨π_{υ,−}^{0,1}\overline{∂}u_{υ}, R_{υ}Xψ⟩⟩ = ⟨⟨\overline{∂}u_{υ}, R_{υ}Xψ⟩⟩ = ∑_{χ_{T} h=1} ∫_{A_{υ,h}^{−}} ⟨\overline{∂}u_{υ}, R_{υ}Xψ⟩. (4.4)

Since q_{υ,ι_{h}}^{−1} ∘ ϕ_{b_{υ},h}^{−1} is holomorphic on Ã_{υ,h}^{−}, Π_{b_{υ},\overline{u}_{υ}}^{−1} is unitary on u_{υ}(A_{υ,h}^{−}), and the inner-product of one-forms is conformally invariant,

∫_{A_{υ,h}^{−}} ⟨\overline{∂}u_{υ}, R_{υ}Xψ⟩ (4.5)

= ∫Ã_{υ,h}^{−} ⟨\overline{∂}(u_{υ} ∘ q_{υ,ι_{h}}^{−1}) ∘ dϕ_{b_{υ},h}^{−1}, R_{υ}Xψ ∘ dq_{υ,ι_{h}}^{−1} ∘ dϕ_{b_{υ},h}^{−1}⟩
= ∫Ã_{υ,h}^{−} ⟨Π_{b_{υ},\overline{u}_{υ}}^{−1} \overline{∂}(u_{υ} ∘ q_{υ,ι_{h}}^{−1}) ∘ dϕ_{b_{υ},h}^{−1}, Xψ ∘ dq_{υ,i_{h}}^{−1} ∘ ϕ_{b_{υ},h}^{−1}⟩ ,

since Π_{b_{υ},\overline{u}_{υ}}^{−1}R_{υ}Xψ = Xψ on A_{υ,h}^{−}. If ι_{h} = ˆ0, we identify F_{h,b_{υ}}^{(0)} = T_{x_{h}}Σ with ℂ in a g_{b_{υ},ˆ0}-unitary way. In all cases, we can then write

ψ ∘ dq_{υ,ι_{h}}^{−1} ∘ dϕ_{b_{υ},h}^{−1} = fd\overline{z}.

enumeration of genus-two curves 387

Since ψ is harmonic and q_{υ,ι_{h}}^{−1} ∘ ϕ_{b_{υ},h}^{−1} is holomorphic on Ã_{υ,h}, f is antiholomorphic. Using the change of variables 2|v_{h}|^{− \frac{1}{2}} z = re^{iθ}, we obtain

∫Ã_{υ,h} ⟨|v_{h}|^{− \frac{1}{2}} (1 − β_{|v_{h}|}(2|z|))^{m−1}D_{T ,h}^{(m)}([b_{υ}, \frac{v_{h}}{z} ])\overline{∂}β|_{2|v_{h}|^{− \frac{1}{2}} z}, (4.6)

Xψ ∘ dq_{υ,ι_{h}}^{−1} ∘ dϕ_{b_{υ},h}^{−1}⟩ = ⟨D_{T ,h}^{(m)}b_{υ}, X⟩v_{h}^{m}

· ∫Ã_{υ,h} \{(1 − β(2|v_{h}|^{− \frac{1}{2}} |z|))^{m−1}β′|_{2|v_{h}|^{− \frac{1}{2}} |z|}\}|v_{h}|^{− \frac{1}{2}} z^{−m} \frac{z}{|z|}\overline{f}

= ⟨D_{T ,h}^{(m)}b_{υ}, X⟩v_{h}^{m}|v_{h}|^{− \frac{m−1}{2}} 2^{m−2}

· \frac{1}{m} ∫_{1}^{2} ∫_{0}^{2π} \{(1 − β(r))^{m}\}′(re^{iθ})^{−(m−1)}\overline{f}(\frac{1}{2}|v_{h}|^{\frac{1}{2}} re^{iθ})dθdr.

Since \overline{f} is holomorphic, for any r > 0,

(4.7)  ∫_{0}^{2π}(re^{iθ})^{−(m−1)}\overline{f} (\frac{1}{2}|v_{h}|^{\frac{1}{2}} re^{iθ}) dθ

= −i ∫_{|z|=r} z^{−m}\overline{f} (\frac{1}{2}|v_{h}|^{\frac{1}{2}} z) dz

= \frac{2π}{(m − 1)!} \frac{d^{(m−1)}}{dz^{(m−1)}} \overline{f} (\frac{1}{2}|v_{h}|^{\frac{1}{2}} z) |_{z=0}

= \frac{2π}{(m − 1)!}2^{−(m−1)}|v_{h}|^{\frac{m−1}{2}} \overline{f}^{⟨m−1⟩}(0).

Since the metric g_{b,ˆ0} is flat near x̃_{h},

(4.8)  \frac{π}{m!}v_{h}^{m}\overline{f}^{⟨m−1⟩}(0) = \overline{\{D_{b_{υ},x̃_{h}(υ)}^{(m)}ψ\}(dq_{υ,ι_{h}}^{−1}|_{x_{h}}dϕ_{b_{υ},h}^{−1}|_{0}v_{h})}
= \overline{\{D_{b_{υ},x̃_{h}(υ)}^{(m)}ψ\}((dϕ_{b_{υ},x_{h̃(T )}}|_{x̃_{h}(υ)})^{−1}ṽ_{h})}.

The claim follows from Equations (4.4)–(4.8) and Lemma 4.2.

Proposition 4.4. If T = (Σ, [N], I; j, \underline{d}) is a simple bubble type with dˆ0 = 0, there exists δ ∈ C^{∞}(M_{T}; ℝ^{+}) such that

π_{υ,−}^{0,1}\overline{∂}u_{υ} = −R̃_{υ} ∑_{m≥1} ∑_{χ_{T} h=1} (D_{T ,h}^{(m)}b)(s_{b,x̃_{h}(υ)}^{(m)}(dϕ_{b,x_{h̃(T )}}|_{x̃_{h}(υ)}^{−1}ṽ_{h}))

∀ υ = [b, (v_{h})_{h∈Î}] ∈ F^{∅}T_{δ}.

Furthermore, the sum is absolutely convergent.

Proof. This proposition follows from Lemma 4.3 and Equation (2.14).

4.2 First-order estimate for ψ_{T ,tν}^{μ}

If T = (Σ, [N], I; j, \underline{d}) is a bubble type as before, we denote by χ(T ) the subset of elements h of I such that χ_{T} h = 1. For any υ ∈ F T and h ∈ χ(T ), let

α_{T ,h}^{(k)}(υ) = (D_{T ,h}^{(k)}b_{υ})s_{b_{υ},x_{h̃(T )}}^{(k)}(ṽ_{h}), α_{T}^{(k)}(υ) = ∑_{h∈χ(T )} α_{T ,h}^{(k)}(υ),

if υ = [(Σ, [N], I; x, (j, y), u), (v_{h})_{h∈Î}].

We denote α_{T ,h}^{(1)} and α_{T}^{(1)} by α_{T ,h} and α_{T}, respectively.

Lemma 4.5. There exist δ, C ∈ C^{∞}(M_{T}; ℝ^{+}) such that for all υ ∈ F^{∅}T_{δ},

∥π_{υ,−}^{0,1}\overline{∂}u_{υ} + R̃_{υ}α_{T} (υ)∥_{2} ≤ C(b_{υ})|υ| ∑_{h∈χ(T )} |υ|_{h}.

Proof. This is immediate from Proposition 4.4, since

∥s_{x̃_{h}(υ)}(dϕ_{b_{υ},x_{h̃(T )}}|_{x̃_{h}(υ)}^{−1}ṽ_{h}) − s_{x_{h̃(T )}}(ṽ_{h})∥_{2} ≤ C(b_{υ})|ϕ_{b_{υ},x_{h̃(T )}} x̃_{h}(υ)|_{b_{υ}} |ṽ_{h}|
≤ C′(b_{υ})|υ||ṽ_{h}|_{b};
∑_{m≥2} |D_{T ,h}^{(m)}b_{υ}||ṽ_{h}|^{m} ≤ C(b_{υ})|ṽ_{h}|^{2},

for all h ∈ Î with χ_{T} h = 1 and υ ∈ F T_{δ} with δ ∈ C^{∞}(M_{T}; ℝ^{+}) sufficiently small.

Lemma 4.6. There exist δ, C ∈ C^{∞}(M_{T} (μ); ℝ^{+}) such that for all υ ∈ F^{∅}T_{δ},

∥ψ_{T ,tν}^{μ}(υ) − (t\overline{ν}_{ev(b_{υ})} + α_{T} (υ))∥_{2} ≤ C(b_{υ})(t + |υ|^{\frac{1}{p}} ) (t + ∑_{h∈χ(T )} |υ|_{h}) ,

where ψ_{T ,tν}^{μ} denotes ψ_{M_{T} ,tν}^{μ}.

Proof. By Lemma 2.2 and Theorem 2.7,

∥π_{υ,−}^{0,1}D_{υ}ξ_{υ,tν}∥_{2} ≤ C(b_{υ}) ( ∑_{h∈χ(T )} |υ|_{h}) ∥D_{υ}ξ_{υ,tν}∥_{υ,p,1}

≤ C′(b_{υ})(t + |υ|^{\frac{1}{p}} ) ∑_{h∈χ(T )} |υ|_{h}.

Combining this estimate with Lemmas 4.1 and 4.5, we obtain

∥ψ_{T ,tν}(υ) − (t\overline{ν}_{ev(b_{υ})} + α_{T} (υ))∥_{2} ≤ C(b_{υ})(t + |υ|^{\frac{1}{p}} ) (t + ∑_{h∈χ(T )} |υ|_{h}) (4.9)

for all υ ∈ F^{∅}T_{δ}, provided δ ∈ C^{∞}(M_{T}; ℝ^{+}) is sufficiently small. On the other hand, if b_{υ} ∈ M_{T} (μ),

∥φ_{T ,tν}^{μ}(υ)∥_{b_{υ}} ≤ C(b_{υ})(t + |υ|^{\frac{1}{p}} ) ⟹ (4.10)

∥(t\overline{ν}_{ev(ϕ_{T}^{μ} φ_{T ,tν}^{μ}(υ))} + α_{T} (Φ_{T}^{μ} φ_{T ,tν}^{μ}(υ)))

− Π_{b_{υ},ϕ_{T}^{μ} φ_{T ,tν}^{μ}(υ)}(t\overline{ν}_{ev(b_{υ})} + α_{T} (υ))∥_{2}

≤ C(b_{υ})(t + |υ|^{\frac{1}{p}} ) (t + ∑_{h∈χ(T )} |υ|_{h}) ,

where φ_{T ,tν}^{μ} = φ_{M_{T} ,tν}^{μ} is the section of Theorem 2.7 for any fixed regularization (Φ_{T} ≡ Id, Φ_{T}^{μ} ) of M_{T} (μ). The claim follows from (4.9) and (4.10).

Our next step is to apply Lemma 3.2 or Corollary 3.6 to the map ψ_{T ,tν}^{μ} whenever possible. In terms of notation of Subsection 3.1, we take

F^{+} = O^{+} = {0}, F^{−} = F T ,

O^{−} = H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, F̃^{−} = ⨁_{h∈χ(T )} ⨂_{i∈Î,i≤h} F_{i}T ;

ϕ_{h}([b, vÎ]) = [b, ⨂_{i∈Î,i≤h} v_{i}] = [b, ṽ_{h}], α^{−}(ϕ(υ)) ≡ α_{T} (υ),

where ϕ_{h} denotes the hth component of ϕ : F^{−} ⟶ F̃^{−}. Note that α^{−} ∈ Γ(M_{T}; F̃^{−∗} ⊗ O^{−}) is well-defined. A priori, α^{−} may not have full rank on every fiber over M_{T} (μ). We will call a subset K ⊂ M_{T} (μ) T -regular if α^{−} has full rank over K. From Theorem 2.7, Lemma 3.2, and Corollary 3.6, we then obtain:

Corollary 4.7. Suppose d is a positive integer, T = (Σ, [N], I; j_{[N]}, \underline{d}) is a simple bubble type, with dˆ0 = 0 and ∑_{i∈I} d_{i} = d, and μ is an N-tuple of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ (Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. If ι_{h} ≠ ˆ0 for some h ∈ Î, for every regular compact subset K of M_{T} (μ), there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), U_{K} ∩ M_{Σ,d,tν}(μ) = ∅. If ι_{h} = ˆ0 for all h ∈ Î, there exists a compact regular subset K_{T} of M_{T} (μ) with the following property. If K is a compact regular subset of M_{T} (μ) containing K_{T}, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for all t ∈ (0, ϵ_{K}), the signed cardinality of U_{K} ∩ M_{Σ,d,tν}(μ) equals to the signed number of zeros of the map

F T |_{M_{T} (μ)} ⟶ H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, υ ⟶ \overline{ν}_{ev(b_{υ})} + α_{T} (υ). (4.11)

Proof. In either case, by Theorem 2.7, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and δ_{K}, ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), there exists a sign-preserving bijection between U_{K} ∩ M_{Σ,d,tν}(μ) and the zeros of ψ_{T ,tν}^{μ} on F^{∅}T_{δ_{K}}|_{U_{K}∩M_{T} (μ)}, provided U_{K} ∩ M_{T} (μ) is precompact in M_{T} (μ). Furthermore, δ_{K} can be required to be arbitrarily small. If K is regular, U_{K} can be chosen so that the closure of U_{K} ∩ M_{T} (μ) in M_{T} (μ) is also regular. Then by Lemma 4.6,

∥ψ_{T ,tν}^{μ}(υ) − (t\overline{ν}_{ev(b_{υ})} + α_{T} (υ))∥_{2}

≤ C_{K}(t + |υ|^{\frac{1}{p}} )(t + |α_{T} (υ)|) ∀υ ∈ F^{∅}T_{δ_{K}} |_{K},

where C_{K} > 0 depends only on K. Thus, the first claim follows from Lemma 3.2. The second follows from Corollary 3.6, provided that for a generic ν the set of zeros of the map in (4.11) is T -regular and finite; see below.

The affine maps of Corollaries 4.7, 4.14, 4.18, and 4.22 extend over the natural compactifications of the spaces M_{T} (μ) and S_{T ;k}(μ) described in Subsection 4.9. Along with counting the zeros of these affine maps in Section 5, we also show that the linear part of each of the affine maps is regular in the sense of Definition 3.9. Thus, by Lemma 3.10 these affine maps have a finite numbers of transverse zeros, which must lie over the subspace of the base where the linear part of the affine map has full rank.

4.3 Consequences of the first-order estimate for ψ_{T ,tν}^{μ}

In this subsection, we show that M_{T} (μ) is T -regular for most bubble types T under consideration, and nearly all of them fall under the first case of Corollary 4.7. We call T effective, if for some generic choice of ν and of the constraints μ_{1}, …, μ_{N}, \overline{⋃_{t<1} M_{Σ,tν,d}(μ)} intersects \overline{M}_{T} (μ). If

K is a compact subset of \overline{M}_{T} (μ), we call K effective if \overline{⋃_{t<1} M_{Σ,tν,d}(μ)}

intersects K.

Lemma 4.8. Let T = (Σ, [N], I; j, \underline{d}) be a simple bubble type. If j_{l} = ˆ0 for some l ∈ [N] and K is a T -regular subset of M_{T} (μ), then K is not effective.

Proof. By Corollary 4.7, it is sufficient to show that the map

\overline{ν} + α_{T} : F T ⟶ H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}

has no zeros for a generic ν. For a generic ν, the zero set of this section is zero-dimensional. However, if j_{l} = ˆ0 for some l ∈ [N], we can move y_{l} ∈ Σ freely, without changing the value of \overline{ν} + α_{T}. Thus, if the zero-set of the section is nonempty, it must be at least one-dimensional, which is not the case for a generic ν.

Lemma 4.9. Let T = (Σ, [N], I; j, \underline{d}) be a bubble type with dˆ0 = 0. If

ng−|Î|−(|Hˆ0T | + |Mˆ0T | + ∑_{i∈Î,d_{i}=0} (|H_{i}T | + |M_{i}T | − 2))≤ n−|χ(T )|,

M_{T} (μ) is T -regular. Furthermore, if the number on the left-hand side above is negative, then M_{T} (μ) is empty.

Proof. (1) The dimension of M(μ) is given by

dim M_{T} (μ) = (d(n+1)+n+N −|Î|)−(d(n+1)−n(g−1)+N) = ng−|Î|.

However, given b = (Σ, [N], I; x, (j, y), u) ∈ M_{T} (μ), we are free to vary x_{h} if ι_{h} = ˆ0 (i.e., x_{h} ∈ Σ) and y_{l} if j_{l} = ˆ0. Similarly, if i ∈ Î, d_{i} = 0, and |H_{i}T | + |M_{i}T | > 2, we can vary |H_{i}T | + |M_{i}T | − 2 marked and singular points on Σ_{b,i}. Thus, the space M_{T} (μ) must have dimension at least

d_{min}(T ) ≡ |Hˆ0T | + |Mˆ0T | + ∑_{i∈Î,d_{i}=0} (|H_{i}T | + |M_{i}T | − 2),

if M_{T} (μ) is nonempty. Therefore, we can assume |χ(T )| ≤ n.

(2) Let h_{1}, …, h_{|χ(T )|} be the elements of χ(T ). The section s_{Σ} ∈ Γ(Σ; T^{∗}Σ ⊗ H_{Σ}^{0,1}) does not vanish; see [3, p. 246]. Thus, the section α^{−} defined above has rank at least k if the section

\overline{D}_{T ;k} ∈ Γ (M_{T} (μ); (⨁_{m≤k} L_{h_{m}}^{∗}T ) ⊗ ev^{∗}T ℙ^{n}) ,

\overline{D}_{T ;k} ([b, c_{\{h_{m}:m≤k\}}]) = ∑_{m≤k} D_{T ,h_{m}} ([b, c_{h_{m}}]) ,

has rank k. We prove inductively that under the assumptions of the lemma this is the case for all k ≤ |χ(T )|. If k = 0, there is nothing to prove. So we can assume that k > 0 and that the statement has been shown to be true for k − 1. The k − 1 statement shows that the image of \overline{D}_{T ;k−1} is a rank k − 1 subbundle of ev^{∗}T ℙ^{n}. Let π_{k−1}^{⊥} denote the orthogonal projection onto the orthogonal complement of this rank (k − 1)-subbundle in ev^{∗}T ℙ^{n} with respect to the standard metric in ℙ^{n}. We need to show that the section

π_{k−1}^{⊥} ∘ D_{T ;k} ∈ Γ(M_{T} (μ); L_{h_{k}}T^{∗} ⊗ π_{k−1}^{⊥}(ev_{T}^{∗} T ℙ^{n}))

does not vanish. By Corollary 6.3, π_{k−1}^{⊥} ∘ D_{T ;k} is transverse to zero for a generic choice of the constraints μ_{1}, …, μ_{N}. Its zero set must have dimension at least d_{min}(T ), if nonempty, since the movements of points described in (1) do not effect π_{k−1}^{⊥} ∘ D_{T ;k}. Thus, π_{k−1}^{⊥} ∘ D_{T ,k} does not vanish if

dim(M_{T} (μ)) − d_{min}(T ) < n − (k − 1).

By the assumption of the lemma, this is the case as long as k ≤ |χ(T )|.

Corollary 4.10. Let T = (Σ, [N], I; j, \underline{d}) be an effective bubble type with dˆ0 = 0. If g = 2 and n = 2, then either:

(1) |Î| = 1 and j_{l} ≠ ˆ0 for all l ∈ [N], or

(2) |Î| = 2, Hˆ0T = Î, and j_{l} ≠ ˆ0 for all l ∈ [N].

Furthermore, in Case (2) α_{T} has full rank over all of M_{T} (μ).

d  \begin{aligned}& dˆ1 \\ dˆ2 \end{aligned}

Figure 1: The two possibilities for T of Corollary 4.10

We illustrate the statement of Corollary 4.10 in Figure 1. We represent each of the potentially effective bubble types T by the domain of any stable map in the space M_{T} (μ). Each disk represents a sphere. We shade the component(s) of the domain on which any (or every) map in M_{T} (μ) is nonconstant. The labels d, dˆ1, and dˆ2 indicate the degree of the map on each of the bubble components; we must have dˆ1+dˆ2 =d. In the case of Figure 1, all marked points must be distributed between the shaded components of the domain.

Due to Corollary 4.10, Corollary 4.7 describes topologically the number of elements of the set M_{Σ,d,tν}(μ) that lie near a compact subset K of M_{Σ,d,0}(μ), provided K is disjoint from the space

S_{T ,1}(μ) ≡ α_{T}^{−1}(0) ⊂ M_{T} (μ),

where T is the bubble type specified by (1) in Corollary 4.10 and by the first diagram in Figure 1. By definition of α_{T}, the set S_{T ,1}(μ) consists of the elements of M_{T} (μ) such that the differential of the bubble map at the attaching node is zero, i.e. the corresponding rational curve in ℙ^{2} has a cusp at the image of Σ. Determining the number of elements of M_{Σ,d,tν}(μ) that lie near S_{T ,1}(μ) requires higher-order estimates. In Subsection 4.4, we determine the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of S_{T ,1}(μ) such that for no element of K the corresponding singular point on Σ is one of the six hyperelliptic points of Σ. Finally, in Subsection 4.5, we determine the number of elements of M_{Σ,d,tν}(μ) that lie near the subset K of S_{T ,1}(μ) such that for every element of K the corresponding singular point on Σ is a hyperelliptic point of Σ.

Proof of Corollary 4.10. (1) By Lemma 4.9, M_{T} (μ) is empty, unless ng − |Î| ≥ 1, i.e., |Î| ≤ 3. Suppose |Î| = 3. If |Hˆ0T | ≥ 2,

ng − |Î| − |Hˆ0T | ≤ 4 − 3 − 2 < 0,

and thus M_{T} (μ) is empty by Lemma 4.9. If |Hˆ0T | = 1,

n − |χ(T )| ≥ 2 − (|Î| − 1) = 0 = ng − |Î| − |Hˆ0T |,

and by Lemma 4.9 the space M_{T} (μ) is T -regular. The space M_{T} (μ) is compact, since by the above M_{T′}(μ) = ∅ if T′ < T . Corollary 4.7 then implies that M_{T} (μ) is not effective, i.e., T is not effective.

(2) Suppose |Î| = 2. If |Hˆ0T | = 2 and j_{l} = ˆ0 for some l ∈ [N],

ng − |Î| − |Hˆ0T | − |Mˆ0T | ≤ 4 − 2 − 2 − 1 < 0,

and thus M_{T} (μ) is empty by Lemma 4.9. If |Hˆ0T | = 1,

n − |χ(T )| = 2 − 1 = ng − |Î| − |Hˆ0T |,

and it follows from Lemma 4.9 and Corollary 4.7, that every compact subset of M_{T} (μ) is not effective. Furthermore, \overline{M}_{T} (μ) − M_{T} (μ) consists of three-bubble strata, all of which are not effective by (1) above. Thus, T is not effective, unless ι_{h} = ˆ0 for all h ∈ Î and j_{l} ≠ ˆ0 for all l ∈ [N]. The second statement about the |Î| = 2 case is immediate from Lemma 4.9.

(3) Finally, suppose |Î| = 1 and j_{l} = ˆ0 for some l ∈ [N]. Then,

n − |χ(T )| = 2 − 1 ≥ ng − |Î| − |Hˆ0T | − |Mˆ0T |,

and thus by Lemmas 4.8 and 4.9, every compact subset of M_{T} (μ) is not effective. Furthermore, \overline{M}_{T} (μ)−M_{T} (μ) consists of two- and threebubble strata that by (1) and (2) are not effective. It follows that T is not effective.

Corollary 4.11. Let T = (Σ, [N], I; j, \underline{d}) be an effective bubble type with dˆ0 = 0. If g = 2 and n = 3, then either:

(1) |Î| = 1,

(2a) |Î| = 2, Hˆ0T = Î, and j_{l} ≠ ˆ0 for all l ∈ [N],

(2b) |Î| = 2, Hˆ0T ≠ Î, and j_{l} ≠ ˆ0 for all l ∈ [N],

(3a) |Î| = 3, Hˆ0T = Î, and j_{l} ≠ ˆ0 for all l ∈ [N], or

(3b) |Î| = 3, ι_{h} = ˆ1 for some ˆ1 ∈ Î and all h ∈ Î − {ˆ1}, dˆ1 = 0, and j_{l} ≠ ˆ0, ˆ1 for all l ∈ [N].

Furthermore, in Case (3a) α_{T} has full rank on all of M_{T} (μ).

\begin{matrix}l_{1} \\ l_{2} \end{matrix}

Figure 2: The five possibilities for T of Corollary 4.11

We illustrate the statement of Corollary 4.11 in Figure 2, using the same conventions as in Figure 1. In the first case, the genus-two Riemann surface Σ may carry some of the marked points. In the remaining four cases, all of the marked points are distributed between the shaded components. In the third diagram, the lightly shaded disk indicates that the restriction of the maps in M_{T} (μ) to the corresponding bubble component may or may not be constant. In the former case, this component must carry at least one marked point.

By the last remark of Corollary 4.11, Corollary 4.7 describes topologically the number of elements of the set M_{Σ,d,tν}(μ) that lie near a compact subset K of M_{T} (μ) for any bubble type T as in (3a) of Corollary 4.11 and in the fourth diagram in Figure 2. If T is a bubble type as in (1) or (2b) of Corollary 4.11 and in the first or third diagram of Figure 2, respectively, Corollary 4.7 describes the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of M_{T} (μ), provided K is disjoint from the space

S_{T ,1}(μ) ≡ α_{T}^{−1}(0) ⊂ M_{T} (μ).

The elements of S_{T ,1}(μ) are characterized geometrically in exactly the same way as in the n = 2 case above. As in the n = 2 case, we give a topological description for the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of S_{T ,1}(μ) in Subsections 4.4 and 4.5.

If T is a bubble type as in (2a) or (3b) of Corollary 4.11 and in the second or last diagram of Figure 2, respectively, Corollary 4.7 describes the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of M_{T} (μ), provided K is disjoint from the space

S_{T ,2}(μ) ≡ α_{T}^{−1}(0) ⊂ M_{T} (μ).

As discussed in the first paragraph of Subsection 4.6, in the first case S_{T ,2}(μ) consists of the stable maps in M_{T} (μ) such that the image of the differentials at the attaching nodes of the two bubble components is the same complex line and the two singular points on Σ are conjugates. The first condition means that the two rational curves form a tacnode at the image of Σ in ℙ^{3}. For T as in (2a) of Corollary 4.11 and in the second diagram of Figure 2, we determine the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of S_{T ,2}(μ) in Subsection 4.6. Finally, if T is as in (3b) of Corollary 4.11 and in the last diagram of Figure 2, S_{T ,2}(μ) consists of the stable maps in M_{T} (μ) such that the image of the differentials at the attaching nodes of the two shaded bubble components is the same complex line. In Subsection 4.7, we determine the number of elements of M_{Σ,d,tν}(μ) that lie near a compact subset K of S_{T ,2}(μ) such that for no element of K the corresponding singular point on Σ is one of the six hyperelliptic points of Σ. In Subsection 4.8, we determine the number of elements of M_{Σ,d,tν}(μ) that lie near the subset K of S_{T ,2}(μ) such that for every element of K the corresponding singular point on Σ is a hyperelliptic point of Σ. We eventually find that only the simplest possible bubble types are effective: that in the first diagram of Figure 2 with no marked points on Σ and those in the second and fourth diagrams in Figure 2; see Subsection 4.9.

Proof of Corollary 4.11. (1) Similarly to the proof of Corollary 4.10, M_{T} (μ) is empty unless |Î| ≤ 5. If |Î| = 5, M_{T} (μ) is compact and |Hˆ0T | = 1. Let ˆ1 ∈ Î be such that ιˆ1 = ˆ0. If dˆ1 > 0,

n − |χ(T )| = 3 − 1 > 0 = ng − |Î| − |Hˆ0T |,

and M_{T} (μ) is not effective by Lemma 4.9 and Corollary 4.7. Suppose dˆ1 = 0. Then |Hˆ1T | ≥ 2; otherwise M_{T} (μ) is empty by Lemma 4.9. It follows that

n − |χ(T )| ≥ 3 − (|Î| − 2) = 0 = ng − |Î| − |Hˆ0T |.

Thus, by Lemma 4.9 and Corollary 4.7, T is not effective.

(2) Suppose |Î| = 4. If |Hˆ0T | ≥ 3, M_{T} (μ) is empty by Lemma 4.9. Let ˆ1 ∈ Î be as above. If |Hˆ0T | = 2,

n − |χ(T )| ≥ 3 − (|Î| − 1) = 0 = ng − |Î| − |Hˆ0T |.

If |Hˆ0T | = 1 and dˆ1 > 0,

n − |χ(T )| = 3 − 1 > 1 = ng − |Î| − |Hˆ0T |.

If |Hˆ0T | = 1, dˆ1 = 0, and |Hˆ1T | = 3,

n − |χ(T )| ≥ 3 − (|Î| − 1) = 0 = ng − |Î| − |Hˆ0T | − (|Hˆ1T | − 2).

Finally, if |Hˆ0T | = 1, dˆ1 = 0, and |Hˆ1T | = 2,

n − |χ(T )| ≥ 3 − (|Î| − 2) = 1 = ng − |Î| − |Hˆ0T |.

Thus, by Corollary 4.7 and Lemma 4.9, in all four cases, no compact subset of M_{T} (μ) is effective. Since \overline{M}_{T} (μ) − M_{T} (μ) consists of fivebubble strata that are not effective by (1) above, it follows that T is not effective.

(3) Suppose |Î| = 3. If Hˆ0T = Î and j_{l} = ˆ0 for some l ∈ [N],

ng − |Î| − |Hˆ0T | − |Mˆ0T | = 6 − 3 − 3 − 1 < 0,

and thus M_{T} (μ) is empty by Lemma 4.9. If |Hˆ0T | = 2,

n − |χ(T )| ≥ 3 − (|Î| − 1) = 1 ≥ ng − |Î| − |Hˆ0T |.

If |Hˆ0T | = 1 and dˆ1 > 0,

n − |χ(T )| = 2 = ng − |Î| − |Hˆ0T |.

If |Hˆ0T | = 1 and |Hˆ1T | = 1,

n − |χ(T )| = 2 = ng − |Î| − |Hˆ0T |.

Thus, in all three cases, by Lemma 4.9 and Corollary 4.7, no compact subset of M_{T} (μ) is effective. Since \overline{M}_{T} (μ) − M_{T} (μ) consists of fourand five-bubble strata that are not effective by (1) and (2) above, T is not effective in these three cases. On the other hand, if |Hˆ0T | = 2, j_{l} = ˆ0 or j_{l} = ˆ1 for some l ∈ [N], and dˆ1 = 0,

n − |χ(T )| ≥ 1 ≥ ng − |Î| − |Hˆ0T | − |Mˆ0T | − (|Hˆ1T | + |Mˆ1T | − 2).

Thus, by Lemmas 4.8 and 4.9, no compact subset of M_{T} (μ) is effective. Similarly to the above, it follows that T is not effective.

(4) Suppose |Î| = 2 and j_{l} = ˆ0 for some l ∈ [N]. If |Hˆ0T | = 2,

n − |χ(T )| ≥ 1 ≥ ng − |Î| − |Hˆ0T | − |Mˆ0T |.

If |Hˆ0T | = 1,

n − |χ(T )| = 2 ≥ ng − |Î| − |Hˆ0T | − |Mˆ0T |.

Thus, in either case, no compact subset of M_{T} (μ) is effective by Lemmas 4.8 and 4.9. Furthermore,

\overline{M}_{T} (μ) − M_{T} (μ) = ⋃_{T′<T} M_{T′}(μ),

where T′ is either a four- or five-bubble strata, or a three bubble-strata T′ = (Σ, [N], I′; j′, \underline{d}′) such that either |Hˆ0T | = 1, or d′ ; ˆ1′ = 0 and j′_{l} = ˆ0 or ˆ1′. By (1)–(3) above, none of such bubble types is effective, and thus T is not effective.

4.4 Second-order estimate for ψ_{T ,tν}^{μ}, Case 1

We now refine the first-order estimate for ψ_{T ,tν}^{μ} along the sets on which the section α^{−} defined above does not have full rank. These are precisely the sets on which the section \overline{D}_{T ,|χ(T |} defined in the proof of Lemma 4.9 does not have full rank.

One set on which \overline{D}_{T ,|χ(T )|} fails to have full rank is the zero set of D_{T ,h_{1}}. If n = 2, 3, by Lemma 4.9, D_{T ,h_{1}} does not vanish unless h_{1} is the only element of the set χ(T ). Thus, we assume that this is the case. We denote the zero-locus of D_{T ,h_{1}} by S_{T ,1} ⊂ M_{T}, which will be abbreviated as S in this subsection. Since D_{T ,h_{1}} is transversal to zero by Corollary 6.3, S is a complex submanifold of M_{T} of codimension n. Its normal bundle N S in M_{T} is the restriction of L_{k_{1}}^{∗}T ⊗ev^{∗}T ℙ^{n} to S_{T ,1}. Let (Φ_{S}, Φ_{S}^{μ}) be a regularization of S_{T ,1}(μ) ≡ S ∩M_{T} (μ). This regularization can be chosen so that

D_{T ,h_{1}} ϕ̃_{S}(b, X) = Π_{b,ϕ̃_{S}(b,X)}X ∀ (b, X) ∈ N S̃ = ev^{∗}T ℙ^{n}, (4.12)

where ϕ̃_{S} is the lift of ϕ_{S} to the preimage S̃ of S and its normal bundle N S̃ in M_{T}^{(0)}; see Subsection 3.8 in [13]. The bundle N S carries a natural norm induced by the g_{ℙ^{n},ev}-metric on ℙ^{n}. Denote by F S and F^{∅}S the bundles described in Subsection 2.4 corresponding to the submanifold S_{T ,1}. Let ˆ1 ∈ Hˆ0T be the unique element such that ˆ1 ≤ h_{1}. If [b; X, υ] ∈ F S = N S ⊕ F T , put

(2)α_{T ;1}(X, υ) = X(b_{υ})s_{Σ,xˆ1} ṽ_{h_{1}} + α_{T ,h_{1}}^{(2)}(υ).

Lemma 4.12. There exist δ, C ∈ C^{∞}(S; ℝ^{+}) such that for all ϖ = [(b; X, υ)] ∈ F^{∅}S_{δ},

∥π_{Φ_{S}(ϖ),−}^{0,1}\overline{∂}u_{Φ_{S}(ϖ)} + R̃_{Φ_{S}(ϖ)}Π_{b,ϕ_{S}(X)}^{(2)}α_{T ;1}(X, υ)∥_{2}
≤ C(b)|υ|(|υ|_{h_{1}}^{2} + |X||υ|_{h_{1}}).

Proof. The proof is almost identical to the proof of Lemma 4.5. The only difference is that we use two terms of the power series of Proposition 4.4. We then make use of the assumption (4.12) on ϕ_{S} and smooth dependence of D_{T ,h_{1}}^{(2)} on X.

Lemma 4.13. There exist δ, C ∈ C^{∞}(S_{T ,1}(μ); ℝ^{+}) such that for all ϖ = [(b; X, υ)] ∈ F^{∅}S_{δ},

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{ev(b)} +^{(2)}α_{T ;1}(X, υ))∥_{2}

≤ C(b)(t + |υ|^{\frac{1}{p}} )(t + |υ|_{h_{1}}^{2} + |X||υ|_{h_{1}}).

Proof. This claim follows from Lemmas 4.1 and 4.12 in a way analogous to the proof of Lemma 4.6. The only difference is that we need to improve the estimate on π_{υ,−}^{0,1}D_{υ}ξ_{υ,tν} made in the proof of Lemma 4.6. Let {ψ_{j}} be an orthonormal basis for H_{Σ}^{0,1}, such that ψ_{1} ∈ H_{Σ}^{+}(x̃_{h_{1}}(υ)), and {X_{i}} an orthonormal basis for T_{ev(ϕ_{S}(X))}ℙ^{n}. By Theorem 2.7, with υ(X) = Φ_{S}(ϖ),

(4.13)  |⟨⟨π_{υ(X),−}^{0,1}D_{υ(X)}ξ_{υ(X),tν}, R_{υ(X)}X_{i}ψ_{j}⟩⟩|
= |⟨⟨ξ_{υ(X),tν}, D_{υ(X)}^{∗}R_{υ(X)}X_{i}ψ_{j}⟩⟩|
≤ C(b)(t + |υ|^{\frac{1}{p}})∥D_{υ(X)}^{∗}R_{υ(X)}X_{j}ψ_{j}∥_{C^{0}}.

Since ξ ∈ Γ̃_{+}(υ), by construction in Subsection 2.3,

⟨⟨ξ_{υ(X),tν}, D_{υ(X)}^{∗}R_{υ(X)}X_{i}ψ_{1}⟩⟩ = 0. (4.14)

On the other hand, since ψ_{2}|_{x̃_{h_{1}}(υ)} = 0 and ∥∇ψ_{2}∥_{g_{ϕ_{S} (X),ˆ0},C^{0}} ≤ C(b), by Equation (2.12)

∥D_{υ(X)}^{∗}R_{υ(X)}X_{i}ψ_{2}∥_{C^{0}(Ã_{υ(X),h_{1}})} ≤ C(b)|υ|_{h_{1}}^{2}, (4.15)

where Ã_{υ(X),h_{1}} is the annulus defined in Lemma 2.2. By Equations (4.13)–(4.15),

|π_{υ(X),−}^{0,1}D_{υ}ξ_{υ(X),tν}| ≤ C(b)(t + |υ|^{\frac{1}{p}})|υ|_{h_{1}}^{2}.

The next step is to apply Lemma 3.2 or Corollary 3.6 whenever possible. Let

F^{+} = ev^{∗}T ℙ^{n} ⊗ ⨂_{i∈Î,i≤h_{1}} F_{i}T , F^{−} = F T ,

F̃^{−} = ( ⨂_{i∈Î,i≤h_{1}} F_{i}T )^{⊗2} , O^{±} = H_{Σ}^{±} ⊗ ev^{∗}T ℙ^{n};

α^{+}([X, υ]) = Xs_{Σ,xˆ1} ṽ_{h_{1}}, ϕ([b, vÎ]) = [b, ṽ_{h_{1}} ⊗ ṽ_{h_{1}}],

α^{−}(ϕ(υ)) ≡ π_{xˆ1(b_{υ})}^{−}α_{T}^{(2)}(υ).

Note that α^{+} ∈ Γ(S; F^{+∗} ⊗ O^{+}), since π^{−} ∘ Xs_{Σ} = 0. Since the map (X, υ) ⟶ (X ⊗ ṽ_{h_{1}}, υ) is injective on F^{∅}T , we can view ψ_{S,tν}^{μ} as a map on an open subset of F^{−} ⊕ F^{+}. Analogously to the first-order case of Subsection 4.2, subset K ⊂ S_{T ,1}(μ) will be called second-order regular if α^{−} has full rank over K.

Corollary 4.14. Suppose d is a positive integer, T = (Σ, [N], I; j, \underline{d}) is a simple bubble type, with dˆ0 = 0 and ∑_{i∈I} d_{i} = d, and μ is an Ntuple of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ(Σ×ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ⊗π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. If |Î| > 1, for every second-order regular compact subset K of S_{T ,1}(μ), there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), U_{K} ∩ M_{Σ,d,tν}(μ) = ∅. If |Î| = 1, there exists a compact regular subset K_{T ,1} of S_{T ,1}(μ) with the following property. If K is a compact subset of S_{T ,1}(μ) containing K_{T ,1}, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for all t ∈ (0, ϵ_{K}), the signed cardinality of U_{K} ∩ M_{Σ,d,tν}(μ) equals to twice the signed number of zeros of the map

T Σ^{⊗2} ⊗ Lˆ1T^{⊗2}|_{S_{T ,1}(μ)} ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.16)

[b, v] ⟶ \overline{ν}_{b}^{−} + α^{(2,−)}([b, v]).

Proof. In either case, by Theorem 2.7, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and δ_{K}, ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), there exists a sign-preserving bijection between U_{K} ∩M_{Σ,d,tν}(μ) and the zeros of ψ_{S,tν}^{μ} on F^{∅}S_{δ_{K}} |_{U_{K}∩S_{T ,1}(μ)}, provided U_{K} ∩S_{T ,1}(μ) is precompact in S_{T ,1}(μ). If K is second-order regular, U_{K} can be chosen so that the closure of U_{K} ∩ S_{T ,1}(μ) in S_{T ,1}(μ) is also second-order regular. Since K is regular and α^{+} is injective on all fibers,

|υ|_{h_{1}}^{2} = |ϕ(υ)| ≤ C_{K} |α^{−}(ϕ(υ))| ⟹

|υ|_{h_{1}}^{2} + |X||υ|_{h_{1}} ≤ C′_{K} |^{(2)}α_{T ;1}(X, υ)| ∀(X, υ) ∈ F^{∅}S_{δ_{K}} |_{K},

where C_{K}, C′_{K} > 0 depend only on K. Thus, by Lemma 4.13,

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{ev(b_{ϖ})} +^{(2)}α_{T ;1}(ϖ))∥_{2}

≤ C_{K}(t + |ϖ|^{\frac{1}{p}} )(t + |^{(2)}α_{T ;1}(ϖ)|) ∀ϖ ∈ F^{∅}S_{δ_{K}} |_{K},

where C_{K} > 0 depends only on K. The first claim now follows from Lemma 3.2. The second follows from Corollary 3.6, provided that for a generic ν the set of zeros of the map in (4.16) is second-order regular and finite; see the last paragraph of Subsection 4.2.

4.5 Third-order estimate for ψ_{T ,tν}^{μ}, Case 1

We continue with the case of Subsection 4.4. Then

α^{−}([b, ṽ_{h_{1}}]) = (D_{T ,h_{1}}^{(2)}b)s_{b,xˆ1}^{(2,−)}(ṽ_{h_{1}}).

By Corollary 6.3, for a generic choice of the constraints μ_{1}, …, μ_{N}, D_{T ,h_{1}}^{(2)} is transversal to zero along S_{T ,1}(μ) if d_{h_{1}} ≥ 2. Since the zero set of D_{T ,h_{1}}^{(2)} must have dimension at least d_{min}(T ) ≥ 1 by the same

argument as in the proof of Lemma 4.9, D_{T ,h_{1}}^{(2)} does not vanish along S_{T ,1}(μ) if d_{h_{1}} ≥ 2. On the other hand, if d_{h_{1}} = 1, S_{T ,1} = ∅, since the differential of any degree-one holomorphic map from S^{2} to ℙ^{n} is nowhere zero. In fact, S_{T ,1}(μ) = ∅ even for d_{h_{1}} = 2, since the image of any degree-two map with a somewhere vanishing differential is a line, and no line intersects μ_{1}, …, μ_{N} if n = 2, 3. Thus, we can assume d_{h_{1}} ≥ 3. It follows that the only way the above homomorphism α^{−} can fail to have full rank on F̃^{−} is if s_{b,xˆ1}^{(2,−)} = 0. While s_{b,xˆ1}^{(2)} depends on the choice of the metric g_{b,ˆ0} on Σ, the section s^{(2,−)} ∈ Γ(Σ; T^{∗}Σ^{⊗2} ⊗ H_{Σ}^{−}) is independent of the metric and is globally defined on Σ. This can be seen by a direct computation. It has transverse zeros at the six branch points of the double cover Σ ⟶ ℙ^{1} induced by s_{Σ}; see [3, p. 246]. Denote by z_{1}, …, z_{6} these six points. Then the set on which α^{−} fails to have full

rank is ⋃_{m∈[6]} S_{T ,1}^{(m)}(μ), where

S_{T ,1}^{(m)} = {b ∈ S_{T ,1} : xˆ1(b) = z_{m}}, S_{T ,1}^{(m)}(μ) = S_{T ,1}^{(m)} ∩ M_{T} (μ).

The sets S_{T ,1}^{(m)} are obviously disjoint.

Since the normal bundle of S_{T ,1}^{(m)} in S_{T ,1} is T_{z_{m}}Σ, the normal bundle

N S of S_{T ,1}^{(m)} in M_{T} (μ) is T_{z_{m}}Σ⊕N S_{1}, where N S_{1} is the normal bundle of S_{T ,1} in M_{T} (μ), as described in the previous subsection. Let (Φ_{S}, Φ_{S}^{μ}) be a regularization of S_{T ,1}^{(m)}(μ) induced by the regularization of S_{T ,1}(μ) described in Subsection 4.4. In particular,

(4.17) D_{T ,h_{1}} ϕ̃_{S}(b, w, X) = Π_{b,ϕ̃_{S}(b,w,X)}X

∀ (b, w, X) ∈ T_{z_{m}}Σ ⊕ N S̃_{1} = T_{z_{m}}Σ ⊕ ev^{∗}T ℙ^{n},

where ϕ̃_{S} is the lift of ϕ_{S}to M_{T}^{(0)}. We can also assume that Φ_{S}^{μ} is given by the g_{ℙ^{n},b}-parallel transport on N_{b}S_{1}. The bundle N S carries a natural norm induced by the g_{ℙ^{n},ev}-metric on ℙ^{n} and g_{·,ˆ0}-metric on Σ. Denote by F S and F^{∅}S the bundles described in Subsection 2.4 corresponding to the submanifold S_{T ,1}^{(m)}. If (b, w, X, υ) ∈ F^{∅}S is sufficiently small, let

x̃ˆ1(w, υ) = x̃ˆ1(ϕ_{S}(w, X, υ)) = x̃ˆ1(ϕ_{S}(w, 0, υ)) ∈ Σ.

We identify a small neighborhood of z_{m} in Σ with a neighborhood of 0 in T_{z_{m}}Σ via the g_{b,ˆ0}-exponential map. Put

α̃(w, X, υ) = (Xb)s_{Σ,x̃ˆ1(w,υ)}(ṽ_{h_{1}})
+ Π_{b,ϕ_{S}(b,X)}^{−1}(D_{T ,h_{1}}^{(2)}ϕ_{S}(b, X))s_{b,x̃ˆ1(w,υ)}^{(2)}(ṽ_{h_{1}})
+ (D_{T ,h_{1}}^{(3)}b)s_{b,z_{m}}^{(3)}(ṽ_{h_{1}}).

If (b, w, X, υ) ∈ F^{∅}S|_{S_{T ,1}^{(m)}(μ)} is sufficiently small, let

α^{μ}(w, X, μ) = (Xb)s_{Σ,x̃ˆ1(w,υ)}(ṽ_{h_{1}}) + (D_{S,tν}^{μ,(2)}(w, X, υ))s_{b,x̃ˆ1(w,υ)}^{(2)}(ṽ_{h_{1}})
+ (D_{T ,h_{1}}^{(3)}b)s_{b,z_{m}}^{(3)}(ṽ_{h_{1}}),

where, with φ_{S,tν}^{μ} as in Theorem 2.7,

D_{S,tν}^{μ,(2)}(w, X, υ) = Π_{ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ),ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}Π_{b,ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}
· (D_{T ,h_{1}}^{(2)}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w, X, υ)) .

Lemma 4.15. There exist δ, C ∈ C^{∞}(S_{T ,1}^{(m)}; ℝ^{+}) such that for all ϖ = [(b, w, X, υ)] ∈ F^{∅}S_{δ},

∥π_{Φ_{S}(ϖ),−}^{0,1}\overline{∂}u_{Φ_{S}(ϖ)} + R̃_{Φ_{S}(ϖ)}Π_{b,ϕ_{S}(X)}α̃(w, X, υ)∥_{2} ≤ C(b)|ϖ||υ|_{h_{1}}^{3}.

Proof. The proof is the same as that of Lemma 4.12, except here we use the first three terms of the expansion of Proposition 4.4. Note that |x̃ˆ1(w, υ)| ≤ C(b)(|w| + |υ|).

Lemma 4.16. There exist δ, C ∈ C^{∞}(S_{T ,1}^{(m)}(μ); ℝ^{+}) such that for all ϖ = (b, w, X, υ) ∈ F^{∅}S_{δ}

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{ev(b)} + α^{μ}(w, X, υ))∥_{2}

≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |υ|_{h_{1}}^{3} + |x̃_{1}(w, υ)||υ|_{h_{1}}^{2}).

Proof. The proof is similar to the proofs of Lemmas 4.6 and 4.13, but we need to obtain an even stronger bound on

∥π_{Φ_{S}(ϖ),−}^{0,1}D_{Φ_{S}(ϖ)}ξ_{Φ_{S}(ϖ),tν}∥_{2}.

Let {ψ_{j}} be an orthonormal basis for H_{Σ}^{0,1} such that ψ_{1} ∈H_{Σ}^{+}(x̃_{h_{1}}(w, υ)), and {X_{i}} an orthonormal basis for T_{ev(ϕ_{S}(X,υ))}ℙ^{n}. Then, as in the proof of Lemma 4.13, with υ(ϖ) = Φ_{S}(ϖ),

⟨⟨D_{Φ_{S}(ϖ)}ξ_{υ(ϖ),tν}, R_{υ(ϖ)}X_{i}ψ_{1}⟩⟩ = 0; (4.18)

|⟨⟨π_{υ(ϖ),−}^{0,1}D_{Φ_{S}(ϖ)}ξ_{υ(ϖ),tν}, R_{υ(ϖ)}X⟩⟩| (4.19)

≤ C(b)(t + |υ|^{\frac{1}{p}} )∥D_{υ(ϖ)}^{∗}R_{υ(ϖ)}X_{i}ψ_{2}∥_{υ(ϖ),1}.

The one-form ψ_{2} vanishes at x̃_{h_{1}}(w, υ) by definition and ∥∇ψ_{2}∥_{g_{b,ˆ0},C^{0}} ≤ C|x̃_{h_{1}}(w, υ)|, since the derivative of the corresponding one-form for z_{m} vanishes. Thus, by Equation (2.12)

∥D_{υ(ϖ)}^{∗}R_{υ(ϖ)}X_{i}ψ_{2}∥_{g_{υ(ϖ)},L^{1}(Ã_{υ(ϖ),h})} (4.20)

≤ C(b)(|x̃_{h_{1}}(w, υ)||υ|_{h_{1}} + |υ|_{h_{1}}^{2})|υ|_{h_{1}},

as needed for our bound. Finally, we use our assumption that Φ_{S}^{μ} is given by the g_{b,ˆ0}-parallel transport on N_{b}S_{1}.

For any (w, X, υ) ∈ F_{b}^{∅}S|_{S_{T ,1}^{(m)}(μ)} sufficiently small, let

Y (w, X, υ) = (Xb)s_{Σ,x̃_{1}(w,υ)}(ṽ_{h_{1}}) + (D_{S,tν}^{μ,(2)}(w, X, υ))s_{b,x̃ˆ1(w,υ)}^{(2,+)}(ṽ_{h_{1}}, ṽ_{h_{1}});

(3)α_{T ;1}^{(m),−}(w, υ)=(D_{T ,h_{1}}^{(2)}b)s_{b,z_{m}}^{(3,−)}(x̃ˆ1(w, υ), ṽ_{h_{1}}, ṽ_{h_{1}})+(D_{T ,h_{1}}^{(3)}b)s_{b,z_{m}}^{(3,−)}(ṽ_{h_{1}});

r_{T ;1}^{+}(υ) = (D_{T ,h_{1}}^{(3)}b)s_{b,z_{m}}^{(3,+)}(ṽ_{h_{1}}), \overline{ν}_{b}^{±} = π_{z_{m}} \overline{ν}_{b}.

Corollary 4.17. There exist δ, C ∈ C^{∞}(S_{T ,1}^{(m)}(μ); ℝ^{+}) such that for all ϖ = [(b, w, X, υ)] ∈ F^{∅}S_{δ}

∥π_{xˆ1(w,υ)}^{+}ψ_{S,tν}^{μ}(w, X, υ) − (tπ_{xˆ1(w,υ)}^{+}\overline{ν}_{b} + Y (w, X, υ) + r_{T ;1}^{+}(υ))∥_{2}

≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |υ|_{h_{1}}^{3} + |x̃_{1}(w, υ)||υ|_{h_{1}}^{2});

∥π_{xˆ1(w,υ)}^{−}ψ_{S,tν}^{μ}(w, X, υ) − (tπ_{z_{m}}^{−}\overline{ν}_{b} +^{(3)}α_{T ;1}^{(m),−}(w, υ))∥_{2}

≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |υ|_{h_{1}}^{3} + |x̃_{1}(w, υ)||υ|_{h_{1}}^{2}).

Proof. The first estimate is clear from Lemma 4.16. For the second, note that since s_{b,z_{m}}^{(2,−)} = 0, |π_{x̃ˆ1(w,υ)}^{−} − π_{z_{m}}^{−}| ≤ C|x̃_{1}(w, υ)|^{2}, and thus

|s_{b,x̃ˆ1(w,υ)}^{(2,−)}(ṽ_{h_{1}}) − s_{b,z_{m}}^{(3,−)}(x̃ˆ1(w, υ), ṽ_{h_{1}}, ṽ_{h_{1}})| ≤ C|x̃ˆ1(w, υ)|^{2}|ṽ_{h_{1}}|^{2} ⟹

|π_{xˆ1(w,υ)}^{−}α^{μ}(w, X, υ) −^{(3)}α_{T ;1}^{−}(w, υ)|

≤ C(b)|(t, w, X, υ)|^{\frac{1}{p}} (|xˆ1(w, υ)||ṽ_{h_{1}}|^{2} + |ṽ_{h_{1}}|^{3}).

Furthermore, |φ_{S,tν}^{μ}(w, X, υ)|_{b} ≤ C(b)(t + |ϖ|^{\frac{1}{p}}).

The next step is to apply Lemma 3.2 and Corollary 3.6. Let

F^{+} = H_{Σ}^{+} ⊗ ev^{∗}T ℙ^{n}, F^{−} = T_{z_{m}}Σ ⊕ F T , O^{±} = H_{Σ}^{±} ⊗ ev^{∗}T ℙ^{n};

F̃^{−} = T_{z_{m}}Σ ⊗ ( ⨂_{i∈Î,i≤h_{1}} F_{i}T )^{⊗2} ⊕ ( ⨂_{i∈Î,i≤h_{1}} F_{i}T )^{⊗3} ;

ϕ([b; w, vÎ]) = [b, x_{ˆ1(w,υ)} ⊗ ṽ_{h_{1}} ⊗ ṽ_{h_{1}}, ṽ_{h_{1}} ⊗ ṽ_{h_{1}} ⊗ ṽ_{h_{1}}];

π^{+}α(w, υ) = r_{T ;1}^{+}(υ), α^{+}(Y) = π_{z_{m}}^{+}Y, α^{−}(ϕ(w, υ)) ≡ ^{(3)}α_{T ,1}^{(m),−}(w, υ).

Note that α^{−} ∈ Γ(S; F̃^{−∗} ⊗ O^{−}) is well-defined. Since the map

(w, X, υ) ⟶ (Y (w, X, υ), w, υ)

is injective on F^{∅}S, we can view ψ_{S,tν}^{μ} as a map on an open subset of F^{−} ⊕ F^{+}.

Corollary 4.18. Suppose d is a positive integer, T =(Σ, [N], I; j, \underline{d}) is a simple bubble type, with dˆ0 = 0 and ∑_{i∈I} d_{i} = d, and μ is an N-tuple

of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ(Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. If |Î| > 1, for every compact subset K of S_{T ,1}^{(m)}(μ), there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), U_{K} ∩ M_{Σ,d,tν}(μ) = ∅. If |Î| = 1, there exists a compact subset K̃_{T ,1}^{(m)}

of S_{T ,1}^{(m)}(μ) with the following property. If K is a compact subset of

S_{T , 1}^{(m)} (μ) containing K̃_{T ,1}^{(m)}, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for all t ∈ (0, ϵ_{K}), the signed cardinality of U_{K} ∩ M_{Σ,d,tν}(μ) equals to three times the signed number of zeros of the map

T_{z_{m}}Σ^{⊗3} ⊗ (Lˆ1T^{⊗2} ⊕ Lˆ1T^{⊗3})|_{S_{T ,1}^{(m)}(μ)} ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.21)

[b, w, vˆ1] ⟶ \overline{ν}_{b}^{−} + (D_{T ,ˆ1}^{(2)}b)s_{b,z_{m}}^{(3,−)}(w) + (D_{T ,ˆ1}^{(3)}b)s_{b,z_{m}}^{(3,−)}(v).

Proof. The proof is similar to the proofs of Corollaries 4.7 and 4.14, but two modifications are needed to be mentioned. First, we need to show that α^{−} always has full rank. Since we are assuming that d_{h_{1}} ≥ 3, the sections D_{T ,h_{1}}^{(1)}, D_{T ,h_{1}}^{(2)}, and D_{T ,h_{1}}^{(3)} over M_{T} have transverse images

in T ℙ^{n}. Thus, the sections of ℙ(ev^{∗}T ℙ^{n}) ⟶ S_{T ,1}^{(m)}(μ) induced by D_{T ,h_{1}}^{(2)}

and D_{T ,h_{1}}^{(3)} are mutually transversal. However, the fiber dimension of

ℙ(ev^{∗}T ℙ^{n}) is n − 1, while the dimension of S_{T ,1}^{(m)}(μ) is n − 2. Thus, the two sections do not intersect and α^{−} has full rank on all fibers over S_{T ,1}^{(m)}(μ). The second difference with the proofs of Corollaries 4.7 and 4.14 is that we replace the section ψ_{S,tν}^{μ} by the map

(w, υ, X) ⟶ π_{z_{m}}^{+}π_{xˆ1(w,υ)}^{+}ψ_{S,tν}^{μ}(w, υ, X) + π_{z_{m}}^{−}π_{xˆ1(w,υ)}^{−}ψ_{S,tν}^{μ}(w, υ, X),

which has exactly the same zeros provided w and υ are sufficiently small (depending only on Σ).

4.6 Second-order estimate for ψ_{T ,tν}^{μ}, Case 2a

We now understand all cases except for (2a) and (3b) of Corollary 4.11. Let {h_{1}, h_{2}} = {ˆ1, ˆ2} in Case (2a) and {ˆ2, ˆ3} in (3b). By dimension count as in the proof of Lemma 4.9, D_{T ,h_{1}} and D_{T ,h_{2}} do not vanish on M_{T} (μ) in these two cases. By Corollary 6.3, π_{b}^{⊥} ∘ D_{T ,h_{2}} is transversal to zero, where π_{b}^{⊥} denotes the projection onto the orthogonal complement E_{1} of the image of D_{T ,h_{1}} in ev^{∗}T ℙ^{n}. Since

α_{T} (υ) = (D_{T ,h_{1}}b_{υ})s_{Σ,x_{h̃_{1}(T )}}(ṽ_{h_{1}}) + (D_{T ,h_{2}}b_{υ})s_{Σ,x_{h̃_{2}(T )}}(ṽ_{h_{2}}),

α_{T} can fail to have the full rank only on the zero set of π_{b}^{⊥} ∘ D_{T ,h_{2}}. Furthermore, s_{Σ,xh̃_{1}} and s_{Σ,xh̃_{2}} must have the same image in H_{Σ}^{0,1}. This

is automatic in Case (3b), since h̃_{1}(T ) = h̃_{2}(T ) = ˆ1, but in Case (2a), this means that xˆ1 and xˆ2 differ by the nontrivial holomorphic automorphism of Σ; see [3, p. 254].

We first treat Case (2a); so we can assume h_{1} = ˆ1, h_{2} = ˆ2. Let S ≡ S_{T ,2} denote the subset of M_{T} on which the section α_{T} has rank one. By Corollary 6.3, this is a complex submanifold of M_{T}. Furthermore, S = S_{0} × S_{1}, where S_{1} is the subspace of U\overline{T} on which the operator \overline{D}_{T ,2}, defined as in the proof of Lemma 4.9, has rank one,

S_{0} = {(xˆ1, −xˆ1) : xˆ1 ∈ Σ^{∗}},

−xˆ1 ∈ Σ denotes the image of xˆ1 under the nontrivial automorphism of Σ, and Σ^{∗} is the subset of Σ which is not fixed by this automorphism, i.e., the complement of the points z_{1}, …, z_{6} described in Subsection 4.5. By Corollary 6.3, S_{1} is a complex submanifold of U\overline{T}. The normal bundle of S in M_{T} is

N S = N S_{0} ⊕ N S_{1}, where N S_{0} = π_{Σ,ˆ2}^{∗}T Σ, N S_{1} = L∗ ; ˆ2T ⊗ E_{1},

and π_{Σ,h} : S_{0} ⊂ Σ × Σ ⟶ Σ is the projection on the hth component. Let (Φ_{S}, Φ_{S}^{μ}) be a regularization of S_{T ,2}(μ) ≡ S ∩ M_{T} (μ). This regularization can be chosen so that

π_{ϕ_{S}(b,X)}^{⊥}D_{T ,ˆ2}ϕ̃_{S}(b, X) = Π_{b,ϕ̃_{S}(b,X)}X ∀(b, X) ∈ N S̃_{1} = E_{1}, (4.22)

where ϕ̃_{S} is the lift of ϕ_{S} to M_{T}^{(0)}. We also assume that Φ_{S}^{μ} is given by the g_{ℙ^{n},b}-parallel transport on N_{b}S_{1}. Since the section s is invariant under the automorphism group of Σ, we identify π_{Σ,ˆ2}^{∗}T Σ|_{S_{0}} with π_{Σ,ˆ1}^{∗}T Σ|_{S_{0}}. If (b; w) ∈ N S_{0} is sufficiently small, let

xˆ2(w) = exp_{b,xˆ2} w.

The bundle N S carries a natural norm induced by the g_{ℙ^{n},ev}-metric on ℙ^{n} and g_{·,ˆ0}-metric on Σ. Denote by F S and F^{∅}S the bundles described in Subsection 2.4 corresponding to the submanifold S_{T ,2}. If (w, X, υ) ∈ F S = N S ⊕ F T , put

α̃(w, X, υ) = Π_{b,ϕ_{S}(b,X)}^{−1}((D_{T ,ˆ1}ϕ_{S}(b, X))s_{Σ,xˆ1}(vˆ1)
+ (D_{T ,ˆ2}ϕ_{S}(b, X))s_{Σ,xˆ2(w)}(vˆ2))
+ ((D_{T ,ˆ1}^{(2)}b)s_{b,xˆ1}^{(2)}(vˆ1) + (D_{T ,ˆ2}^{(2)}b)s_{b,xˆ1}^{(2)}(vˆ2)).

If (w, X, υ) ∈ F^{∅}S|_{S_{T ,2}(μ)} is sufficiently small, let

α^{μ}(w, X, υ) = ((D_{S,tν,ˆ1}^{μ}(w, X, υ))s_{Σ,xˆ1}(vˆ1)
+ (D_{S,tν,ˆ2}^{μ}(w, X, υ))s_{Σ,xˆ2(w)}(vˆ2))
+ ((D_{T ,ˆ1}^{(2)}b)s_{b,xˆ1}^{(2)}(vˆ1) + (D_{T ,ˆ2}^{(2)}b)s_{b,xˆ1}^{(2)}(vˆ2)),

where, with φ_{S,tν}^{μ} as in Theorem 2.7,

D_{S,tν,h}^{μ}(w, X, υ) = Π_{ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ),ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}Π_{b,ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}(Z),
where Z = D_{T ,h}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w, X, υ).

Lemma 4.19. There exist δ, C ∈ C^{∞}(S; ℝ^{+}) such that for all ϖ = [(b, w, X, υ)] ∈ F^{∅}S_{δ},

∥π_{Φ_{S}(ϖ),−}^{0,1}\overline{∂}u_{Φ_{S}(ϖ)} + R̃_{Φ_{S}(ϖ)}P i_{b,ϕ_{S}(X)}α̃(w, X, υ)∥_{2} ≤ C(b)|ϖ||υ|^{2}.

Proof. The proof is analogous to the proof of Lemma 4.15; here we use Proposition 4.4 with two terms for h = ˆ1 and two terms for h = ˆ2.

Lemma 4.20. There exist δ, C ∈ C^{∞}(S_{T ,2}(μ); ℝ^{+}) such that for all ϖ = [(b, w, X, υ)] ∈ F^{∅}S_{δ},

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{ev(b)} + α^{μ}(w, X, υ))∥_{2}
≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |υ|^{2} + |w||vˆ2|).

Proof. As in the proof of Lemmas 4.13 and 4.16, we need to obtain an appropriate estimate on

∥D_{Φ_{S}(ϖ)}^{∗}R_{Φ_{S}(ϖ)}X_{i}ψ_{2}∥_{L^{1}},

where ψ_{2} is a (0, 1)-form vanishing at xˆ1 and with norm 1. From Equation (2.11), we see that the L^{1}-norm over the small annulus centered at xˆ1 is bounded by C(b)|vˆ1|^{2}; see also the proof of Lemma 4.13. Furthermore, since xˆ2 is “dual” to xˆ1, ψ_{2} also vanishes at xˆ2. Thus, the L^{1}-norm over the small annulus centered at xˆ2(w) is bounded by C(b)(|w| + |vˆ2|)|vˆ2| as can be seen from Equation (2.11).

Let s_{b,x}^{(2,+)} ∈ T_{x}^{∗}Σ be given by s_{b,x}^{(2,+)}(v, v) = s_{b,x}^{(2,+)}(v)s_{Σ,x}(v). For any b ∈ S_{T ,2}(μ), define

κ(b) ∈ L∗ ; ˆ2T ⊗ Lˆ1T − {0} and μ(b) ∈ L∗ ; ˆ2T ⊗ Lˆ1T by

(D_{T ,ˆ2}b) = κ(b)(D_{T ,ˆ1}b), π_{b}(D_{T ,ˆ1}^{(2)}b) = μ(b)(D_{T ,ˆ1}^{(1)}b),

where π_{b} : ev^{∗}T ℙ^{n} ⟶ Im(D_{T ,ˆ1}) is the orthogonal projection map. If (w, X, υ) ∈ F^{∅}S|_{S_{T ,2}(μ)} is sufficiently small, let κ̃(w, X, υ) ∈ ℂ^{∗} be given by

π_{ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}(D_{T ,ˆ2}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w, X, υ))

= κ̃(w, X, υ)(D_{T ,ˆ1}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w, X, υ)).

enumeration of genus-two curves 409

Note that by Theorem 2.7, |κ̃(w, X, υ) − κ(b)| ≤ C(b)(t + |ϖ|^{\frac{1}{p}}). Let

Y^{t}(w, X, υ) = (D_{S,tν,ˆ1}^{μ}(w, X, υ))s_{Σ,xˆ1}(vˆ1 + κ̃(w, X, υ)vˆ2
+ μ(b)s_{Σ,xˆ1}^{(2,+)}(vˆ1)vˆ1),
Y^{⊥}(X, vˆ2) = Xs_{Σ,xˆ1}(vˆ2);
(2)α_{T ;2}^{−}(w, vˆ2) = (D_{T ,ˆ1}b)s_{b,xˆ1}^{(2,−)}(w, vˆ2) + (D_{T ,ˆ1}^{(2)}b)s_{b,xˆ1}^{(2,−)}(κ(b)vˆ2)
+ (D_{T ,ˆ2}^{(2)}b)s_{b,xˆ1}^{(2,−)}(vˆ2);
r_{T ;2}^{+}(w, υ) = (D_{T ,ˆ1}^{(1)}(b))s_{b,xˆ1}^{(2,+)}(w, vˆ2) + π_{b}^{⊥}(D_{T ,ˆ1}^{(2)}(b))s_{b,xˆ1}^{(2,+)}(κ(b)vˆ2)
+ (D_{T ,ˆ2}^{(2)}(b))s_{b,xˆ1}^{(2,+)}(vˆ2).

Let Y = Y^{t} + Y^{⊥} and \overline{ν}_{b}^{±} = π_{xˆ1}^{±}\overline{ν}_{b}.

Corollary 4.21. There exist δ, C ∈ C^{∞}(S_{T ,2}(μ); ℝ^{+}) such that for all ϖ = [(b, w, X, υ)] ∈ F^{∅}S_{δ},

∥π_{xˆ1}^{+}ψ_{S,tν}^{μ}(ϖ)−(t\overline{ν}_{b}^{+} + Y (w, X, υ) + r_{T ;2}^{+}(w, υ))∥_{2}
≤ C(b)(t + |ϖ|^{\frac{1}{p}})(|υ|^{2} + |w||vˆ2| + |Y |);
∥π_{xˆ1}^{−}ψ_{S,tν}^{μ}(ϖ)−(t\overline{ν}_{b}^{−} +^{(2)}α_{T ;2}^{−}(w, vˆ2))∥_{2}
≤ C(b)(t + |ϖ|^{\frac{1}{p}})(|υ|^{2} + |w||vˆ2| + |Y |).

Proof. The proof is similar to that of Corollary 4.17, but we use

|s_{Σ,xˆ2(w)}(vˆ2) − (s_{Σ,xˆ1}(vˆ2) + s_{b,xˆ1}^{(2)}(w, vˆ2))| ≤ C(b)|w|^{2}|vˆ2|. We also use |D_{S,tν,ˆ1}^{μ}(w, X, υ)| ≥ C(b)^{−1}.

The next step is to apply Corollary 3.6. Let

F^{+} = H_{Σ}^{+} ⊗ ev^{∗}T ℙ^{n}, F^{−} = π_{Σ,ˆ1}^{∗}T Σ ⊕ Fˆ2T ,

O^{±} = H_{Σ}^{±} ⊗ ev^{∗}T ℙ^{n}, F̃^{−} = π_{Σ,ˆ1}^{∗}T Σ ⊗ Fˆ2T ⊕ Fˆ2T^{⊗2};

ϕ([b; w, vˆ2]) = [b, w ⊗ vˆ2, vˆ2 ⊗ vˆ2], α^{−}(ϕ(w, vˆ2)) ≡ ^{(2)}α_{T ,2}^{−}(w, vˆ2),

π^{+}r(w, υ) = r_{T ;1}^{+}(w, υ).

Note that α^{−} ∈ Γ(S; F̃^{−∗} ⊗ O^{−}) is well-defined. Since the map (w, X, υ) ⟶ (Y (w, X, υ), w, vˆ2) is injective on F^{∅}S as long as δ ∈ C^{∞}(S_{T ,2}(μ); ℝ^{+}) is sufficiently small, we can view ψ_{S,tν}^{μ} as a map on an open subset of F^{−} ⊕ F^{+}.

Corollary 4.22. Suppose d is a positive integer, T = (Σ, [N], I; j, \underline{d}) is a simple bubble type, with Î = {ˆ1, ˆ2}, Mˆ0T = ∅, dˆ0 = 0, and ∑_{i∈I} d_{i} = d, and μ is an N-tuple of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ(Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. Then there exists a compact subset K̃_{T ,2} of S_{T ,2}(μ) with the following property. If K is a compact subset of S_{T ,2}(μ) containing K̃_{T ,1}, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), the signed cardinality of U_{K} ∩ M_{Σ,d,tν}(μ) equals to twice the signed number of zeros of the map

π_{Σ}^{∗}T Σ^{⊗2} ⊗ (Lˆ2\overline{T} ⊕ Lˆ2\overline{T}^{⊗2})|_{Σ^{∗}×S_{\overline{T} ;2}(μ)} ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.23)

[(x, b); (w, v)] ⟶ \overline{ν}_{b}^{−} + (D_{\overline{T} ,ˆ2}b)s_{x}^{(2,−)}(w, v) + (D_{\overline{T} ,ˆ1}^{(2)}b)s_{x}^{(2,−)}(κ(b)v)

+(D_{\overline{T} ,ˆ2}^{(2)}b)s_{x}^{(2,−)}(v).

Proof. The proof is similar to that of Corollary 4.14. We only need to see that the section α^{−} defined above has rank two. If dˆ1 = dˆ2 = 1, the space S_{T ,2}(μ) = ∅, since any two tangent lines in ℙ^{n} agree, and no line passes through all of the constraints μ_{1}, …, μ_{N} if n = 3. Thus, it can be assumed that dˆ1 ≥ 2. Note that S_{T ,2}(μ) is one-dimensional, with the only dimension coming from the singular point xˆ1 ∈ Σ. Thus, by Corollary 6.3, if the constraints μ_{1}, …, μ_{N} are in general position, the image of D_{T ;ˆ1}^{(2)} does not lie in the linear span of D_{T ,ˆ2}b and D_{T ,ˆ2}^{(2)}b. Furthermore, D_{T ,ˆ2}b ≠ 0.

4.7 Second-order estimate for ψ_{T ,tν}^{μ}, Case 2b

We now treat Case (3b) of Corollary 4.11; we can assume h_{1} = ˆ2, h_{2} = ˆ3. Let S ≡ S_{T ,2} denote the subset of M_{T} on which the operator \overline{D}_{T ,2} of Lemma 4.9 has rank one. Similarly to the case of Subsection 4.6, S is a regular submanifold of M_{T} with normal bundle N S = L∗ ; ˆ3T ⊗E_{1}. As before, we can choose a regularization (Φ_{S}, Φ_{S}^{μ}) of S_{T ,2}(μ) ≡ S ∩ M_{T} (μ) such that

π_{ϕ_{S}(b,X)}^{⊥}D_{T ,ˆ3}ϕ̃_{S}(b, X) = Π_{b,ϕ̃_{S}(b,X)}X ∀(b, X) ∈ N S̃_{1} = E_{1}, (4.24) where ϕ̃_{S} is the lift of ϕ_{S} to M_{T}^{(0)}, and Φ_{S}^{μ} is given by the g_{ℙ^{n},b}-parallel transport on N_{b}S. Denote by F S and F^{∅}S the bundles described in Subsection 2.4 corresponding to the submanifold S_{T ,2}. If (X, υ) is a sufficiently small element of F S = N S ⊕ F T , let

α̃(X, υ) = (D_{T ,ˆ2}ϕ_{S}(b, X))s_{Σ,x̃ˆ2(υ)}(ṽˆ2) + (D_{T ,ˆ3}ϕ_{S}(b, X))s_{Σ,x̃ˆ3(υ)}(ṽˆ3);

α^{μ}(X, υ) = (D_{S,tν,ˆ2}^{μ}(X, υ))s_{Σ,x̃ˆ2(υ)}(ṽˆ2) + (D_{S,tν,ˆ3}^{μ}(X, υ))s_{Σ,xˆ3(υ)}(vˆ3),

where, with φ_{S,tν}^{μ} as in Theorem 2.7,

D_{S,tν,h}^{μ}(X, υ) = Π_{ϕ_{S}^{μ}φ_{S,tν}^{μ}(X,υ),ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(X,υ)}^{−1}Π_{b,ϕ_{S}^{μ}φ_{S,tν}^{μ}(X,υ)}^{−1}
· (D_{T ,h}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(X, υ)).

Lemma 4.23. There exist δ, C ∈ C^{∞}(S_{T ,2}; ℝ^{+}) such that for all ϖ = [(b, X, υ)] ∈ F^{∅}S_{δ},

∥π_{Φ_{S}(ϖ),−}^{0,1}\overline{∂}u_{Φ_{S}(ϖ)} + R̃_{Φ_{S}(ϖ)}α̃(X, υ)∥_{2} ≤ C(b)(|ṽˆ2|^{2} + |ṽˆ3|^{2}).

Proof. This lemma is immediate from Proposition 4.4 applied with one term for each h = ˆ2, ˆ3.

Lemma 4.24. There exist δ, C ∈ C^{∞}(S_{T ,2}(μ); ℝ^{+}) such that for all ϖ = [(b, X, υ)] ∈ F^{∅}S_{δ},

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{b} + α^{μ}(X, υ))∥_{2} ≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |vˆ1|(|ṽˆ2| + |ṽˆ3|)).

Proof. As usually, we only need to obtain a good bound on

∥D_{Φ_{S}(ϖ)}^{∗}R_{Φ_{S}(ϖ)}X_{i}ψ_{2}∥_{L^{1}},

where the notation is as in the proof of Lemma 4.20. By Equation (2.11), the L^{1}-norm on the small annulus centered at x̃ˆ2(υ) is bounded by |ṽˆ2|^{2}. Since g_{b,ˆ0}-distance between x̃ˆ2(υ) and x̃ˆ3(υ) is bounded by C(b)|vˆ1|, the L_{1}-norm over the annulus centered at x̃ˆ3(υ) is bounded by |vˆ1||ṽˆ3|.

For any b ∈ S_{T ,2}(μ), let κ(b) ∈ Lˆ3T^{∗} ⊗ Lˆ2T be given by D_{T ,ˆ3}b = κ(b)(D_{T ,ˆ2}b). For (X, υ) ∈ F^{∅}S|_{S_{T ,2}(μ)} sufficiently small, we define the nonzero element κ̃(X, υ) of Lˆ3T^{∗} ⊗ Lˆ2T by

π_{ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(X,υ)}(D_{T ,ˆ3}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(X, υ))

= κ̃(X, υ)(D_{T ,ˆ2}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(X, υ)).

Note that by Theorem 2.7, |κ̃(X, υ) − κ(b)| ≤ C(b)(t + |ϖ|^{\frac{1}{p}}). Let

Y^{t}(X, υ) = (D_{S,tν,ˆ2}^{μ}(X, υ))(s_{Σ,xˆ1}(ṽˆ2 + κ̃(X, υ)ṽˆ3)
+ s_{b,xˆ1}^{(2,+)}(vˆ1, xˆ2ṽˆ2 + xˆ3κ̃(b)ṽˆ3));
Y^{⊥}(X, υ) = Xs_{Σ,xˆ1}(ṽˆ3),
(2)α_{T ;2}^{−}(υ) = (D_{T ,ˆ2}b)s_{b,xˆ1}^{(2,−)}(vˆ1, xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3).

Let Y = Y^{t} + Y^{⊥} and \overline{ν}_{b}^{±} = π_{xˆ1}^{±}\overline{ν}_{b}.

Corollary 4.25. There exist δ, C ∈ C^{∞}(S_{T ,2}(μ); ℝ^{+}) such that for all ϖ = [(b, X, υ)] ∈ F^{∅}S_{δ},

∥π_{xˆ1}ψ_{S,tν}^{μ}(X, ϖ) − (t\overline{ν}_{b}^{+} + Y (X, υ))∥_{2}

≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |vˆ1|(|ṽˆ2| + |ṽˆ3|) + |Y^{⊥}(X, υ)|);

∥π_{xˆ1}^{−}ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{b}^{−} +^{(2)}α_{T ;2}^{−}(υ))∥_{2}

≤ C(b)(t + |ϖ|^{\frac{1}{p}})(t + |vˆ1|(|ṽˆ2| + |ṽˆ3|)).

Proof. This claim is proved similarly to Corollary 4.21.

The next step is to apply Lemma 3.2. Let

F^{+} = H_{Σ}^{+} ⊗ E_{1}, F^{−} = F T ,
O^{±} = H_{Σ}^{±} ⊗ ev^{∗}T ℙ^{n}, F̃^{−} = π_{Σ}^{∗}T Σ ⊗ Fˆ2T ;

ϕ([b; υ]) = [b, vˆ1 ⊗ (xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3)], α^{−}(ϕ(υ)) ≡ ^{(2)}α_{T ,2}^{−}(υ),

α(X, υ) = Y (X, υ) +^{(2)}α_{T ,2}^{−}(υ).

Note that α^{−} ∈ Γ(S; F̃^{−∗} ⊗ O^{−}) is well-defined. Since the map

(X, υ) ⟶ (Y^{⊥}(X, υ), υ)

is injective on F^{∅}S, we can view ψ_{S,tν}^{μ} as a map on an open subset of F^{+} ⊕ F^{−}.

Corollary 4.26. Suppose d is a positive integer, T = (Σ, [N], I; j, \underline{d}) is a simple bubble type, with Î = {ˆ1, ˆ2, ˆ3}, Hˆ1T = {ˆ2, ˆ3}, dˆ0 = 0, and ∑_{i∈I} d_{i} = d, and μ is an N-tuple of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ(Σ×ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ⊗π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. For every compact subset K of S_{T ,2}(μ), such that xˆ1(b) ∈ Σ^{∗} for all b ∈ K, there exist a neighborhood U_{K} of K in \overline{C}_{(d;[N])}^{∞}(Σ; μ), where and ϵ_{K} > 0 such that for any t ∈ (0, ϵ_{K}), U_{K} ∩ M_{Σ,d,tν}(μ) = ∅.

Proof. The set S_{T ,2}^{∗}(μ) ≡ {b ∈ S_{T ,2}(μ) : xˆ1 ∈ Σ^{∗}} is an open subset of S_{T ,2}(μ) on which the section α^{−} has full rank, since D_{T ,ˆ2} does not vanish on S_{T ,2}(μ). Note that the dimension of S_{T ,2}(μ) is 1, the rank of F̃^{−} is also 1, while the rank O^{−} is 3. Thus, the claim follows from Theorem 2.7, Lemma 3.2, and Corollary 4.25, provided

|vˆ1|(|ṽˆ2| + |ṽˆ3|) ≤ C(b)(|vˆ1||xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3| + |Y^{t}(X, υ)|)

for some C ∈ C^{∞}(S_{T ,2}^{∗}(μ); ℝ^{+}). By definition of Y^{t}(X, υ),

|ṽˆ2 + κ(b)ṽˆ3| ≤ |Y^{t}(X, υ)| + C(b)|xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3|.

Since xˆ2 ≠ xˆ3,

|vˆ1|(|ṽˆ2| + |ṽˆ3|) ≤ C(b)|vˆ1|(|ṽˆ2 + κ(b)ṽˆ3| + |xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3|)
≤ C′(b)|vˆ1|(|xˆ2ṽˆ2 + xˆ3κ(b)ṽˆ3| + |Y^{t}(X, υ)|).

4.8 Third-order estimate for ψ_{T ,tν}^{μ}, Case 2

It remains to consider gluing along the subset S_{T ,2}^{(m)}(μ) of S_{T ,2}(μ) consisting of bubble maps b such that xˆ1(b) = z_{m}, one of the six distinguished points of Σ. Let

S = S_{T ,2}^{(m)} = {b ∈ S_{T ,2} : xˆ1(b) = z_{m}}.

The normal bundle of S_{T ,2}^{(m)} in M_{T} is N S = T_{z_{m}}Σ⊕N S_{1}, where N S_{1} is the normal bundle of S_{T ,2} in M_{T} described in the previous subsection. Let (Φ_{S}, Φ_{S}^{μ}) be a regularization of S_{T ,2}^{(m)}(μ) induced by the regularization of S_{T ,2}(μ) described in Subsection 4.7. In particular,

π_{ϕ_{S}(b,X)}^{⊥}D_{T ,ˆ3}ϕ̃_{S}(b, w, X) = Π_{b,ϕ̃_{S}(b,w,X)}X

∀(b, w, X) ∈ T_{z_{m}}Σ ⊕ N S̃_{1} = T_{z_{m}}Σ ⊕ E_{1},

where ϕ̃_{S} is the lift of ϕ_{S} to M_{T}^{(0)}. We also assume that Φ_{S}^{μ} is given by the g_{ℙ^{n},b}-parallel transport on N_{b}S_{1}. The bundle N S carries a natural norm induced by the g_{ℙ^{n},ev}-metric on ℙ^{n} and g_{·,ˆ0}-metric on Σ. Denote by F S and F^{∅}S the bundles described in Subsection 2.4 corresponding to the submanifold S_{T ,2}^{(m)}. If (b, w, X, υ) ∈ F^{∅}S is sufficiently small, let

x̃_{h}(w, υ) = x̃_{h}(ϕ_{S}(w, X, υ)) = x̃_{h}(ϕ_{S}(w, 0, υ)) ∈ Σ, h = ˆ2, ˆ3.

We identify a small neighborhood of z_{m} in Σ with a neighborhood of 0 in T_{z_{m}}Σ via the g_{b,ˆ0}-exponential map. Put

α̃(w, X, υ) = Π_{b,ϕ_{S}(b,X)}^{−1}((D_{T ,ˆ2}ϕ_{S}(b, X))s_{Σ,x̃ˆ2(w,υ)}(ṽˆ2)
+ (D_{T ,ˆ3}ϕ_{S}(b, X))s_{Σ,x̃ˆ3(w,υ)}(ṽˆ3)

+ (D_{T ,ˆ2}^{(2)}ϕ_{S}(b, X))s_{b,z_{m}}^{(2)}(ṽˆ2) + (D_{T ,ˆ3}^{(2)}ϕ_{S}(b, X))s_{b,z_{m}}^{(2)}(ṽˆ3));

α^{μ}(w, X, υ) = (D_{S,tν,ˆ2}^{μ}(w, X, υ))s_{Σ,x̃ˆ2(w,υ)}(ṽˆ2)
+ (D_{S,tν,ˆ3}^{μ}(w, X, υ))s_{Σ,x̃ˆ3(w,υ)}(vˆ3)
+ (D_{S,tν,ˆ2}^{μ,(2)}b)s_{b,z_{m}}^{(2)}(ṽˆ2) + (D_{S,tν,ˆ3}^{μ,(2)}b)s_{b,z_{m}}^{(2)}(ṽˆ3),

where, with φ_{S,tν}^{μ} as in Theorem 2.7,

D_{S,tν,h}^{μ,(k)}(w, X, υ) = Π_{ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ),ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}Π_{b,ϕ_{S}^{μ}φ_{S,tν}^{μ}(w,X,υ)}^{−1}(Z)
where Z = D_{T ,h}^{(k)}ϕ_{S}Φ_{S}^{μ}φ_{S,tν}^{μ}(w, X, υ).

With κ(b) as in the previous subsection, let

α^{+}(υ) = (D_{T ,ˆ2}b)s_{Σ,z_{m}}(ṽˆ2 + κ(b)ṽˆ3),
α− ; ˆ2(w, υ) = (D_{T ,ˆ2}b)s_{b,z_{m}}^{(3,−)}(x̃ˆ2(w, υ), (xˆ2 − xˆ3)vˆ1, ṽˆ2),
α− ; ˆ3(w, υ) = (D_{T ,ˆ3}b)s_{b,z_{m}}^{(3,−)}(x̃ˆ3(w, υ), (xˆ3 − xˆ2)vˆ1, ṽˆ3).

Lemma 4.27. There exist δ, C ∈ C^{∞}(S_{T ,2}^{(m)}; ℝ^{+}) such that for all ϖ = (b, w, X, υ) ∈ F^{∅}S_{δ},

∥π_{Φ_{S}(ϖ),−}^{0,1}\overline{∂}u_{Φ_{S}(ϖ)} − R̃_{Φ_{S}(ϖ)}α̃(w, X, υ)∥_{2} ≤ C(b)|ϖ|(|ṽˆ2|^{2} + |ṽˆ3|^{2}).

Proof. This lemma follows from Proposition 4.4 applied with firstand second-order terms.

Lemma 4.28. There exist δ, C > 0 such that for all ϖ = (b, w, X, υ) ∈ F^{∅}S_{δ}|_{S_{T ;2}^{(m)}(μ)},

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{b} + α^{μ}(w, X, υ))∥_{2}

≤ C(t + |ϖ|^{\frac{1}{p}})(t + (|vˆ1|^{2} + |vˆ1||w|)(|ṽˆ2| + |ṽˆ3|)).

Proof. Note that the space S_{T ,2}^{(m)}(μ) is zero-dimensional and compact if n = 3. As before, we need to bound

∥D_{Φ_{S}(ϖ)}^{∗}R_{Φ_{S}(ϖ)}X_{i}ψ_{2}∥_{L^{1}},

where the notation is as in the proof of Lemma 4.20. By Equation (2.11), the L^{1}-norm on the annulus centered at x̃ˆ2 = x̃ˆ2(w, υ) is bounded by (|x̃ˆ2||ṽˆ2| + |ṽˆ2|^{2})|ṽˆ2|, while the norm over the other annulus is bounded by (|x̃ˆ2||vˆ1| + |vˆ1|^{2})|ṽˆ3|, since the g_{b,ˆ0}-distance between x̃ˆ2 and x̃ˆ3 is bounded by C|vˆ1|. See the proof of Lemma 4.16 for more detail. The claim follows from x̃ˆ2 = w + xˆ2vˆ1.

Lemma 4.29. There exist δ, C > 0 such that for all ϖ = (b, w, X, υ) ∈ F^{∅}S_{δ}|_{S_{T ;2}^{(m)}(μ)},

∥α^{μ}(w, X, υ) − α^{+}(w, υ)∥_{2}

≤ C(t + |ϖ|^{\frac{1}{p}})(|ṽˆ2| + |ṽˆ3|);

∥π_{x̃ˆ2(w,υ)}^{−}α^{μ}(w, X, υ) − α− ; ˆ3(w, υ)∥

≤ C(t + |ϖ|^{\frac{1}{p}})(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|);

∥π_{x̃ˆ3(w,υ)}^{−}α^{μ}(w, X, υ) − α− ; ˆ2(w, υ)∥

≤ C(t + |ϖ|^{\frac{1}{p}})(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|).

Proof. The first bound is clear from the definition of α^{μ}, since

(D_{T ,ˆ3}b) = κ(b)(D_{T ,ˆ2}b), |φ(w, X, υ)|_{b} ≤ C(t + |ϖ|^{\frac{1}{p}}).

Since s_{b,z_{m}}^{(2,−)} = 0,

|π− ; x̃ˆ2s(2) ; x̃_{h}(ṽ_{h})| ≤ C(|x̃ˆ2| + |vˆ1|)|ṽ_{h}|^{2}. (4.25)

where x̃_{h} = x̃_{h}(w, υ). Since x̃ˆ3 − x̃ˆ2 = (xˆ3 − xˆ2)vˆ1,

|s_{b,x̃ˆ3}(ṽˆ3) − (s_{b,x̃ˆ2}(ṽˆ3) + s_{b,x̃ˆ2}^{(2)}((xˆ3 − xˆ2)vˆ1, ṽˆ3)

+ s_{b,x̃ˆ2}^{(3)}((xˆ3 − xˆ2)vˆ1, (vˆ3 − vˆ2)vˆ1, ṽˆ3))| ≤ C|vˆ1|^{3}|ṽˆ3|.

Since π− ; x̃ˆ2s_{Σ,x̃ˆ2} = 0 and s_{b,z_{m}}^{(2,−)} = 0,

|π− ; x̃ˆ2s_{b,x̃ˆ2}^{(2)}((xˆ3 − xˆ2)vˆ1, ṽˆ3) − s_{b,z_{m}}^{(3,−)}(x̃ˆ2, (xˆ3 − xˆ2)vˆ1, ṽˆ3)| ≤ C|x̃ˆ2|^{2}|vˆ1||ṽˆ3|;

|π− ; x̃ˆ2s_{b,x̃ˆ2}^{(3)}((xˆ3 − xˆ2)vˆ1, (xˆ3 − xˆ2)vˆ1, ṽˆ3)

− s_{b,z_{m}}^{(3,−)}((xˆ3 − xˆ2)vˆ1, (xˆ3 − xˆ2)vˆ1, ṽˆ3)| ≤ C|x̃ˆ2||vˆ1|^{2}|ṽˆ3|.

Putting the last three equations together, we see that

(4.26) |π− ; x̃ˆ2s_{b,x̃ˆ3}(ṽˆ3) − s_{b,z_{m}}^{(3,−)}(x̃ˆ3, (xˆ3 − xˆ2)vˆ1, ṽˆ3)|
≤ C(|x̃ˆ2| + |vˆ1|)(|x̃ˆ2||v_{1}| + |vˆ1|^{2})|ṽˆ3|.

The second bound follows from Equations (4.25) and (4.26). The last estimate is proved similarly.

Corollary 4.30. There exist δ, C > 0 such that for all ϖ = (b, w, X, υ) ∈ F^{∅}S_{δ}|_{S_{T ;2}^{(m)}(μ)},

∥ψ_{S,tν}^{μ}(ϖ) − (t\overline{ν}_{b} + α^{μ}(ϖ))∥_{2} ≤ C(t + |ϖ|^{\frac{1}{p}})(t + |α^{μ}(ϖ)|).

Proof. In light of Lemma 4.28, it is sufficient to show that

(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|) ≤ C|α^{μ}(w, X, υ)| (4.27)

for some C > 0. Since (D_{T ,ˆ2}b)s_{Σ,z_{m}}, (D_{T ,ˆ2}b)s_{b,z_{m}}^{(3,−)} and (D_{T ,ˆ3}b)s_{b,z_{m}}^{(3,−)} are nonzero, by Lemma 4.29

|ṽˆ2 + κ(b)ṽˆ3| ≤ C(|α^{μ}(w, X, υ)| + (t + |ϖ|^{\frac{1}{p}})(|ṽˆ2| + |ṽˆ3|));

|x̃_{h}||vˆ1||ṽ_{h}| ≤ C(|α^{μ}(w, X, υ)| + (t + |ϖ|^{\frac{1}{p}})(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|)).

Since κ(b) ≠ 0, xˆ2 ≠ xˆ3, and x̃_{h} = w + x_{h}vˆ1, we obtain

(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|) ≤ C(|x̃ˆ2| + |x̃ˆ3|)|vˆ1|(|ṽˆ2| + |ṽˆ3|) (4.28)

≤ C′(|x̃ˆ2||vˆ1|(|ṽˆ2| + |ṽˆ2 + κ(b)ṽˆ3|) + |x̃ˆ3||vˆ1|(|ṽˆ3| + |ṽˆ3 + κ(b)ṽˆ3|))

≤ C′′(|α^{μ}(w, X, υ)| + (t + |ϖ|^{\frac{1}{p}})(|vˆ1| + |w|)|vˆ1|(|ṽˆ2| + |ṽˆ3|)).

If δ is sufficiently small, estimate (4.27) follows from (4.28).

The next step is to apply Lemma 3.2. Let

F^{+} = L∗ ; ˆ3T ⊗ E_{1}, F^{−} = T_{z_{m}}Σ ⊕ F T ,

O^{±} = H_{Σ}^{±} ⊗ ev^{∗}T ℙ^{n}, F̃^{−} = π_{Σ}^{∗}T Σ^{⊗3} ⊗ Lˆ2T^{⊗3}; ϕ([b, w, υ]) = [b, (w + xˆ2vˆ1) ⊗ ((xˆ2 − xˆ2)vˆ1) ⊗ ṽˆ2];

α^{−}(ϕ(w, υ)) ≡ α− ; ˆ2(w, υ), α(X, w, υ) = α^{μ}(X, w, υ).

Note that α^{−} ∈ Γ(S; F̃^{−∗} ⊗ O^{−}) is well-defined.

Corollary 4.31. Suppose d is a positive integer, T = (Σ, [N], I; j, \underline{d}) is a simple bubble type, with Î = {ˆ1, ˆ2, ˆ3}, Hˆ1T = {ˆ2, ˆ3}, dˆ0 = 0, and ∑_{i∈I} d_{i} = d, and μ is an N-tuple of constraints in general position such that

codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

Let ν ∈ Γ(Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T ℙ^{n}) be a generic section. There exist a neighborhood U of S_{T ,2}^{(m)}(μ) in \overline{C}_{(d;[N])}^{∞}(Σ; μ), and ϵ > 0 such that for any t ∈ (0, ϵ), U ∩ M_{Σ,d,tν}(μ) = ∅.

Proof. Analogously to the proof of Corollary 4.18, we apply Lemma 3.2 to the map

(w, υ, X) ⟶ π_{z_{m}}^{+}π_{xˆ3(w,υ)}^{+}ψ_{S,tν}^{μ}(w, υ, X) + π_{z_{m}}^{−}π_{xˆ3(w,υ)}^{−}ψ_{S,tν}^{μ}(w, υ, X)

instead of ψ_{S,tν}^{μ}. The claim then follows from Theorem 2.7, Lemma 3.2, and Corollary 4.30.

4.9 Summary of Section 4

We conclude Section 4 by reviewing the main results so far. Throughout this subsection,

T = (Σ, [N], I; j, \underline{d})

is a simple bubble type, with d = ∑ d_{h} and dˆ0 = 0, and μ is an N-tuple of constraints in general position such that codim_{ℂ}μ = d(n + 1) − n(g − 1) + N.

If |Î| > n, by Corollaries 4.10 and 4.11, there exist a neighborhood U_{T} of \overline{M}_{T} (μ) in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{T} > 0 such that for all t ∈ (0, ϵ_{T}), U_{T} ∩ M_{Σ,d,tν}(μ) = ∅. This is also true if Hˆ0T ≠ Î or Mˆ0T ≠ ∅. If n = 2, this statement is just Corollary 4.10. If n = 3, we only need to consider Cases (1), (2b), and (3b) of Corollary 4.11. Case (3b) follows from Corollaries 4.7, 4.26, and 4.31. The claim for Case (2b) is obtained from Corollaries 4.7, 4.14, 4.18 and the same claim for Case (3b). Finally, in Case (1), we use Corollaries 4.7, 4.26, and 4.31, the statement of Corollary 4.11 for |Î| ≥ 2, and the just stated result for Case (2b).

If |Î| ≤ n, Hˆ0T = Î, and Mˆ0T = ∅, i.e., T is a primitive bubble type, by the previous paragraph and Corollaries 4.7, 4.14, 4.18, and 4.22, there exist a neighborhood U_{T} of \overline{M}_{T} (μ) in \overline{C}_{(d;[N])}^{∞}(Σ; μ) and ϵ_{T} > 0 such that for all t ∈ (0, ϵ_{T}), the signed cardinality n_{T} (μ) of U_{T} ∩ M_{Σ,d,tν}(μ) is the sum of the numbers given by these four corollaries applied to T . If |Î| = 1,

n_{1}(μ) ≡ n_{T} (μ) = n_{1}^{(1)}(μ) + 2n_{1}^{(2)}(μ) + 18n_{1}^{(3)}(μ), (4.29)

where the numbers n_{1}^{(k)}(μ) are described as follows. The number n_{1}^{(1)}(μ) is the signed number of zeros of the affine map

ψ_{1}^{(1)} : T Σ ⊗ Lˆ1\overline{T} ⟶ H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, (4.30)

ψ_{1}^{(1)}(x, [b, vˆ1]) = \overline{ν}_{b} + (D_{T ,ˆ1}b)s_{Σ,x}(vˆ1),

where the bundles are considered over Σ×\overline{U}\overline{T} (μ) = \overline{M}_{T} (μ) and ˆ1 is the unique element of Î. Note that this number is the same as the number of zeros of the map in (4.11), since Σ × \overline{U}\overline{T} (μ) − M_{T} (μ) is a finite union of smooth manifolds of dimension less than the dimension of M_{T} (μ). Thus, if ν is generic, ψ_{1}^{(1)} has no zeros over Σ × \overline{U}\overline{T} (μ) − M_{T} (μ). The number n_{1}^{(2)}(μ) is the signed number of zeros of the affine map

ψ_{1}^{(2)} : T Σ^{⊗2} ⊗ Lˆ1\overline{T}^{⊗2} ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.31)

ψ_{2}^{(2)}(x, [b, vˆ1]) = \overline{ν}_{b}^{−} + (D_{T ,ˆ1}^{(2)}b)s_{Σ,x}^{(2,−)}(vˆ1),

where the bundles are considered over Σ×\overline{S}_{1}(μ) and \overline{S}_{1}(μ) is the closure in \overline{U}\overline{T} (μ) of the space

S_{1}(μ) = \{b ∈ U\overline{T} (μ) : D_{T ,ˆ1}|_{b} = 0\}. (4.32)

If n = 2, S_{1}(μ) is a finite set and thus \overline{S}_{1}(μ) = S_{1}(μ). If n = 3, S_{1}(μ) is one-dimensional over ℂ. The boundary \overline{S}_{1}(μ) − S_{1}(μ) is a finite set, as can be seen from the estimate on D_{\overline{T} ,ˆ1} of Theorem 2.8. Thus, in either case, the maps in (4.31) and (4.16) have the same zeros. Finally, the number n_{1}^{(3)}(μ) is the signed number of zeros of the affine map

ψ_{1}^{(3)} : T Σ^{⊗3} ⊗ (Lˆ1\overline{T}^{⊗2} ⊕ Lˆ1\overline{T}^{⊗3}) ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.33)

ψ_{1}^{(3)}(x, [b, vˆ1, wˆ1]) = \overline{ν}_{b}^{−} + (D_{T ,ˆ1}^{(2)}b)s_{b,z_{m}}^{(3,−)}(vˆ1) + (D_{T ,ˆ1}^{(3)})s_{b,z_{m}}^{(3,−)}(wˆ1),

where the bundles are considered over \overline{S}_{1}(μ) and z_{m} is one of the six distinguished points of Σ. By the same argument as above, this number is precisely the number of zeros of the map in (4.21).

If |Î| = 2 and n = 2, n_{T} (μ) = n_{T}^{(1)}(μ) is the signed number of zeros of the affine map

ψ_{T}^{(1)} : T Σˆ1 ⊗ Lˆ1\overline{T} ⊕ T Σˆ2 ⊗ Lˆ2\overline{T} ⟶ H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, (4.34)

ψ_{T}^{(1)}(xˆ1, xˆ2, [b, vˆ1, vˆ2]) = \overline{ν}_{b} + (D_{T ,ˆ1}b)s_{Σ,xˆ1}(vˆ1) + (D_{T ,ˆ2}b)s_{Σ,xˆ2}(vˆ2),

where the bundles are considered over Σ^{2} × \overline{U}\overline{T} (μ) = Σˆ1 × Σˆ2 × \overline{U}\overline{T} (μ) and ˆ1, ˆ2 are the two elements of Î. By the same argument as before, the number n_{T}^{(1)}(μ) is the same as the number of zeros of the map (4.7). If |Î| = 2 and n = 3,

n_{T} (μ) = n_{T}^{(1)}(μ) + 2n_{T}^{(2)}(μ), (4.35)

where n_{T}^{(1)}(μ) is defined the same way as in the n = 2 case, while n_{T}^{(2)}(μ) is the signed number of zeros of the affine map

ψ_{T}^{(2)} : T Σ^{⊗2} ⊗ (Lˆ2\overline{T} ⊕ Lˆ2\overline{T}^{⊗2}) ⟶ H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{n}, (4.36)

ψ_{T}^{(2)}(x, [b, vˆ2, wˆ2]) = \overline{ν}_{b}^{−} + (D_{T ,ˆ2}b)s_{Σ,x}^{(2,−)}(wˆ2) + (D_{T ,ˆ1}^{(2)}b)s_{Σ,x}^{(2,−)}(κ(b)vˆ2)

+(D_{T ,ˆ2}^{(2)}b)s_{Σ,x}^{(2,−)}(vˆ2),

where the bundles are viewed over Σ × S\overline{T} (μ),

S\overline{T} (μ) = \{b ∈ U\overline{T} (μ) : π_{[b]}^{⊥} ∘ D_{T ,ˆ2}|_{b} = 0\}, (4.37)

E_{1} is the quotient of ev^{∗}T ℙ^{n} by Im(D_{T ,ˆ1}), π^{⊥} : ev^{∗}T ℙ^{n} ⟶ E_{1} is the projection map, and κ(b) ∈ L∗ ; ˆ2\overline{T} ⊗ Lˆ1\overline{T} is a nonzero homomorphism. Note that S\overline{T} (μ) is a finite set with our choice of constraints. Finally, if |Î| = 3 and n = 3, n_{T} (μ) = n_{T}^{(1)}(μ) is the signed number of zeros of the affine map

ψ_{T}^{(1)} : T Σˆ1 ⊗ Lˆ1\overline{T} ⊕ T Σˆ2 ⊗ Lˆ2\overline{T} ⊕ T Σˆ3 ⊗ Lˆ3\overline{T} ⟶ H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, (4.38)

ψ_{T}^{(1)}(xˆ1, xˆ2, xˆ3, [b, vˆ1, vˆ2, vˆ3]) = \overline{ν}_{b} + (D_{T ,ˆ1}b)s_{Σ,xˆ1}(vˆ1)

+ (D_{T ,ˆ2}b)s_{Σ,xˆ2}(vˆ2) + (D_{T ,ˆ3}b)s_{Σ,xˆ3}(vˆ3),

where the bundles are considered over Σ^{3}×\overline{U}\overline{T} (μ) = Σˆ1×Σˆ2×Σˆ3×\overline{U}\overline{T} (μ) and ˆ1, ˆ2, ˆ3 are the three elements of Î. As before, the number n_{T}^{(1)}(μ) is precisely the number of zeros of the map (4.7). If m ≥ 2 and k ≥ 1, we denote by n_{m}^{(k)}(μ) the sum of the numbers n_{T}^{(k)}(μ) over all equivalence classes of primitive bubble types T with |Î| = m.

5. Computations

5.1 The numbers n_{m}^{(1)}(μ) with m = n

Our goal now is to compute the numbers n_{T}^{(k)}(μ) for any primitive bubble type T = (Σ, [N], I; j, \underline{d}), and thus the genus-two enumerative invariants for ℙ^{2} and ℙ^{3}. Most of this section is devoted to expressing the numbers n_{T}^{(k)}(μ) in terms of intersection numbers of tautological classes of various spaces of stable rational maps that pass through the constraints μ. These are shown to be computable in [9]. The procedure for counting the zeros of affine maps between vector bundles is described in Section 3. We start with the easiest cases.

Lemma 5.1. If T = (Σ, [N], I; j, \underline{d}) is a primitive bubble type with |Î| = n and μ is an N-tuple of constraints in general position such that

codim_{ℂ}μ = (n + 1) ∑_{i∈I} d_{i} − n + N,

the set \overline{U}\overline{T} (μ) is finite and n_{T}^{(1)}(μ) = 2^{n}|\overline{U}\overline{T} (μ)|.

Proof. The first statement is clear by dimension counting. By Equations (4.34) and (4.38), we need to apply Lemma 3.14 with

\overline{M} = \begin{cases}Σˆ1 × Σˆ2 × \overline{U}\overline{T} (μ), & if n = 2; \\ Σˆ1 × Σˆ2 × Σˆ3 × \overline{U}\overline{T} (μ), & if n = 3; \end{cases}

E = \begin{cases}T Σˆ1 ⊗ Lˆ1\overline{T} ⊕ T Σˆ2 ⊗ Lˆ2\overline{T} , & if n = 2; \\ T Σˆ1 ⊗ Lˆ1\overline{T} ⊕ T Σˆ2 ⊗ Lˆ2\overline{T} ⊕ T Σˆ3 ⊗ Lˆ3\overline{T} , & if n = 3, \end{cases}

O = H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{n}, and α given by (4.34) and (4.38). By Lemma 4.9, α ∈ Γ(\overline{M}; E^{∗}⊗O) has full rank on every fiber of E. Thus by Lemma 3.14,

n_{T}^{(1)}(μ) = ⟨e(O/α(E)), [\overline{M}]⟩ = ⟨c(O)c(E)^{−1}, [\overline{M}]⟩. (5.1)

Since \overline{U}\overline{T} (μ) is a finite set,

E ≈ \begin{cases}T Σˆ1 ⊕ T Σˆ2, & if n = 2; \\ T Σˆ1 ⊕ T Σˆ2 ⊕ T Σˆ3, & if n = 3; \end{cases} O ≈ \overline{M} × ℂ^{2n}.

Let y_{h} = c_{1}(T Σ_{h}). Thus, if n = 2, by (5.1)

n_{T}^{(1)}(μ) = ⟨(1 + (yˆ1 + yˆ2) + yˆ1yˆ2)^{−1}, [\overline{M}]⟩
= ⟨yˆ1yˆ2, [Σˆ1 × Σˆ2]⟩|\overline{U}\overline{T} (μ)| = 4|\overline{U}\overline{T} (μ)|,

since ⟨y_{h}, [Σ_{h}]⟩ = −2. If n = 3, we similarly obtain

n_{T}^{(1)}(μ) = ⟨−yˆ1yˆ2yˆ3, [Σˆ1 × Σˆ2 × Σˆ3]⟩|\overline{U}\overline{T} (μ)| = 8|\overline{U}\overline{T} (μ)|,

as claimed.

Let τ_{n}(μ) denote the sum of the numbers |\overline{U}\overline{T} (μ)| taken over all equivalence classes of primitive bubble types T with |Î| = n. This is the number of n-component connected curves of total degree d passing through the constraints μ_{1}, …, μ_{N} in ℙ^{n} with a choice of a node which belongs to all n components. From Lemma 5.1, we immediately conclude:

Corollary 5.2. If n = 2, n_{2}^{(1)}(μ) = 4τ_{2}(μ). If n = 3, n_{3}^{(1)}(μ) = 8τ_{3}(μ).

5.2 The numbers n_{m}^{(2)}(μ) and n_{m}^{(3)}(μ) with m = n − 1

In this subsection, we describe the numbers n_{T}^{(2)}(μ) and n_{T}^{(3)}(μ) with |Î| = n − 1 topologically. The similarity between these cases is that U\overline{T} (μ) is two-dimensional (over ℂ), while S\overline{T} (μ) is a finite set; see Subsection 4.9 for notation.

The numbers n_{T}^{(2)}(μ) with |Î| = n − 1 = 1 and |Î| = n − 1 = 2 are the signed cardinalities of the zero sets of the affine maps in (4.31) and (4.36), respectively. By Subsections 4.4 and 4.6, the linear part α of the affine map ψ_{T}^{(2)} has full rank in these cases, except over the zero set of s_{Σ}^{(2,−)}. In order to simplify our computations, we replace s_{Σ}^{(2,−)} by another section that has no zeros on Σ, but so that the corresponding affine maps have the same number of zeros as the maps in (4.31) and (4.36). The section

s_{Σ}^{(2,−)} ∈ Γ(Σ; T^{∗}Σ^{⊗2} ⊗ H_{Σ}^{−})

has transverse zeros at the points z_{1}, …, z_{6} ∈ Σ; see Subsection 4.5. Thus, it induces a nonvanishing section

s_{Σ}^{(2,−)} ∈ Γ(Σ; T̃Σ^{∗} ⊗ H_{Σ}^{−}), where T̃Σ = T Σ^{⊗2} ⊗ O(z_{1}) ⊗ ⋯ ⊗ O(z_{6})

and O(z_{m}) denotes the holomorphic line bundle corresponding to the divisor z_{m} on Σ. The bundles T̃Σ and T Σ^{⊗2} can be identified on Σ^{∗}, the complement of the six points, in such a way that s_{Σ}^{(2,−)} = ηs_{Σ}^{(2,−)} on Σ^{∗} for some η ∈ C^{∞}(Σ^{∗}; ℝ^{+}). Let ψ̃_{T}^{(2)} denote the affine maps obtained by replacing T Σ^{⊗2} and s_{Σ}^{(2,−)} by T̃Σ and s_{Σ}^{(2,−)}, respectively, in (4.31) and (4.36) (depending on T ). Since ψ_{T}^{(2)} and ψ̃_{T}^{(2)} have no zeros over {z_{m}} if ν is generic and s_{Σ}^{(2,−)} and s_{Σ}^{(2,−)} differ by a nonzero multiple on Σ^{∗}, there is a sign-preserving bijection between the zeros of ψ_{T}^{(2)} and of ψ̃_{T}^{(2)}. Furthermore, the linear part of ψ̃_{T}^{(2)} has full rank on every fiber.

Denote by S_{2}(μ) the union of the spaces S\overline{T} (μ) defined by Equation (4.37) taken over all equivalence classes of appropriate bubble types T . This set can be identified with the degree-d two-component rational curves in ℙ^{3} that are connected at a tacnode and pass through the constraints μ. Similarly, in the n = 2 case, the set S_{1}(μ) corresponds to the degree-d cuspidal rational curves passing through the constraints.

Lemma 5.3. If n = 2, n_{1}^{(2)}(μ) = 2|S_{1}(μ)| and n_{1}^{(3)}(μ) = |S_{1}(μ)|. If n = 3, n_{2}^{(2)}(μ) = 2|S_{2}(μ)|.

Proof. Let T = (Σ, [N], I; j, \underline{d}) be a bubble type that contributes to one of these numbers. By dimension counting and Corollary 6.3, S\overline{T} (μ) is zero-dimensional and compact. Thus, in all cases the bundles L_{h}\overline{T} and ev^{∗}T ℙ^{n} of Equations (4.31), (4.33) and (4.36) are trivial. If n = 2 and k = 2, we are in the case of (4.31). By the above, we can apply Lemma 3.14 with

E = T̃Σ, O = H_{Σ}^{−} ⊕ H_{Σ}^{−},

and α ∈ Γ(Σ × S\overline{T} (μ); E^{∗} ⊗ O) that has full rank. We obtain

n_{T}^{(2)}(μ) = ⟨c_{1}(O) − c_{1}(E), [Σ × S\overline{T} (μ)]⟩
= (4 + (4 − 6))|S\overline{T} (μ)| = 2|S\overline{T} (μ)|.

If n = 3 and k = 2, we are in the case of (4.36) and apply Lemma 3.14 with

E = T̃Σ ⊕ T̃Σ, O = H_{Σ}^{−} ⊕ H_{Σ}^{−} ⊕ H_{Σ}^{−},

and α ∈ Γ(Σ × S\overline{T} (μ); E^{∗} ⊗ O) that again has full rank. Thus,

n_{T}^{(2)}(μ) = ⟨c_{1}(O) − c_{1}(E), [Σ × S\overline{T} (μ)]⟩ = (6 − 4)|S\overline{T} (μ)| = 2|S\overline{T} (μ)|.

Finally, if n = 2 and k = 3, we are in the case of (4.33). Note that all the bundles involved are trivial and the linear part of ψ_{T}^{(2)} is an isomorphism on every fiber. Thus, n_{T}^{(3)}(μ) = |U\overline{T} (μ)|.

The next step is to compute the cardinalities of the sets S_{n−1}(μ). In order to simplify our answers, it is convenient to introduce cohomology classes c_{1}(L_{k}^{∗}T ) closely related to c_{1}(L_{k}^{∗}T ). Suppose T = (S^{2}, M, I; j, \underline{d}) is a bubble type. and \{T_{k} = (S^{2}, M_{k}, I_{k}; j_{k}, \underline{d}_{k})\} are the corresponding simple types; see [13]. For any k ∈ I − Î and nonempty subset M_{0} of M_{k}T , we define bubble types T (M_{0}) and T /M_{0} as follows. Let

T /M_{0} = (S^{2}, I, M − M_{0}; j|_{M−M_{0}}, \underline{d}).

Let T (M_{0}) ≡ (S^{2}, M, Î⊔_{k} ˆ1; j′, \underline{d}′) be given by

j′_{l} = \begin{cases}k, & if l ∈ M_{0}; \\ ˆ1, & if l ∈ M_{k}T − M_{0}; \\ j_{l}, & otherwise; \end{cases} d′_{i} = \begin{cases}0, & if i = k; \\ d_{k}, & if i = ˆ1; \\ d_{i}, & otherwise. \end{cases}

The tuples T /M_{0} and T (M_{0}) are bubble types as long as d_{k} ≠ 0 or M_{0} ≠ Mˆ0T . Then,

\overline{U}_{T (M_{0})}(μ) = \overline{M}_{0,\{ˆ1\}⊔M_{0}} × \overline{U}_{T /M_{0}} ( ⋂_{l∈M_{0}} μ_{l}; μ) , (5.2)

where \overline{M}_{0,\{ˆ1\}⊔M_{0}} denotes the Deligne-Mumford moduli space of rational curves with ({ˆ0, ˆ1}⊔M_{0})-marked points. If l ∈ M_{k}T for some k ∈ I −Î, we denote T ({l}) by T (l). If T is a basic bubble type, by Theorem 2.8 and decomposition (5.2), \overline{U}_{T (M_{0})}(μ) is an oriented topological suborbifold of \overline{U}_{T} (μ) of (real) codimension two. Thus,

c_{1}(L_{k}^{∗}T ) ≡ c_{1}(L_{k}^{∗}T ) − ∑_{M_{0}⊂M_{k},M_{0}≠∅} P D_{\overline{U}_{T} (μ)}[\overline{U}_{T (M_{0})}(μ)] ∈ H^{2}(\overline{U}_{T} (μ)), (5.3)

where P D_{\overline{U}_{T} (μ)}[\overline{U}_{T (M_{0})}(μ)] denotes the Poincare Dual of [\overline{U}_{T (M_{0})}(μ)] in \overline{U}_{T} (μ), is a well-defined cohomology class. Since our constraints μ are disjoint, \overline{U}_{T (M_{0})}(μ) = ∅ if |M_{0}| ≥ 2. Furthermore, it is well-known in algebraic geometry that for any l ∈ M_{k} the normal bundle of \overline{U}_{T (l)}(μ) in \overline{U}_{T} (μ) is Lˆ1T (l); see [9]. Thus, if μ is an M-tuple of disjoint constraints,

(5.4)  [\overline{U}_{T (l)}(μ)] ∩ c_{1}(L_{k}^{∗}T ) = [\overline{U}_{T (l)}(μ)] ∩ c_{1}(L∗ ; ˆ1T (l))

= [\overline{U}_{T (l)}(μ)] ∩ c_{1}(L∗ ; ˆ1T (l)),

since L_{k}T |_{\overline{U}_{T (l)}} is the trivial line bundle. The above fact from algebraic geometry is only used to simplify notation and is not really needed for our computations. In addition, (5.4) can deduced from Subsection 5.7.

In the n = 3 case, we denote by \overline{V}_{2}(μ) the disjoint union of the spaces \overline{U}_{T} (μ) taken over equivalence classes of basic bubble types T = (S^{2}, M, I; j, \underline{d}) with |I| = 2. While the components of \overline{V}_{2}(μ) are unordered, we can still define the chern classes

c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗}), c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗}), c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}) ∈ H^{∗}(\overline{V}_{2}(μ)).

In the notation of the previous paragraph, c_{1}(L_{i}^{∗}) denotes the cohomology class c_{1}(L_{k_{i}}^{∗}T_{k_{i}}), where we write I = {k_{1}, k_{2}}. If T^{∗} = (S^{2}, M, {ˆ0}; ˆ0, d), we denote by \overline{V}_{1}(μ) the space \overline{U}_{T^{∗}}(μ) and by c_{1}(L^{∗}) ∈ H^{2}(\overline{V}_{1}(μ)) the cohomology class c_{1}(L∗ ; ˆ0T^{∗}).

Lemma 5.4. If d ≥ 1, the number of rational degree-d cuspidal curves passing through a tuple μ of 3d − 2 points in general position in ℙ^{2} is given by

|S_{1}(μ)| = ⟨3a^{2} + 3ac_{1}(L^{∗}) + c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − τ_{2}(μ),

where a = ev^{∗}(O(1)).

Proof. (1) This result is well-known in algebraic geometry; see [12]. Nevertheless, for the sake of completeness, we include a proof. Let T^{∗} be as above. By definition, S_{1}(μ) is the intersection of the zero set of the section

D ≡ D_{T^{∗}} ∈ Γ(\overline{V}_{1}(μ); L^{∗} ⊗ ev^{∗}T ℙ^{2}), where L = Lˆ0T^{∗},

with V_{1}(μ) = U_{T^{∗}}(μ). Thus, by Corollary 3.13, with ∂\overline{V}_{1}(μ) = \overline{V}_{1}(μ) − V_{1}(μ),

(5.5) |S_{1}(μ)| = ⟨c_{2}(L^{∗} ⊗ ev^{∗}T ℙ^{2}), [\overline{V}_{1}(μ)]⟩ − C_{∂\overline{V}_{1}(μ)}(D)
= ⟨3a^{2} + 3ac_{1}(L^{∗}) + c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − C_{∂\overline{V}_{1}(μ)}(D).

(2) Suppose T = (S^{2}, [N], I; j, \underline{d}) < T^{∗}, where N = 3d − 2, is a bubble type such that D vanishes somewhere on U_{T} (μ). Since the complex dimension of U_{T} (μ) is at most one, by Corollary 6.3 dˆ0 = 0. Let

ρ_{T} ∈ Γ(U_{T} (μ); Polyn(FT ; F̃T )) and
α_{T} ∈ Γ(U_{T} (μ); Hom(F̃T ; L^{∗} ⊗ ev^{∗}T ℙ^{n}))

be the sections defined in Equation (2.21). Recall that with appropriate identifications

|D(γ_{T}^{μ} (υ)) − α_{T} (ρ_{T} (υ))| ≤ C(b_{υ})|υ|^{\frac{1}{p}} |ρ_{T} (υ)| ∀υ ∈ FT_{δ}, (5.6)

where δ, C ∈ C^{∞}(U_{T} (μ); ℝ^{+}) and γ_{T}^{μ} : FT_{δ} ⟶ \overline{V}_{1}(μ) is an identification of neighborhoods of U_{T} (μ), which is smooth on the preimage of V_{1}(μ). Note that |Î| ∈ {1, 2} if U_{T} (μ) is nonempty. By the proof of Lemma 4.9, α_{T} has full rank on every fiber F̃T ⟶ U_{T} (μ). Thus, by Equation (5.6) and Corollary 3.13,

C_{U_{T} (μ)}(D) = 0 if Hˆ0T ≠ Î.

(3) Suppose |Hˆ0T | = |Î| = 1. Then T = T^{∗}(l) for some l ∈ [N] and F̃T = FT ≈ Lˆ1T . Since α_{T} ∘ ρ_{T} has constant rank over \overline{U}_{T} (μ), by Corollary 3.13 and Lemma 3.14,

C_{U_{T} (μ)}(D) = ⟨c_{1}(L^{∗} ⊗ ev^{∗}T ℙ^{2}) − c_{1}(Lˆ1T ), [\overline{U}_{T} (μ)]⟩

= ⟨3a + c_{1}(L∗ ; ˆ1T ), [\overline{U}_{T} (μ)]⟩.

If |Hˆ0T | = |Î| = 2, α_{T} ∘ ρ_{T} is an isomorphism on every fiber. Thus, C_{U_{T} (μ)}(D) = |U_{T} (μ)| by Corollary 3.13. Combining these contributions to the euler class of L^{∗} ⊗ ev^{∗}T ℙ^{2} gives

C_{∂\overline{U}}(D_{T^{∗}}) = ∑_{l∈[N]} ⟨3a + c_{1}(L∗ ; ˆ1T^{∗}(l)), [U_{T^{∗}(l)}(μ)]⟩ +  ∑_{[T ],|Hˆ0T |=|Î|=2}|\overline{U}_{T} (μ)| (5.7)

= ∑_{l∈[3d−2]} ⟨3a + c_{1}(L∗ ; ˆ1T^{∗}(l)), [U_{T^{∗}(l)}(μ)]⟩ + τ_{2}(μ).

The claim follows by plugging Equation (5.7) into (5.5) and using Equations (5.3) and (5.4).

Lemma 5.5. If d ≥ 1, the number of two-component rational degree-d curves connected at a tacnode and passing through a tuple μ of p points and q lines in general position in ℙ^{3}, where 2p + q = 4d − 3, is given by

|S_{2}(μ)| = ⟨6a^{2} + 4a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})) + (c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗}))

+ c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩ − 3τ_{3}(μ).

Proof. (1) Let T^{∗} = (S^{2}, [N], I^{∗}; j^{∗}, \underline{d}^{∗}) be a basic bubble type such that I^{∗} = {k_{1}, k_{2}} is a two-element set, d_{k_{1}}^{∗}, d_{k_{2}}^{∗} > 0, d_{k_{1}}^{∗} + d_{k_{2}}^{∗} = d, and N = p + q. Denote by T_{1}^{∗} and T_{2}^{∗} the corresponding simple types. The proof is similar to that of Lemma 5.4, but we pass to the projectivization ℙE (over ℂ) of the bundle

E = L_{1} ⊕ L_{2} ⟶ \overline{U}_{T^{∗}}(μ), where L_{i} = L_{k_{i}}T_{i}^{∗}.

The section \overline{D}_{T^{∗},2} of Lemma 4.9 induces a section D ∈ Γ(ℙE; γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3}) such that S_{T^{∗}}(μ) corresponds to the intersection of the zero set of D with ℙE|_{U_{T ^{∗}}(μ)}. If ℙE′ denotes the restriction of ℙE to ∂\overline{U} ≡ \overline{U}_{T^{∗}}(μ) − U_{T^{∗}}(μ), by Corollary 3.13,

(5.8) |\overline{S}_{T^{∗}}(μ)| = ⟨c_{3}(γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3}), [ℙE]⟩ − C_{ℙE′}(D)
= ⟨6a^{2} + 4a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})) + (c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗}))
+ c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{U}_{T^{∗}}(μ)]⟩ − C_{ℙE′}(D).

The second equality above is obtained by applying (3.16).

(2) Suppose T = (S^{2}, [N], I; j, \underline{d}) < T^{∗} is a bubble type such that D vanishes somewhere on ℙE|_{U_{T} (μ)}. Let T_{1} and T_{2} be the corresponding simple types. Since the constraints are disjoint, up to interchanging the indices, we must have

T_{1} = T_{1}^{∗}, T_{2} = (S^{2}, M_{2}, I_{2}; j|_{M_{2}}, \underline{d}|_{I_{2}}) < T_{2}^{∗} with d_{k_{2}} = 0.

Furthermore, D_{T_{1}^{∗},k_{1}} does not vanish on U_{T} (μ); see the proof of Lemma 4.9. Thus, D vanishes only the subspace

Z_{T} ≡ ℙL_{2}|_{U_{T} (μ)} = {(b, L_{2}|_{b}) : b ∈ U_{T} (μ)}.

The map γ_{T}^{μ} of Theorem 2.8 induces an identification of a neighborhood of 0 in

FS ≡ π_{E}^{∗}FT ⊕ π_{E}^{∗}L_{2}^{∗} ⊗ π_{E}^{∗}L_{1} ⟶ Z_{T}

with a neighborhood of Z_{T} in ℙE. Similarly to the n = 2 case, with appropriate identifications,

|D(γ_{T}^{μ} (υ, u)) − α̃_{T} (ρ̃_{T} (υ, u))| ≤ C(b_{υ})|υ|^{\frac{1}{p}} |ρ_{T} (υ)| ∀(υ, u) ∈ FS_{δ}, (5.9)

where ρ̃_{T} (υ, u) = (ρ_{T} (υ), u) ∈ F̃S ≡ π^{∗}F̃T ⊕ π_{E}^{∗}L_{2}^{∗} ⊗ π_{E}^{∗}L_{1} ⟶ Z_{T} ,

and α̃_{T} has full rank on every fiber by (2.21) and Lemma 4.9. Thus, similarly to the proof of Lemma 5.4, and C_{ℙE′|_{Z_{T}}} (D) = 0 if H_{k_{2}}T ≠ Î_{2}, and only two cases remain to be considered.

(3) If |H_{k_{2}}T | = |Î_{2}| = 1, α̃_{T} ∘ ρ̃_{T} has full rank over all of \overline{Z}_{T}. Thus, by Corollary 3.13 and Lemma 3.14,

(5.10)  C_{Z_{T}} (D) = ⟨c_{1}(γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3}) − c_{1}(FS), [\overline{Z}_{T} ]⟩

= ⟨4a + c_{1}(L∗ ; ˆ1T_{2}) + c_{1}(L_{1}^{∗}), [\overline{U}_{T} (μ)]⟩;

note that c_{1}(γ_{E}^{∗}) = c_{1}(L_{2}^{∗}) = 0 over \overline{U}_{T} (μ). If |H_{k_{2}}T | = |Î_{2}| = 2, α̃_{T} ∘ρ̃_{T} is an isomorphism on every fiber, and thus

C_{ℙE′|_{Z_{T}}} (D) = |Z_{T} | = |U_{T} (μ)|. (5.11)

Note that the sum of |U_{T} (μ)| over all equivalence classes of bubble types T^{∗} and T < T^{∗} is 3τ_{3}(μ), since one of the three components of the image of each bubble map in U_{T} (μ) is distinguished by the bubble type T . As before, we now sum up Equations (5.10) and (5.11) over all equivalence classes of bubble types T < T^{∗} of the appropriate form, plug the result back into (5.8) and use Equations (5.3) and (5.4). The claim follows by summing the result over all equivalence classes of basic simple bubble types T^{∗}.

5.3 The numbers n_{m}^{(1)} with m = n − 1

In this subsection, we give topological formulas for the numbers n_{T}^{(1)} with |Î| = n − 1. As before, the reason these two cases are similar is that the complex dimension of U_{T} (μ) is two.

Lemma 5.6. If n = 2, n_{1}^{(1)}(μ) = 2⟨6a^{2} + 3ac_{1}(L^{∗}), [\overline{V}_{1}(μ)]⟩.

Proof. (1) Let N, T^{∗}, L, D be as in the proof of Lemma 5.4. Since s_{Σ} does not vanish on Σ, by Equation (4.30) and Lemma 3.14,

n_{1}^{(1)}(μ) = ∑_{k=0}^{k=3} ⟨c_{k}(O)c_{1}^{3−k}(T^{∗}Σ ⊗ L^{∗}), [Σ × \overline{U}_{T^{∗}}(μ)]⟩ − C_{Σ×D^{−1}(0)}(α^{⊥}), (5.12)

= 2⟨15a^{2} + 12ac_{1}(L^{∗}) + 3c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − C_{Σ×D^{−1}(0)}(α^{⊥}),

where O = H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{2} and α ∈ Γ(Σ × \overline{V}_{1}(μ); T^{∗}Σ ⊗ L^{∗} ⊗ O) is the linear part of the affine map ψ_{1}^{(1)} of (4.30).

(2) We first compute C_{Σ×S_{1}(μ)}(α^{⊥}). Since V_{1}(μ) is a complex manifold and D is transverse to the zero set in L^{∗} ⊗ ev^{∗}T ℙ^{2} by Corollary 6.3, we can identify a neighborhood of 0 in

F ≡ L^{∗} ⊗ ev^{∗}T ℙ^{2} ⟶ S_{1}(μ)

with a neighborhood of S_{1}(μ) in V_{1}(μ) via a map γ in such a way that

Π_{b,γ(b,X)}^{−1}(Dγ(b, X)) = X ∀(b, X) ∈ F_{δ}. (5.13)

Then with appropriate identifications,

α^{⊥}(γ(X)) = π^{⊥} ∘ Xs_{Σ} ≡ α_{S}(X),

where π^{⊥} : O ⟶ O^{⊥} is the quotient projection map. In particular, α_{S} has full rank if \overline{ν}_{b} ∉ H_{Σ}^{+} ⊗ ev^{∗}T ℙ^{2} for all b ∈ S_{1}(μ), i.e., ν is generic. Furthermore,

(T^{∗}Σ ⊗ L^{∗} ⊗ O^{⊥})/(Im α_{S}) ≈ T^{∗}Σ ⊗ ((H_{Σ}^{−} ⊗ ℂ^{2})/ℂ).

Thus, by Corollary 3.13,

C_{Σ×S_{1}(μ)}(α^{⊥}) = ⟨e(T^{∗}Σ^{⊗3}), [Σ × S_{1}(μ)]⟩ = 6|S_{1}(μ)|. (5.14)

(3) It remains to compute the contribution to C_{Σ×D^{−1}(0)}(α^{⊥}) from Σ × (\overline{V}_{1}(μ) − V_{1}(μ)). Suppose

T = (S^{2}, [N], I; j, \underline{d}) < T^{∗}

is a bubble type such that D vanishes somewhere on U_{T} (μ). As in the proof of Lemma 5.4, |Î| ∈ {1, 2} and dˆ0 = 0. Furthermore,

|α^{⊥}(x, γ_{T}^{μ} (υ)) − α̃_{T} (x, b; ρ_{T} (υ))| ≤ C(b_{υ})|υ|^{\frac{1}{p}} |ρ_{T} (υ)| (x, b; υ) ∈ FT_{δ},

where α̃_{T} = π^{⊥} ∘ (s_{Σ} ⊗ α_{T}). If ν is generic, α̃_{T} has full rank on every fiber, since s_{Σ} has no zeros. Thus, by Corollary 3.13,

C_{Σ×U_{T} (μ)}(α^{⊥}) = 0 if Hˆ0T ≠ Î.

If |Hˆ0T | = |Î| = 1, α̃_{T} ∘ ρ̃_{T} has full rank over all Σ × \overline{U}_{T} (μ), and thus by Corollary 3.13 and Lemma 3.14

(5.15)  C_{Σ×U_{T} (μ)}(α^{⊥}) = ⟨c(T^{∗}Σ ⊗ O^{⊥})c(Lˆ1T )^{−1}, [Σ × \overline{U}_{T} (μ)]⟩

= 2⟨12a + 3c_{1}(L∗ ; ˆ1T ), [\overline{U}_{T} (μ)]⟩.

If |Hˆ0T | = |Î| = 2,

(T^{∗}Σ ⊗ L^{∗} ⊗ O^{⊥})/(Im α̃_{T} ∘ ρ̃_{T} ) ≈ T^{∗}Σ ⊗ ((H_{Σ}^{−} ⊗ ℂ^{2})/ℂ).

Thus, similarly to the computation in (2) above,

C_{Σ×U_{T} (μ)}(α^{⊥}) = 6|U_{T} (μ)|. (5.16)

Summing Equations (5.15) and (5.16) over all equivalence classes of T < T^{∗}, we obtain

C_{Σ×(\overline{V}_{1}(μ)−V_{1}(μ))}(α^{⊥}) = 2 ∑_{l∈[N]} ⟨12a + 3c_{1}(L∗ ; ˆ1T ), [U_{T^{∗}(l)}(μ)]⟩ + 6τ_{2}(μ). (5.17)

The claim follows by plugging (5.14) and (5.17) into (5.12) and using (5.3), (5.4), and Lemma 5.4.

Lemma 5.7. If n = 3, n_{2}^{(1)}(μ) = 4⟨10a^{2} + 4a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})) +

c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩.

Proof. (1) We use the same notation as in the proof of Lemma 5.5. By Section 4.9 and Equation (4.34), n_{2}^{(1)}(μ) = N(α_{2}), where

α_{2} ∈ Γ(Σ^{2} × \overline{V}_{2}(μ); Hom(Ẽ; O)), Ẽ = T Σ_{1} ⊗ L_{1} ⊕ T Σ_{2} ⊗ L_{2},

O = H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{3},

α_{2}(x_{1}, x_{2}, b; v_{1} ⊗ υ_{1}, v_{2} ⊗ υ_{2}) = (Dυ_{1})(s_{Σ,x_{1}}v_{1}) + (Dυ_{2})(s_{Σ,x_{2}}v_{2}).

Here the bundles L_{i} ⟶ \overline{V}_{2}(μ) and the sections D_{i} ∈ Γ(\overline{V}_{2}(μ); L_{i}^{∗} ⊗ ev^{∗}T ℙ^{2}) are defined as follows. If b ∈ U_{T^{∗}}(μ) ⊂ V_{2}(μ), T^{∗} = (S^{2}, [N], I^{∗}; j^{∗}, \underline{d}^{∗}), and I^{∗} = \{k_{1}, k_{2}\}, we let L_{i}|_{b} = L_{k_{i}}T and D_{i} = D_{T ,k_{i}}. These bundles and sections are well-defined once we fix a representative for each equivalence class of such bubble types T^{∗} and order the elements of the corresponding set I^{∗}.

(2) By Lemma 3.14,

(5.18) n_{1}^{(1)}(μ) = ∑_{k=0}^{k=5} ⟨c_{k}(O)λ5−k ; Ẽ, [ℙẼ]⟩ − C_{α^{−1}(0)}(α^{⊥}),

= 4⟨28a^{2} + 16a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})) + 3(c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗}))

+ 4c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩ − C_{α^{−1}(0)}(α^{⊥}),

where α̃ ∈ Γ(ℙẼ; γ∗ ; Ẽ ⊗ O) is the section induced by α_{2}. Let

Σ^{(±)} = {(x_{1}, x_{2}) ∈ Σ_{1}^{∗} × Σ_{2}^{∗} : x_{1} = ±x_{2}}, Σ^{(0)} = {(z_{m}, z_{m}) : m ∈ [6]};

S_{2}^{(±)} = Σ^{(±)} × S_{2}(μ), S_{2}^{(0)} = Σ^{(0)} × S_{2}(μ),

where +x_{2} ≡ x_{2} and −x_{2} is the image of x_{2} under the nontrivial automorphism of Σ. The zero set of α̃ is the union of a section of ℙẼ over S_{2}^{(±)}, S_{2}^{(0)}, and Σ^{2} × U_{T} (μ), where T is as in the proof of Lemma 5.5.

(3) The above section over Σ^{2} × U_{T} (μ) is given by

Z_{T} ≡ α^{−1}(0) ∩ ℙẼ|_{Σ^{2}×U_{T} (μ)}

= {(x_{1}, x_{2}, b, T_{x_{1}}Σ_{1} ⊗ L_{1}|_{b}) : (x_{1}, x_{2}, b) ∈ Σ^{2} × U_{T} (μ)}.

The map γ_{T}^{μ} of Theorem 2.8 induces identifications of neighborhoods of Z_{T} in

FS = π∗ ; Ẽ(FT ⊕ T^{∗}Σ_{1} ⊗ L_{1}^{∗} ⊗ T Σ_{2} ⊗ L_{2})

and in ℙẼ as well as of appropriate bundles such that

|α̃(γ_{T}^{μ} (υ, u)) − α̃_{T} (ρ_{T} (υ), u)| ≤ C(b_{υ})|υ|^{\frac{1}{p}} |ρ_{T} (υ)| ∀(υ, u) ∈ FS_{δ},

where

α̃_{T} ∈ Γ(Z_{T}; Hom(F̃S; γ∗ ; Ẽ ⊗ O)),

F̃S = π∗ ; Ẽ(F̃T ⊕ T^{∗}Σ_{1} ⊗ L_{1}^{∗} ⊗ T Σ_{2} ⊗ L_{2}),

α̃_{T} (x_{1}, x_{2}, b; υ̃, u) = {α_{T} (υ̃)} ⊗ s_{x_{1}} + (D_{2} ⊗ s_{x_{2}}) ∘ u.

By the proof of Lemma 4.9, α̃_{T} is nondegenerate. The same is true of α^{⊥} as long as \overline{ν} ∈ Γ(ℙẼ; O) is generic. Thus, if Î ≠ Hˆ0T , Z_{T} is α^{⊥}-hollow and C_{Z_{T}} (α^{⊥}) = 0 by Corollary 3.13. If |H_{k_{1}}T | = |Î| = 1, i.e., T = T^{∗}(l) for some l ∈ [N], α̃_{T} has full rank on \overline{Z}_{T} ≈ Σ^{2} × \overline{U}_{T} (μ). Thus, by Corollary 3.13,

(5.19)  C_{Z_{T}} (α^{⊥}) = ⟨c(γ∗ ; Ẽ ⊗ O^{⊥})c(FS)^{−1}, [\overline{Z}_{T}]⟩

= 4⟨16a + 4c_{1}(L_{2}^{∗}) + 3c_{1}(L∗ ; ˆ1T ), [\overline{U}_{T} (μ)]⟩,

since F S ≈ Lˆ1T ⊕ T^{∗}Σ_{1} ⊗ T Σ_{2} ⊗ L_{2}. If |H_{k_{1}}T | = |Î| = 2, we similarly obtain

C_{Z_{T}} (α^{⊥}) = ⟨c(γ∗ ; Ẽ ⊗ O^{⊥})c(FS)^{−1}, [\overline{Z}_{T}]⟩ = 12|U_{T} (μ)|. (5.20)

Note that F S ≈ ℂ^{2} ⊕ T^{∗}Σ_{1} ⊗ T Σ_{2} in this case. Summing up Equations (5.19) and (5.20) over all equivalence classes of bubble types T of the appropriate form and using (5.3) and (5.4), we obtain

C_{ℙẼ|∂V_{2}(μ)}(α^{⊥}) = 4 ∑_{[T^{∗}]} ∑_{l∈M_{i}^{∗},i≠j} ⟨16a + 3c_{1}(L_{i}^{∗}) + 4c_{1}(L_{j}^{∗}), [U_{T^{∗}(l)}(μ)]⟩ (5.21)
− 36|V_{3}(μ)|,

where the outer sum is taken over equivalence classes of bubbles T^{∗} as in (1) above.

(4) It remains to compute C_{ℙẼ|S_{2}^{±}}(α^{⊥}) and C_{ℙẼ|S_{2}^{(0)}}(α^{⊥}). Note that

α^{−1}(0) ∩ ℙẼ|_{S_{2}^{(±)}}

= Z_{2}^{(±)} ≡ {(x, ±x, b; [v ⊗ υ_{1}, v ⊗ υ_{2}]) ∈ ℙẼ|_{S_{2}^{(±)}} : D|_{(b;[υ_{1},υ_{2}])} = 0},

where D is the section of γ_{E}^{∗} ⊗ev^{∗}T ℙ^{3} defined in the proof of Lemma 5.5. Identify neighborhoods of Z_{2}^{(±)} in

FS ≡ T Σ ⊕ γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3} ≈ T Σ ⊕ ℂ^{3}

and in ℙẼ via a map γ in such a way that

|α̃(γ(w, X)) − α_{S}(w, X)| ≤ C(x, b)(|w| + |X|)|w ∀(w, X) ∈ FS_{δ},

where α_{S} ∈ Γ(Z_{2;2}; Hom(FS; γ∗ ; Ẽ ⊗ O)),

{α_{S}(w, X)}(v ⊗ υ) = (Xυ)(s_{x}v) + (D_{2}υ_{2})(s_{b,x}^{(2)}(w, v)) ∈ O,
if v ∈ T_{x}Σ, υ = (υ_{1}, υ_{2}) ∈ γ_{E}.

Since s_{x}^{(2)} = π_{x}^{−}∘s_{b,x}^{(2)} does not vanish on Σ^{∗}, α_{S} has full rank on Z_{2;2}^{(±)} and

extends over \overline{Z}_{2}^{(±)} ≈ Σ × S_{2}(μ). This extension is a regular polynomial in the sense of Definition 3.9. Furthermore,

π⊥ ; \overline{ν}α_{S} : γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3} ⟶ γ∗ ; Ẽ ⊗ π⊥ ; \overline{ν}(H_{Σ}^{+} ⊗ ev^{∗}T ℙ^{3})

is an isomorphism. Thus, by Corollary 3.13, C_{Z_{2}^{(±)}}(α^{⊥}) = N(α_{S}^{−}), where

α_{S}^{−} ∈ Γ(\overline{Z}_{2}; Hom(T Σ; O_{2})),

O_{2} = γ∗ ; Ẽ ⊗ (H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{3})^{⊥} ≈ T^{∗}Σ ⊗ (H_{Σ}^{−} ⊗ ℂ^{3})^{⊥},

\{α_{S}^{−}(w)\}(v ⊗ υ_{2}) = π_{π_{x}^{−}\overline{ν}}^{⊥}((D_{2}υ_{2})s_{x}^{(2,−)}(w, v)).

As in the previous section, we can replace T Σ with

T̃′Σ ≡ T Σ ⊗ O(z_{1}) ⊗ ⋯ ⊗ O(z_{6})

and s^{(2,−)} with s^{(2,−)} ∈Γ(Σ; T̃′Σ^{∗} ⊗ H_{Σ}^{−}) above to obtain a non-vanishing linear map α_{S}^{−} such that N(α_{S}^{−}) = N(α_{S}^{−}). Thus, by Lemma 3.14,

(5.22)  C_{Z_{2}^{(+)}∪Z_{2}^{(−)}}(α^{⊥}) = 2⟨c_{1}(O_{2}) − c_{1}(T̃′Σ), [Z_{2}^{(+)}]⟩
= 2(10 − 4)|S_{2}(μ)| = 12|S_{2}(μ)|.

(5) We next show that C_{ℙẼ|S_{2}^{(0)}}(α^{⊥}) = 0. Similarly to (4),

α^{−1}(0) ∩ ℙẼ|_{S_{2}^{(0)}} = Z_{2}^{(0)} ≡ {(z_{m}, z_{m}, b; [v ⊗ υ_{1}, v ⊗ υ_{2}]) ∈

ℙẼ|_{S_{2}^{(0)}} : D|_{(b;[υ_{1},υ_{2}])} = 0}.

We can identify neighborhoods of Z_{2}^{(0)} in

FS ≡ T Σ_{1} ⊕ T Σ_{2} ⊕ γ_{E}^{∗} ⊗ ev^{∗}T ℙ^{3} ≈ ℂ^{2} ⊕ ℂ^{3}

and in ℙẼ via a map γ in such a way that

|π_{w_{1}}^{−} ∘ α̃(γ(w_{1}, w_{2}, X)) − α_{S}^{−}(w_{1}, w_{2}, X)|
≤ C|X, w_{1}, w_{2}||w_{1}||w_{1} − w_{2}| ∀(w_{1}, w_{2}, X) ∈ FS_{δ},

where

{α_{S}^{−}(w_{1}, w_{2})}(v ⊗ υ_{1}, v ⊗ υ_{2})
= (D_{2}υ_{2})s_{z_{m}}^{(3)}(w_{1}, w_{2} − w_{1}, v) ∈ H_{Σ}^{−}(z_{m}) ⊗ ev^{∗}T ℙ^{3}.

Since the rank of (H_{Σ}^{−}(z_{m}) ⊗ ev^{∗}T ℙ^{3})/ℂπ_{z_{m}}^{−}\overline{ν} is two, while the rank of T_{z_{m}}Σ_{1}⊗T_{z_{m}}Σ_{2} is one, it follows that Z_{2}^{(0)} is α^{⊥}-hollow, and C_{ℙẼ|S_{2}^{(0)}}(α^{⊥}) = 0 by Corollary 3.13. The lemma is obtained by plugging (5.21) and (5.22) into (5.18), using (5.3) and (5.4), and Lemma 5.5.

5.4 Behavior of D^{(2)} and D^{(3)} near \overline{S}_{1}(μ) − S_{1}(μ)

If n = 3, the space S_{1}(μ) is not compact. In order to be able to compute the numbers n_{1}^{(k)}(μ), we thus must understand the structure of \overline{S}_{1}(μ) as well as the behavior of D_{T^{∗},ˆ0}^{(2)} and D_{T^{∗},ˆ0}^{(3)}, where T^{∗} = (S^{2}, [N], {ˆ0}; ˆ0, d), near \overline{S}_{1}(μ) − S_{1}(μ).

If T = (S^{2}, [N], I; j, \underline{d}) < T^{∗}, from Theorem 2.8 one should expect that the normal bundle, or cone, FS of S_{T} (μ) ≡ U_{T} (μ)∩\overline{S}_{1}(μ) in \overline{S}_{1}(μ) is the closure of the set

{[υ = (b, (v_{h})_{h∈Î})] ∈ F^{(∅)}T |_{S_{T} (μ)} : ∑_{h∈χ(T )} ∏_{i∈Î,i≤h} v_{i} (D_{T ,h}b) = 0} (5.23)

in FT. The next lemma shows that this is indeed the case. By a dimension-counting argument, if the set in (5.23) is not empty, either |χ(T )| = 1 or χ(T ) = {h_{1}, h_{2}} is a two-element set, ι_{h_{1}} = ι_{h_{2}}, D_{T ,h_{1}}b ≠ 0, and D_{T ,h_{2}}b ≠ 0. In the first case FS = FT |_{S_{T} (μ)}, while in the second FS is a codimension-one subbundle of FT |_{S_{T} (μ)}.

Let N S ⟶ FS denote the normal bundle of FS in FT ⟶ U_{T} (μ). While for the purposes of Lemma 5.8, we can use any identification of neighborhoods of FS in N S and in FT ⟶ U_{T} (μ), in order to simplify the statement of Lemma 5.10, we choose a fairly natural one. More precisely, denote by FS^{⊥} a subspace of FT |_{S_{T} (μ)} complementary to FS and by π_{S} : N S^{(1)} ⟶ S_{T} (μ) the normal bundle of S_{T} (μ) in U_{T} (μ). Choose a norm on N S^{(1)} and an identification ϕ_{S} : N S_{δ}^{(1)} ⟶ S_{T} (μ) of neighborhoods of S_{T} (μ) in N S^{(1)} and in U_{T} (μ). Let Φ_{S} : π_{S}^{∗}FT ⟶ FT be a lift of ϕ_{S} such that Φ_{S} restricts to the identity over S_{T} (μ) ⊂ N S_{δ}^{(1)}. Let π : FT ⟶ S_{T} (μ) be the bundle projection. Then

N S = π^{∗}N S^{(1)} ⊕ FS^{⊥}, and ϕ̃_{S} : N S_{δ} ⟶ FT ,
ϕ̃_{S}((b, v), (X, v^{⊥})) = Φ_{S}((b, X), v + v^{⊥}),

is an identification of neighborhoods of FS in N S and FT ⟶ U_{T} (μ).

Lemma 5.8. For every bubble type T = (S^{2}, [N], I; j, \underline{d}) < T^{∗}, there exist δ, C ∈ C^{∞}(S_{T} (μ); ℝ^{+}) and a section φ_{S} ∈ Γ(FS_{δ}; N S) such that

∥φ_{S}(υ)∥ ≤ C(b_{υ})|υ|^{\frac{1}{p}} , ∥φ_{FS}_{⊥}(υ)∥ ≤ C(b_{υ})|υ|^{1+ \frac{1}{p}} ,

where φ_{FS}_{⊥} denotes the FS^{⊥}-component of φ_{S}, and the map

γ_{S} : FS_{δ} ⟶ \overline{S}_{1}(μ), γ_{S}(υ) = γ_{T}^{μ} (ϕ̃_{S}φ_{S}(υ)),

is a homeomorphism onto an open neighborhood of S_{T} (μ) in \overline{S}_{1}(μ), which is smooth and orientation-preserving on the preimage of S_{1}(μ).

Proof. (1) The proof is similar to that of Lemma 3.32 in [13], and so we only describe the differences. If S_{T} (μ) ≠ ∅, T must have one of the three forms described by Lemma 5.10. In Case (1), we apply Subsection 3.7 in [13], which contains an application of the Implicit Function Theorem, to D_{T^{∗},ˆ0} instead of the evaluation maps. By Theorem 2.8,

|Π_{b,γ_{T}^{μ} (υ)}^{−1}(D_{T^{∗},ˆ0}γ_{T}^{μ} (υ)) − (D_{T^{∗},ˆ0}b_{υ})| ≤ C′(b)|υ|^{\frac{1}{p}} ∀υ ∈ FS_{δ}.

This estimate suffices for applying an argument similar to the proof of Lemma 3.32 in [13].

(2) In Case (2) of Lemma 5.10, instead of the section D_{T^{∗},ˆ0} of L∗ ; ˆ0T^{∗}⊗ ev^{∗}T ℙ^{3}, we consider the section D̃ of (Lˆ0T^{∗} ⊗ FT )^{∗} ⊗ ev^{∗}T ℙ^{3} on a neighborhood of U_{T} (μ) in \overline{U}_{T^{∗}}(μ) defined by

D̃|_{γ_{T}^{μ} (b,vˆ1)}(vˆ0, vˆ1) = D_{T^{∗},ˆ0}|_{γ_{T}^{μ} (b,vˆ1)}(vˆ0) ∈ ev^{∗}T ℙ^{3}.

This section is well-defined outside of U_{T} (μ) and by Theorem 2.8 extends over U_{T} (μ) by

D̃|_{b}(vˆ0 ⊗ vˆ1) = vˆ0vˆ1(D_{T ,ˆ1}b).

The restriction of this section to U_{T} (μ) vanishes transversally at S_{T} (μ) by Corollary 6.3, while its zero set on U_{T^{∗}}(μ) is the same as the zero set of D_{T^{∗},ˆ0}. By Theorem 2.8, with appropriate identifications,

|D̃|_{γ_{T}^{μ} (b,υ)} − D̃|_{b}| ≤ C′(b)|υ|^{\frac{1}{p}} ∀υ ∈ FS_{δ}.

(3) In the final case of Lemma 5.10, we replace D_{T^{∗},ˆ0} by a bundle section over the blowup of FT along U_{T} (μ). Let

Ω_{T} = {(b, v, ℓ) : (b, v) ∈ FT , v ∈ ℓ ∈ ℙFT |_{b}},
Ω_{T}^{∗} = {(b, v, ℓ) ∈ Ω_{T} : v ≠ 0}, E_{T} = Ω_{T} − Ω_{T}^{∗} .

Denote by γ ⟶ Ω_{T} the tautological line bundle. The normal bundle ÑS of γ ⟶ ℙFS in γ ⟶ E_{T} is given by

ÑS = π_{γ}^{∗}π_{FT}^{∗} N S^{(1)} ⊕ π_{γ}^{∗}(γ^{∗} ⊗ π_{FT}^{∗} FS^{⊥}),
ϕ̃_{ÑS}((b, ℓ, v), X, σ) = (ϕ_{S}(b, X), [Φ_{S}(v + σ(v))], v + σ(v)),

where π_{γ} : γ ⟶ Ω_{T} is the bundle projection map. The bundle Lˆ0T^{∗} pulls back to a bundle L̃ over a neighborhood Ω_{δ} of E_{T} in Ω_{T}. We define a section D̃ of (L̃ ⊗ γ)^{∗} ⊗ ev^{∗}T ℙ^{3} over Ω_{δ} by

D̃|_{(b,vˆ1,vˆ2,ℓ)}(vˆ0, vˆ1, vˆ2) = D_{T^{∗},ˆ0}|_{γ_{T}^{μ} (b,vˆ1,vˆ2)}(vˆ0) ∈ ev^{∗}T ℙ^{3}.

This section is well-defined outside of E_{T} (μ) and by Theorem 2.8 extends over E_{T} (μ) by

D̃|_{b}(vˆ0, vˆ1, vˆ2) = vˆ0(vˆ1(D_{T ,ˆ1}b) + vˆ2(D_{T ,ˆ2}b)).

The restriction of this section to E_{T} (μ) vanishes transversally at ℙFS ⟶ S_{T} (μ) by Corollary 6.3, while its zero set on Ω^{∗} corresponds to the zero set of D_{T^{∗},ˆ0} on γ_{T}^{μ} (FT_{δ} − U_{T} (μ)). By Theorem 2.8, with appropriate identifications,

|D̃|_{(b,vˆ1,vˆ2,ℓ)} − D̃|_{(b,ℓ)}| ≤ C′(b)|υ|^{\frac{1}{p}} .

Thus, we can apply the arguments of Lemma 3.32 in [13] to D̃ to describe its zero set near E_{T}. We obtain a section φ̃_{S} ∈ Γ(γ_{δ}|_{ℙFS}; ÑS) such that ∥φ̃_{S}(υ)∥ ≤ C(b_{υ})|υ|^{\frac{1}{p}}, and the map

γ̃_{S} : γ_{δ}|_{ℙFS} ⟶ Ω_{T} , γ̃_{S}(υ) = ϕ̃_{ÑS}(φ_{S}(υ)),

is a homeomorphism onto an open neighborhood of ℙFS in D̃^{−1}(0). This section φ̃_{S} induces the required section φ_{S} with the claimed properties.

Corollary 5.9. For every bubble type T = (S^{2}, [N], I; j, \underline{d}) < T^{∗}, there exist δ ∈ C^{∞}(S_{T} (μ); ℝ^{+}) and a map

γ_{S} : (N S^{(1)} ⊕ FT )_{δ}|_{S_{T} (μ)} ⟶ \overline{U}_{T^{∗}}(μ)

such that γ_{S} is a homeomorphism onto an open neighborhood of S_{T} (μ) in \overline{U}_{T^{∗}}(μ), which is smooth and orientation-preserving on the preimage of U_{T^{∗}}(μ), and with appropriate identifications,

Dγ_{S}(X, υ) = \begin{cases}X, & in Case (1) with X ∈ L^{∗} ⊗ ev^{∗}T ℙ^{3}; \\ Xυˆ1, & in Case (2) with X ∈ L∗ ; ˆ1T ⊗ ev^{∗}T ℙ^{3}, \end{cases}

where the cases are the ones described by Lemma 5.10.

Proof. The proof is just a modification of the proof of Lemma 5.8. We work with the sections \overline{D} ≡ D and \overline{D} ≡ D̃ in Cases (1) and (2), respectively. Choose an identification γ : N S_{δ}^{(1)} ⟶ U_{T} (μ) of neighborhoods of S_{T} (μ) in N S^{(1)} and in U_{T} (μ) as well as of the appropriate line bundle over these neighborhoods such that

\overline{D}|_{(b,X)} = X ∀X ∈ N S_{δ}^{(1)}.

By the same argument as in the proof of Lemma 5.8, for any (Y, υ) ∈ NS^{(1)} ⊕ FT, there exists a unique Z ∈ NS^{(1)}, such that

\overline{D}|_{γ_{T}^{μ} (Φ_{S}(X+Y;υ))} = \overline{D}|_{X} = X.

Furthermore, |Z| ≤ C(b)(|Y | + |υ|^{\frac{1}{p}} ).

Lemma 5.10. If d ≥ 1, μ is a tuple of p points and q lines in general position in ℙ^{3} with 2p + q = 4d − 3, and N = p + q, the set \overline{S}_{1}(μ) − S_{1}(μ), is finite. Furthermore, if

T = (S^{2}, [N], I; j, \underline{d}) < T^{∗} and S_{T} (μ) ≠ ∅,

then one of the following cases holds:

(1) Î = {ˆ1}, dˆ0 > 0, and the images of D_{T^{∗},ˆ0}^{(2)} and D_{T^{∗},ˆ0}^{(3)} are linearly independent in every fiber of ev^{∗}T ℙ^{3} over S_{T} (μ).

(2) Î = {ˆ1}, dˆ0 = 0, dˆ1 = d, and for all υ = [b, vˆ1] ∈ FS_{δ},

|Π_{b,γ_{S}(υ)}^{−1}(D^{(2)}γ_{S}(υ)) − v2 ˆ1(D_{T ,ˆ1}^{(2)}b)| ≤ C|vˆ1|^{2+ \frac{1}{p}};

|Π_{b,γ_{S}(υ)}^{−1}((D^{(3)}γ_{S}(υ))− 3xˆ1(D^{(2)}γ_{S}(υ))) − v3 ˆ1(D_{T ,ˆ1}^{(3)}b)| ≤C|vˆ1|^{3+ \frac{1}{p}} .

(3) Î = {ˆ1, ˆ2}, dˆ0 = 0, and for all υ = [b, vˆ1, vˆ2] ∈ FS

|Π_{b,γ_{S}(υ)}^{−1}(D^{(2)}γ_{S}(υ)) − 2(xˆ1vˆ1(D_{T ,ˆ1}b) + xˆ2vˆ2(D_{T ,ˆ2}b))| ≤ C|υ|^{1+ \frac{1}{p}};

|Π_{b,γ_{S}(υ)}^{−1}(2(D^{(3)}γ_{S}(υ)) − 3(xˆ1 + xˆ2)(D^{(2)}γ_{S}(υ)))
−3(xˆ1 − xˆ2)(v2 ˆ1(D_{T ,ˆ1}^{(2)}b) − v2 ˆ2(D_{T ,ˆ2}^{(2)}b))| ≤ C|υ|^{2+ \frac{1}{p}} .

Proof. (1) The statement about the possible structures of T is easily seen from Theorem 2.8 and dimension count. The finiteness claim then also follows by dimension count. In Case (1), if dˆ0 ≥ 3, by Corollary 6.3, the images of D_{T^{∗},ˆ0}^{(2)} and D_{T^{∗},ˆ0}^{(3)} are transversal and thus linearly independent over the finite set S_{T} (μ). On the other hand, if dˆ0 < 3, S_{T} (μ) = ∅; see Subsection 4.5.

(2) The four inequalities in the lemma will be obtained by refining the proof of the analytic estimate of Theorem 2.8. We use the same notation. Combining Equations (2.22), (2.23), (2.24), and (2.26), we obtain

(5.24)  (D^{(m)}γ̃_{T} (υ)) = m ∑_{h∈χ(T )} ∑_{k=1}^{k=m} \frac{a_{k,h}(υ)}{k}v^{k}(D_{T ,h}^{(k)}b)

− \frac{m}{2πi} ∑_{h∈χ(T )} ∫_{A_{h}^{−}(υ)} ξ_{υ}w^{m−1}dw,

where the integral is computed by using the same trivializations as before. This equality holds for any bubble type. If γ_{T} (υ) ∈ S_{1}(μ) and T is as in (2) of the lemma, (5.24) with m = 1, 2, 3 gives

(5.25)  0 = vˆ1(D_{T ,ˆ1}^{(1)}b) − \frac{1}{2πi} ∫_{|xˆ1−w|=ϵ} ξ_{υ}dw;

(D^{(2)}γ̃_{T} (υ)) = 2xˆ1vˆ1(D_{T ,ˆ1}^{(1)}b) + v2 ˆ1(D_{T ,ˆ1}^{(2)}b) − \frac{1}{πi} ∫_{|xˆ1−w|=ϵ} ξ_{υ}wdw; (5.26)

(D^{(3)}γ̃_{T} (υ)) = 3x2 ˆ1vˆ1(D_{T ,ˆ1}^{(1)}b) + 3xˆ1v2 ˆ1(D_{T ,ˆ1}^{(2)}b) (5.27)

+ 2v3 ˆ1(D_{T ,ˆ1}^{(3)}b) − \frac{3}{2πi} ∫_{|xˆ1−w|=ϵ} ξ_{υ}w^{2}dw.

where ϵ = 4δ(b_{υ})^{−1}|vˆ1|. Subtracting 2xˆ1 times the first equation from the second, we obtain

|(D^{(2)}γ̃_{T} (υ)) − v2 ˆ1(D_{T ,ˆ1}^{(2)}b)| ≤ C(b)|vˆ1|^{2+ \frac{1}{p}} . (5.28)

Similarly, subtracting 3xˆ1 times (5.26) from and adding 3x2 ˆ1 times (5.25) to (5.27), we obtain

|((D^{(3)}γ̃_{T} (υ)) − 3xˆ1(D^{(2)}γ̃_{T} (υ))) − 2v3 ˆ1(D_{T ,ˆ1}^{(3)}b)| ≤ C(b)|vˆ1|^{3+ \frac{1}{p}} . (5.29)

If υ ∈ FS is sufficiently small, the claim in Case (2) follows from Equations (5.28) and (5.29) along with Lemma 5.8 and our choice of ϕ̃_{S}. Note that if υ ∈ FS, we have to apply (5.28) and (5.29) with υ replaced by Φ_{T}^{μ} φ_{T}^{μ} ϕ̃_{S}φ_{S}(υ), where Φ_{T}^{μ} and φ_{T}^{μ} are as in Subsection 3.9 of [13]. However, applying the bounds on φ_{T}^{μ} and ϕ̃_{S}, we obtain the claimed estimates.

(3) In Case (3), we proceed similarly. The analog of Equation (5.26) gives

|(D^{(2)}γ̃_{T} (υ)) − 2(xˆ1vˆ1(D_{T ,ˆ1}b) + xˆ2vˆ2(D_{T ,ˆ2}b))| ≤ C(b)|υ|^{1+ \frac{1}{p}} .

Subtracting 3(xˆ1 +xˆ2) times the analog of (5.26) from and adding 6xˆ1xˆ2 times the analog of (5.25) to twice the analog of (5.27), we obtain

|(2(D^{(3)}γ̃_{T} (υ)) − 3(xˆ1 + xˆ2)(D^{(2)}γ̃_{T} (υ)))

− 3(xˆ1 − xˆ2)(v2 ˆ1(D_{T ,ˆ1}^{(2)}b) − v2 ˆ2(D_{T ,ˆ2}^{(2)}b))| ≤ C(b)|υ|^{2+ \frac{1}{p}} .

The estimates of Case (3) follow from the last two equations and Lemma 5.8. The finer bound on φ_{FS}_{⊥} of Lemma 5.8 is essential here.

5.5 The numbers n_{1}^{(2)}(μ) and n_{1}^{(3)}(μ) in the n = 3 case

In this subsection, we express the numbers n_{1}^{(2)}(μ) and n_{1}^{(3)}(μ) in the n = 3 case in terms of intersection numbers on the spaces \overline{V}_{1}(μ), \overline{V}_{2}(μ), \overline{V}_{3}(μ).

Lemma 5.11. If n=3, n_{1}^{(2)}(μ) = 4⟨2a + c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩−2|S_{2}(μ)|.

Proof. (1) We continue with the notation of the previous subsection. The number n_{1}^{(2)}(μ) is the number of zeros of the affine map in (4.31). As in the proof of Lemma 5.3, we can replace s_{Σ}^{(2,−)} by s_{Σ}^{(2,−)}. Since the linear part of the new affine map does not vanish on Σ × S_{1}(μ) (see Subsection 4.4), by Lemma 3.14,

n_{1}^{(2)}(μ) = ∑_{k=0}^{k=2} ⟨c_{1}^{2−k}(T̃Σ^{∗} ⊗ L^{∗⊗2})c_{k}(O), [Σ × \overline{S}_{1}(μ)]⟩ − C_{Σ×∂\overline{S}_{1}}(α^{⊥}) (5.30)
= 4⟨2a + c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ − C_{Σ×∂\overline{S}_{1}}(α^{⊥}),

where O = H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{3}, ∂\overline{S}_{1} = \overline{S}_{1}(μ) − S_{1}(μ), and α is the linear part of the affine map in (4.31), with s_{Σ}^{(2,−)} replaced by s_{Σ}^{(2,−)}.

(2) If T = (S^{2}, [N], I; j, \underline{d}) < T^{∗} and S_{T} (μ) ≠ ∅, T must have one of the three forms given by Lemma 5.10. Since D^{(2)} does not vanish on S_{T} (μ) in Case (1) of Lemma 5.10, C_{Σ×S_{T} (μ)}(α^{⊥}) = 0 in this case. In Case (2), i.e., T = T^{∗}(l) for some l ∈ [N], D_{T ,ˆ1}^{(2)} does not vanish over S_{T} (μ); see Subsection 4.5. Thus, by Corollaries 3.13, 3.6, and the first estimate of Lemma 5.10, C_{Σ×S_{T} (μ)}(α^{⊥}) is twice the number of Lemma 3.14 corresponding to

\overline{M} = Σ×S_{T} (μ), E_{2} = FT^{⊗2} ≈ ℂ, O_{2} = T̃Σ^{∗} ⊗L^{∗} ⊗O^{⊥} ≈ T̃Σ^{∗} ⊗O^{⊥},

and α_{2} ∈ Γ(\overline{M}; E_{2}^{∗} ⊗ O_{2}) that has full rank on every fiber. It follows that

C_{Σ×S_{T} (μ)}(α^{⊥}) = 2⟨c_{1}(O_{2}) − c_{1}(E_{2}), [Σ × S_{T} (μ)]⟩ = 4|S_{T} (μ)|. (5.31)

(3) Suppose T is as in Case (3) of Lemma 5.10. Since xˆ1 ≠ xˆ2, D_{T ,ˆ1} and D_{T ,ˆ2} do not vanish on S_{T} (μ) = U_{T} (μ) ∩ S_{2}(μ) (see Subsection 4.6), and D_{T ,ˆ1} + D_{T ,ˆ2} vanishes on FS, xˆ1D_{T ,ˆ1} + xˆ2D_{T ,ˆ2} does not vanish on FS. Thus, the third estimate of Lemma 5.10, Corollary 3.13, and Lemma 3.14,

C_{Σ×S_{T} (μ)}(α^{⊥}) = ⟨c_{1}(O_{2}) − c_{1}(E), [Σ × S_{T} (μ)]⟩ = 2|S_{T} (μ)|. (5.32)

Summing up Equations (5.31) and (5.32) over all appropriate bubble types T < T^{∗} and substituting the result into (5.30), we obtain the claim.

Lemma 5.12. If n=3, n_{1}^{(3)}(μ) = ⟨4a + 5c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩−3|S_{2}(μ)|.

Proof. (1) We continue with the notation of Lemma 5.11. The number n_{1}^{(3)}(μ) is the number of zeros of the affine map in (4.33). Let

E = L^{⊗2} ⊕ L^{⊗3} ⟶ \overline{S}_{1}(μ).

Since the linear part α of the affine map has full rank on S_{1}(μ) (see Subsection 4.5),

(5.33)  n_{1}^{(3)}(μ) = ∑_{k=0}^{k=2} ⟨λ_{E}^{2−k}c_{k}(O), [ℙE]⟩ − C_{ℙE|∂\overline{S}_{1}}(α_{E}^{⊥})
= ⟨4a + 5c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ − C_{ℙE|∂\overline{S}_{1}}(α_{E}^{⊥}),

where O = ev^{∗}T ℙ^{3}.

(2) As in the proof of Lemma 5.11, C_{ℙE|S_{T} (μ)}(α_{E}^{⊥}) = 0 for bubble types T of Case (1) of Lemma 5.10. Suppose T = T^{∗}(l) for some l ∈ [N], i.e., we are in Case (2) of Lemma 5.10. The normal bundle of ℙE|_{S_{T} (μ)} in ℙE is π_{E}^{∗}FT ≈ ℂ. By the first two estimates of Lemma 5.10, with appropriate identifications,

|α_{E}^{⊥}(γ_{S}(b, vˆ1)) − α̃_{T} (b, vˆ1)| ≤ C|vˆ1|^{2+ \frac{1}{p}} ∀(b, vˆ1) ∈ FT_{δ},

for some α̃_{T} ∈ Γ(ℙE|_{S_{T} (μ)}; FT^{∗⊗2} ⊗ γ_{E}^{∗} ⊗ O^{⊥}) which vanishes only on

Z_{T} ≡ {(b, [v, w]) ∈ ℙE|_{S_{T} (μ)} : v − 3xˆ1w = 0}.

Thus, by Corollaries 3.13 and 3.6 and Lemma 3.14,

C_{ℙE|(S_{T} (μ)−Z_{T} )}(α_{E}^{⊥}) = 2(⟨c_{1}(γ_{E}^{∗} ⊗ O^{⊥}) − c_{1}(FT), [ℙE|_{S_{T} (μ)}]⟩
− C_{Z_{T}} (α_{T}^{⊥}))
= 4|S_{T} (μ)| − 2C_{Z_{T}} (α_{T}^{⊥}).

By the first two estimates of Lemma 5.10, C_{Z_{T}} (α_{T}^{⊥}) = |Z_{T} | = |S_{T} (μ)|. Since the images of D_{T ,ˆ1}^{(2)} and D_{T ,ˆ1}^{(3)} are linearly independent in every fiber of ev^{∗}T ℙ^{3} over S_{T} (μ), by the first two estimates of Lemma 5.10 and Corollary 3.13, C_{Z_{T}} (α_{E}^{⊥}) = 3|Z_{T} |. Thus,

C_{ℙE|S_{T} (μ)}(α_{E}^{⊥}) = (4|S_{T} (μ)| − 2|S_{T} (μ)|) + 3|S_{T} (μ)| = 5|S_{T} (μ)|. (5.34)

(3) Suppose T is as in Case (3). By the last two estimates of Lemma 5.10, with appropriate identifications,

|α_{E}^{⊥}(γ_{S}(b, v)) − α̃_{T} (b, v)| ≤ C|vˆ1|^{1+ \frac{1}{p}} ∀(b, v) ∈ F S_{δ},

for some α̃_{T} ∈ Γ(ℙE|_{S_{T} (μ)}; F S^{∗} ⊗ γ_{E}^{∗} ⊗ O^{⊥}) which vanishes only on

Z_{T} ≡ {(b, [v, w]) ∈ ℙE|_{S_{T} (μ)} : 2v − 3(xˆ1 + xˆ2)w = 0}.

Thus, by Corollary 3.13 and Lemma 3.14,

C_{ℙE|(S_{T} (μ)−Z_{T} )}(α_{E}^{⊥}) = ⟨c_{1}(γ_{E}^{∗} ⊗ O^{⊥}) − c_{1}(F S), [ℙE|_{S_{T} (μ)}]⟩ − C_{Z_{T}} (α_{T}^{⊥})
= 2|S_{T} (μ)| − C_{Z_{T}} (α_{T}^{⊥}).

By the last two estimates of Lemma 5.10, C_{Z_{T}} (α_{T}^{⊥}) = |Z_{T} |. Finally, by Lemma 5.10 and Corollary 3.13, C_{Z_{T}} (α_{E}^{⊥}) = 2|Z_{T} |. Thus,

C_{ℙE|S_{T} (μ)}(α^{⊥}) = (2|S_{T} (μ)| − |S_{T} (μ)|) + 2|S_{T} (μ)| = 3|S_{T} (μ)|. (5.35)

The claim follows by summing up Equations (5.34) and (5.35) over the appropriate equivalence classes of bubble types T < T^{∗} and plugging the result back into (5.33).

The next step is to relate ⟨a, [\overline{S}_{1}(μ)]⟩ and ⟨c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ to intersection numbers on the spaces \overline{V}_{1}(μ), \overline{V}_{2}(μ), and \overline{V}_{3}(μ). The approach is similar to the proof of Lemma 5.4, but first we need to interpret ⟨a, [\overline{S}_{1}(μ)]⟩ and ⟨c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ as the zero sets of some bundle sections. In our case, the spaces \overline{U}_{T^{∗}}(μ) and \overline{U}_{T^{∗}(l)}(μ) for all l ∈ [N] are topological manifolds (not just orbifolds). Thus, c_{1}(L^{∗}) represents the first chern class of some line bundle L^{∗} ⟶ \overline{U}_{T} (μ). It is well-known in algebraic geometry that a slightly weaker statement is in fact true for any choice of constraints, and

L^{∗} = L^{∗} ⊗ O (− ∑_{l∈[N]} \overline{U}_{T^{∗}(l)}) .

Let V_{1} = ev^{∗}O(1) ⟶ \overline{U}_{T^{∗}}(μ), V_{2} = L^{∗} ⟶ \overline{U}_{T^{∗}}(μ), and η_{i} = c_{1}(V_{i}). Choose sections s_{i} ∈ Γ(\overline{U}_{T^{∗}}(μ); V_{i}) such that s_{i} is smooth and transversal to the zero set on all smooth strata U_{T} (μ) ⊂ \overline{U}_{T^{∗}}(μ) and on S_{T} (μ) ⊂ \overline{S}_{1}(μ). The second condition implies that s_{i} does not vanish on the finite set ∂\overline{S}_{1}.

Lemma 5.13. If d ≥ 1, μ is a tuple of p points and q lines in general position in ℙ^{3} with 2p + q = 4d − 3,

⟨a, [\overline{S}_{1}(μ)]⟩ = ⟨6a^{3}c_{1}(L^{∗}) + 4a^{2}c_{1}^{2}(L^{∗}) + ac_{1}^{3}(L^{∗}), [\overline{V}_{1}(μ)]⟩
− ⟨4a^{2} + a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})), [\overline{V}_{2}(μ)]⟩;
⟨c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ = ⟨4a^{3}c_{1}(L^{∗}) + 6a^{2}c_{1}^{2}(L^{∗})
+ 4ac_{1}^{3}(L^{∗}) + c_{1}^{4}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − τ_{3}(μ).

Proof. (1) Similarly to the proof of Lemma 5.4,

⟨η_{i}, [\overline{S}_{1}(μ)]⟩ = ⟨η_{i}c_{3}(L^{∗} ⊗ ev^{∗}T ℙ^{3}), [\overline{V}_{1}(μ)]⟩ − C_{∂\overline{V}_{1}(μ)}(D ⊕ s_{i}). (5.36)

Suppose T = (S^{2}, [N], I; j, \underline{d}) < T^{∗} is a bubble type such that U_{T} (μ) ≠ ∅. If dˆ0 ≠ 0, by our assumptions on s_{i}, D ⊕ s_{i} does not vanish on U_{T} (μ). Thus, for the purposes of computing C_{∂\overline{V}_{1}(μ)}(D ⊕ s_{i}), we can assume dˆ0 = 0.

(2) In order to compute the numbers C_{U_{T} (μ)}(D ⊕ s_{i}), we slightly modify the approach of Subsection 3.2, since we have a great amount of flexibility in choosing the section s_{i}. We consider a family ψ_{t} = (tν + D, s_{i}) of sections of L^{∗} ⊗ ev^{∗}T ℙ^{3} ⊕ V_{i}, with ν generic with respect to D. Let π : FT ⟶ U_{T} (μ) be the bundle projection map and fix an identification of γ_{T}^{μ∗}V_{i} ⟶ FT_{δ} with π^{∗}V_{i}. It can be assumed that the section s_{i} has been chosen so that γ_{T}^{μ∗}s_{i} ∈ Γ(FT_{δ}; π^{∗}V_{i}) is constant on the fibers of FT_{δ} over an open subset K_{T} of U_{T} (μ) that contains all of the finitely many zeros of the affine map

FT ⟶ L^{∗} ⊗ ev^{∗}T ℙ^{3} ⊕ V_{i}, (b, υ) ⟶ (\overline{ν}_{b} + α_{T} (ρ_{T} (υ)), s_{i}(b)),

over \overline{U}_{T} (μ), where α_{T} and ρ_{T} are as in (2.21). Note that by our assumptions on s_{i}, the images of {D_{T ,h} : h ∈ χ(T )} are linearly independent in every fiber of ev^{∗}T ℙ^{n} over s_{i}^{−1}(0). Thus by Theorem 2.8, Corollary 3.13, and Lemma 3.2, C_{U_{T} (μ)}(D ⊕ s_{i}) = 0 if Hˆ0T ≠ Î. Furthermore, if Hˆ0T = Î, C_{U_{T} (μ)}(D ⊕s_{i}) is the number of zeros of the affine map

FT ⟶ L^{∗} ⊗ ev^{∗}T ℙ^{3}, υ = (b, v) ⟶ \overline{ν}_{T ,b} + α_{T} (υ), (5.37)

over s_{i}^{−1}(0) ∩ \overline{U}_{T} (μ), where \overline{ν}_{T} ∈ Γ(\overline{U}_{T} (μ); L^{∗} ⊗ ev^{∗}T ℙ^{3}) is a generic section. Thus, by Lemma 3.14,

C_{U_{T} (μ)}(D ⊕ s_{i}) = ∑_{k=0}^{k=2} ⟨λ_{FT}^{2−k}c_{k}(L^{∗} ⊗ ev^{∗}T ℙ^{3}), [ℙFT |_{s_{i}^{−1}(0)∩\overline{U}_{T} (μ)}]⟩ (5.38)

− C_{ℙFT |s_{i}^{−1}(0)∩∂\overline{U}_{T}} (α_{FT}^{⊥}),

= ∑_{k=0}^{k=2} ⟨λ_{FT}^{2−k}c_{k}(L^{∗} ⊗ ev^{∗}T ℙ^{3})η_{i}, [ℙFT]⟩

− C_{ℙFT |s_{i}^{−1}(0)∩∂\overline{U}_{T}} (α_{FT}^{⊥}),

where ∂\overline{U}_{T} = \overline{U}_{T} (μ) − U_{T} (μ) and α_{FT} ∈ Γ(ℙFT; γ_{FT}^{∗} ⊗ L^{∗} ⊗ ev^{∗}T ℙ^{3}) is the section induced by α_{T}.

(3) Suppose i = 1, i.e., η_{i} = a. If T = T^{∗}(l) and T′ = (S^{2}, [N], I′; j, \underline{d}′) < T is a bubble type such that s_{1}^{−1}(0) ∩ U_{T′}(μ) ∩ α_{T}^{−1}(0) ≠ ∅, T′ must have the form

|I′ − I| = 2, Hˆ1T′ = {ˆ2, ˆ3}, d′ ; ˆ1 = 0, d′ ; ˆ2 ≠ 0, d′ ; ˆ3 ≠ 0.

By Theorem 2.8 applied to \overline{T}′ < \overline{T} , and Corollary 3.13,

C_{s_{1}^{−1}(0)∩U_{T ′}(μ)}(α_{FT}^{⊥}) = |U_{T′}(μ) ∩ s_{1}^{−1}(0)| = ⟨a, [\overline{U}_{T′}(μ)]⟩.

Thus, summing up Equation (5.38) over T = T^{∗}(l) with l ∈ [N], we obtain

∑_{l∈[N]} C_{U_{T ^{∗}(l)}(μ)}(D ⊕ s_{1}) (5.39)

= ∑_{l∈[N]} ⟨6a^{3} + 4a^{2}c_{1}(L∗ ; ˆ1T^{∗}(l)) + ac_{1}^{2}(L∗ ; ˆ1T^{∗}(l)), [\overline{U}_{T^{∗}(l)}(μ)]⟩ − τ_{2}^{(1)}(μ),

where τ_{2}^{(1)} (μ) is the number of two-component connected degree-d curves passing through the constraints with the node at the intersection of one of the constraints with a generic plane in ℙ^{3}. If |Hˆ0T | = |Î| = 2, |Mˆ0T | = 0, and T′ is as above, up to equivalence of bubble types,

|I′ − I| = 1, ι′ ; ˆ3 = ˆ1, d′ ; ˆ1 = 0, d′ ; ˆ2 ≠ 0, d′ ; ˆ3 ≠ 0,

i.e., \overline{T}′ = \overline{T} (l) for some l ∈ [N]. By Theorem 2.8 applied to \overline{T}′ < \overline{T} and Corollary 3.13,

C_{ℙFT |s_{1}^{−1}(0)∩U_{T ′}(μ)}(α_{FT}^{⊥}) = |U_{T′}(μ) ∩ s_{1}^{−1}(0)| = ⟨a, [\overline{U}_{T′}(μ)]⟩.

Thus, summing up Equation (5.38) over T with |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 0, we obtain

∑_{|Hˆ0T |=|Î|=2,|Mˆ0T |=0}⟨a(4a + c_{1}(L∗ ; ˆ1T ) + c_{1}(L∗ ; ˆ2T )), [\overline{U}_{T} (μ)]⟩ − 2τ_{2}^{(1)}(μ) (5.40)
= ⟨4a^{2} + a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})), [\overline{V}_{2}(μ)]⟩.

If |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 1, α_{T} has full rank on all of \overline{U}_{T} (μ). Thus, by Corollary 3.13,

C_{U_{T} (μ)}(D ⊕ s_{i}) = ⟨c_{1}(L^{∗} ⊗ ev^{∗}T ℙ^{3}) − c_{1}(FT), [\overline{U}_{T} (μ) ∩ s_{1}^{−1}(0)]⟩
= |U\overline{T} (μ)|.

Here we used FT = L^{∗} ⊗ (Lˆ1 ⊕ Lˆ2) ≈ L^{∗} ⊕ L^{∗} and Corollary 5.22. Thus, summing up Equation (5.38) over T with |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 1 gives

∑_{|Hˆ0T |=|Î|=2,|Mˆ0T |=1} C_{U_{T} (μ)}(D ⊕ s_{i}) = τ_{2}^{(1)}(μ). (5.41)

Finally, if |Hˆ0T | = |Î| = 3, η_{1}|_{\overline{U}′_{T} (μ)} = 0. The first claim follows by plugging the sum of Equations (5.39)–(5.41) into (5.36). See also Equations (5.3) and (5.4).

(4) Suppose η_{i} = c_{1}(L^{∗}). We continue as in (3) above. If T = T^{∗}(l), α_{T} does not vanish anywhere on s_{2}^{−1}(0) ∩ \overline{U}_{T} (μ). Thus, by Corollary 3.13,

∑_{l∈[N]}C_{U_{T ^{∗}(l)}(μ)}(D ⊕ s_{2}) (5.42)

= ∑_{l∈[N]} ⟨c(L^{∗} ⊗ ev^{∗}T ℙ^{3})c(Lˆ1T )^{−1}, [\overline{U}_{T^{∗}(l)}(μ) ∩ s_{2}^{−1}(0)]⟩

= ∑_{l∈[N]} ⟨c_{1}(L^{∗})(6a^{2} + 4ac_{1}(L∗ ; ˆ1T^{∗}(l)) + c_{1}^{2}(L∗ ; ˆ1T^{∗}(l))), [\overline{U}_{T^{∗}(l)}(μ)]⟩.

If |Hˆ0T | = |Î| = 3, α_{T} again does not vanish anywhere on s_{2}^{−1}(0) ∩ \overline{U}_{T} (μ), and thus

C_{U_{T} (μ)}(D ⊕ s_{2}) = |\overline{U}_{T} (μ) ∩ s_{2}^{−1}(0)| = ⟨c_{1}(L^{∗}), [\overline{U}_{T} (μ)]⟩ = |\overline{U}\overline{T} (μ)|. (5.43)

Here we used Corollary 5.22 again. Note that if |Hˆ0T | = |Î| = 2, η_{2}|_{\overline{U}_{T} (μ)} = 0. This is immediate in the case |Mˆ0T | = 0 and follows from Corollary 5.22 and (5.3) in the case |Mˆ0T | = 1. The second claim of the lemma is obtained by summing (5.43) over all equivalence classes of bubble types T < T^{∗} with |Hˆ0T | = |Î| = 3, and plugging the result along with (5.42) into (5.36). Note that

a^{3}|_{\overline{U}_{T ^{∗}(l)}(μ)} = 0 ∀ l ∈ [N] ⟹

⟨4a^{3}c_{1}(L^{∗}), [\overline{V}_{1}(μ)]⟩ = ⟨4a^{3}c_{1}(L^{∗}), [\overline{V}_{1}(μ)]⟩.

5.6 The number n_{1}^{(1)}(μ) in the n = 3 case

We finally compute the remaining number n_{1}^{(1)}(μ). The computation parallels the proof of Lemma 5.6.

Lemma 5.14. If n = 3,

n_{1}^{(1)}(μ) = 4⟨10a^{3}c_{1}(L^{∗}) + 3a^{2}c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − 12τ_{2}^{(2)}(μ),

where τ_{2}^{(2)}(μ) denotes the number of two-component connected degree-d curves that pass through the constraints and with the node on a generic line in ℙ^{3}.

Proof. (1) We use the same notation as in the proof of Lemma 5.6. By Equation (4.30) and Lemma 3.14,

n_{1}^{(1)}(μ) = ∑_{k=0}^{k=5} ⟨c_{k}(O)c_{1}^{5−k}(T^{∗}Σ ⊗ L^{∗}), [Σ × \overline{U}_{T^{∗}}(μ)]⟩ − C_{Σ×D^{−1}(0)}(α^{⊥}), (5.44)
= 2⟨112a^{3}c_{1}(L^{∗}) + 84a^{2}c_{1}^{2}(L^{∗}) + 32ac_{1}^{3}(L^{∗})
+ 5c_{1}^{4}(L^{∗}), [\overline{U}_{T^{∗}}(μ)]⟩ − C_{Σ×D^{−1}(0)}(α^{⊥}),

where O = H_{Σ}^{0,1} ⊗ ev^{∗}T ℙ^{3} and α ∈ Γ(Σ × \overline{U}_{T^{∗}}(μ); T^{∗}Σ ⊗ L^{∗} ⊗ O) is the linear part of the affine map ψ_{1}^{(1)} of (4.30). Let O_{2} = T^{∗}Σ ⊗ L^{∗} ⊗ O^{⊥}.

(2) Similarly to (2) of the proof of Lemma 5.4,

C_{Σ×S_{1}(μ)}(α^{⊥}) = ⟨e(T^{∗}Σ ⊗ L^{∗} ⊗ (H_{Σ}^{−} ⊗ ev^{∗}T ℙ^{3}/ℂ)), [Σ × \overline{S}_{1}(μ)]⟩ (5.45)
= 2⟨12a + 5c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩.

Suppose T = (S^{2}, [N], I; j, \underline{d}) < T^{∗} is a bubble type such that S_{T} (μ) ≠ ∅. By Lemma 5.10, there are three possibilities for the structure of T , but C_{Σ×D^{−1}(0)}(α^{⊥}) = 0 in all three cases. This claim follows from Corollaries 5.9 and 3.13 and Lemma 3.2.

(3) As before, if T < T^{∗} and C_{Σ×(U_{T} (μ)−S_{T} (μ))}(α^{⊥}) ≠ 0, dˆ0 = 0 and Hˆ0T = Î. In such a case,

C_{Σ×(U_{T} (μ)−S_{T} (μ))}(α^{⊥}) = ∑_{k=0}^{k=4} ⟨λ_{FT}^{4−k}c_{k}(O_{2}), [ℙFT ]⟩ − C_{α_{FT}^{−1} (0)}(α_{FT}^{⊥}), (5.46)

where α̃_{FT} ∈ Γ(ℙFT; γ_{FT}^{∗} ⊗ O_{2}) is the section induced by the section

π^{⊥} ∘ (α_{T} ∘ s_{Σ}) ∈ Γ(Σ × \overline{U}_{T} (μ); FT^{∗} ⊗ O_{2}).

(4) If T = T^{∗}(l) for some l ∈ [N], F T ≈ Lˆ1T over \overline{U}_{T} (μ), and α_{T}^{−1}(0) = Σ × D_{T ,ˆ1}^{−1}(0). Thus, by (5.46),

C_{Σ×(U_{T} (μ)−S_{T} (μ))}(α^{⊥}) = 2⟨112a^{3} + 84a^{2}c_{1}(L∗ ; ˆ1T ) + 32ac_{1}^{2}(L∗ ; ˆ1T )

+ 5c_{1}^{3}(L∗ ; ˆ1T ), [\overline{U}_{T} (μ)]⟩

− C_{Σ×D_{T ,ˆ1}^{−1}(0)}(α_{FT}^{⊥}).

By Corollaries 6.3 and 3.13,

C_{Σ×S_{T} (μ)}(α_{FT}^{⊥}) = ⟨c(FT^{∗} ⊗ O_{2}^{⊥})c(L_{T ,ˆ1}^{∗} ⊗ ev^{∗}T ℙ^{3})^{−1}, [Σ × S_{T} (μ)]⟩
= 10|S_{T} (μ)|.

On the other hand, if T′ = (S^{2}, [N], I′; j′, \underline{d}′) < T , we apply Theorem 2.8 to \overline{T}′ < \overline{T} . Then for the same reason as before, C_{Σ×U_{T ′}} (α_{FT}^{⊥}) = 0 unless dˆ1 = 0 and Hˆ1T′ = I′ − I, i.e.,

Î′ = {ˆ1, ˆ2, ˆ3}, ι′ ; ˆ2 = ˆ1, ι′ ; ˆ3 = ˆ1, d′ ; ˆ1 = 0, d′ ; ˆ2 ≠ 0, d′ ; ˆ3 ≠ 0.

In such a case, with E = Lˆ2T′ ⊕ Lˆ3T′, by Corollary 3.13,

C_{Σ×U_{T ′}(μ)}(α_{FT}^{⊥}) = ⟨c(F T^{∗} ⊗ O_{2})c(E)^{−1}, [Σ × \overline{U}_{T′}(μ)]⟩

= ⟨32a + 5(c_{1}(L∗ ; ˆ2T′) + c_{1}(L∗ ; ˆ3T′)), [\overline{U}_{T′}(μ)]⟩.

Summing Equation (5.46) over T = T^{∗}(l), we thus obtain

∑_{l∈[N]} C_{Σ×(U_{T ^{∗}(l)}(μ)−S_{T ^{∗}(l)}(μ))}(α^{⊥}) (5.47)

= −64τ_{2}^{(1)}(μ) − 10⟨c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗}), [\overline{V}_{2,1}(μ)]⟩

+ 2 ∑_{l∈[N]} (⟨112a^{3} + 84a^{2}c_{1}(L∗ ; ˆ1) + 32ac_{1}^{2}(L∗ ; ˆ1) + 5c_{1}^{3}(L∗ ; ˆ1), [\overline{U}_{T^{∗}(l)}(μ)]⟩

− 5|\overline{U}_{T^{∗}(l)} ∩ \overline{S}_{1}(μ)|),

where Lˆ1 = Lˆ1T^{∗}(l), \overline{V}_{2,1}(μ) = ⋃_{l∈[N]} \overline{V}_{2,1;l}(μ), and \overline{V}_{2,1;l}(μ) denotes

the union of the spaces \overline{U}_{T} (μ) taken over all equivalence classes of basic bubble types T = (S^{2}, [N]−\{l\}, \{ˆ1, ˆ2\}; j, \underline{d}) with dˆ1, dˆ2 > 0 and dˆ1+dˆ2 = d.

(5) If |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 0, FT ≈ Lˆ1T ⊕ Lˆ2T over \overline{U}_{T} (μ) and α_{FT}^{−1} (0) consists of a section Z_{T} of ℙF T over Σ × S_{T} (μ) and the spaces Σ × \overline{U}_{T′}(μ), with \overline{T}′ corresponding to the bubble types T described in (2) in the proof of Lemma 5.5. By Corollaries 6.3 and 3.13,

C_{Σ×S_{T} (μ)}(α_{FT}^{⊥}) = ⟨c(γ_{FT}^{∗} ⊗ O_{2}^{⊥})c(ℂ^{3})^{−1}, [Σ × S_{T} (μ)]⟩ = 10|S_{T} (μ)|.

On the other hand, if T′ =(S^{2}, [N], I′; j′, \underline{d}′)<T and C_{ℙFT |Σ×U′_{T} (μ)}(α_{FT}^{⊥}) ≠ 0,

|I′ − I| ∈ {1, 2}, Hˆ2T = I′ − I, d′ ; ˆ2 = 0, d′_{h} ≠ 0 if h ∈ Î′ − {ˆ2}.

If |I′ − I| = 1, by Corollary 3.13,

C_{ℙFT |Σ×U′_{T} (μ)}(α_{FT}^{⊥}) = ⟨c(O_{2})c(Lˆ1T′ ⊕ Lˆ3T′)^{−1}, [Σ × \overline{U}_{T′}(μ)]⟩

= 2⟨32a + 5(c_{1}(L∗ ; ˆ1T′) + c_{1}(L∗ ; ˆ3T′)), [\overline{U}_{T′}(μ)]⟩;

see the proof of Lemma 5.13 for more details. If |I′ − I| = 2, by Corollary 3.13,

C_{ℙFT |Σ×U′_{T} (μ)}(α_{FT}^{⊥}) = ⟨c_{1}(O_{2}) − c_{1}(ℂ^{3}), [Σ × \overline{U}_{T′}(μ)]⟩ = 10|\overline{U}_{T′}(μ)|.

Thus, summing Equation (5.46) over T < T^{∗} with |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 0, we obtain

∑_{[T ]} C_{Σ×(U_{T} (μ)−S_{T} (μ))}(α^{⊥}) (5.48)

= −30τ_{3}(μ) − 10|S_{2}(μ)| + 2⟨84a^{2} + 32a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗}))

+ 5(c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗})) + 5c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩.

(6) If |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 1, α̃_{FT} does not vanish on Σ × \overline{U}\overline{T} (μ). Thus, by Corollary 3.13,

C_{Σ×U_{T} (μ)}(α^{⊥}) = ⟨c(O_{2})c(L^{∗} ⊗ (Lˆ1T ⊕ Lˆ2T ))^{−1}, [Σ × \overline{U}_{T} (μ)]⟩

= 2⟨32a + 5(c_{1}(L∗ ; ˆ1T ) + c_{1}(L∗ ; ˆ2T )), [\overline{U}\overline{T} (μ)]⟩.

Here we used the decomposition (5.2) and Corollary 5.22. Summing Equation (5.46) over T < T^{∗} with |Hˆ0T | = |Î| = 2 and |Mˆ0T | = 1, we obtain

∑_{[T ]} C_{Σ×U_{T} (μ)}(α^{⊥}) = 64τ_{2}^{(1)}(μ) + 10⟨c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗}), [\overline{V}_{2,1}(μ)]⟩. (5.49)

(7) Finally, if |Hˆ0T | = |Î| = 3, FT ≈ L^{∗} ⊕ L^{∗} ⊕ L^{∗} over \overline{U}_{T} (μ), and α̃_{FT} again does not vanish over Σ × \overline{U}\overline{T} (μ). Then by Corollary 3.13,

C_{Σ×U_{T} (μ)}(α^{⊥}) = ⟨c(O_{2})c(L^{∗} ⊕ L^{∗} ⊕ L^{∗})^{−1}, [Σ × \overline{U}_{T} (μ)]⟩ = 10|\overline{U}\overline{T} (μ)|.

Thus, summing Equation (5.46) over T < T^{∗} with |Hˆ0T | = |Î| = 3, we obtain

∑_{|Hˆ0T |=3} C_{Σ×U_{T} (μ)}(α^{⊥}) = 10τ_{3}(μ). (5.50)

From Equations (5.44), (5.45), and (5.47)–(5.50), we conclude that

n_{1}^{(1)}(μ) = 2⟨112a^{3}c_{1}(L^{∗}) + 84a^{2}c_{1}^{2}(L^{∗}) + 32ac_{1}^{3}(L^{∗}) + 5c_{1}^{4}(L^{∗}), [\overline{V}_{1}(μ)]⟩ (5.51)
− 2⟨84a^{2} + 32a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗})) + 5(c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗}))
+ 5c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩
− 2⟨12a + 5c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩ + 10|S_{2}(μ)| + 20τ_{3}(μ).

The claim follows by using Lemma 5.5 and 5.13.

5.7 Computation of Chern classes

In this subsection, we show that all intersection numbers of the spaces \overline{V}_{k}(μ) involving powers of a and powers of c_{1}(L_{i}^{∗}) are computable. We can then conclude that the numbers n_{m}^{(k)}(μ) are computable. The computability of intersection numbers of tautological classes of \overline{V}_{k}(μ), which include a and c_{1}(L_{i}^{∗}), has been shown in [9]. For the sake of completeness, a slightly different approach is presented below.

If dˆ0 and dˆ1 are nonnegative integers and μ is an N-tuple of any generic constraints in ℙ^{n}, let \overline{M}_{(dˆ0,dˆ1)}(μ) denote the union of the spaces \overline{U}_{T} (μ), where T is a simple bubble type of the form

T = (S^{2}, [N], {ˆ0, ˆ1}; j, {dˆ0, dˆ1}).

Then \overline{M}_{T ,(dˆ0,dˆ1)}(μ) is a complex codimension-one homology class in the space \overline{V}_{1}(μ) with d = dˆ0 + dˆ1. If d > 0, let

∑_{dˆ0+dˆ1=d}^{≥} f(dˆ0, dˆ1) = ∑_{dˆ0,dˆ1≥0^{dˆ0+dˆ1=d}} f(dˆ0, dˆ1),

∑_{dˆ0+dˆ1=d}^{>} f(dˆ0, dˆ1) = ∑_{dˆ0,dˆ1>0^{dˆ0+dˆ1=d}} f(dˆ0, dˆ1),

whenever f is any function defined on the appropriate subset of ℤ × ℤ.

Lemma 5.15. Let T^{∗} = (S^{2}, [N], {ˆ0}; ˆ0, d) be a bubble type with d > 0. Then in H^{∗}(\overline{U}_{T^{∗}}(μ)),

c_{1}(L^{∗}) = \frac{1}{d^{2}} (H − 2da + ∑_{dˆ0+dˆ1=d}^{≥} d2 ˆ1\overline{M}_{(dˆ0,dˆ1)}(μ)) ,

where H denotes the subset of elements in \overline{U}_{T^{∗}}(μ) that pass through a generic codimension-two linear subspace of ℙ^{n}.

Proof. (1) We restate the proof of [4] in terms of the line bundle L^{∗⊗d^{2}} ⟶ \overline{U}_{T^{∗}}(μ), instead of passing to a cover of \overline{U}_{T^{∗}}(μ). Define a section ψ ∈ Γ(\overline{U}_{T^{∗}}(μ); L^{∗⊗d^{2}}) as follows. Let H_{0} and H_{1} be two fixed hyperplanes in ℙ^{n}, generic with respect to the constraints μ_{1}, … μ_{N}. Suppose

[b] = [(S^{2}, [N], ˆ0; , (ˆ0, y), uˆ0)] ∈ U_{T^{∗}}(μ)

is such that uˆ0 is transversal to H_{0} and H_{1}. Then,

u−1 ; ˆ0(H_{i}) = {[x_{1}^{(i)}, y_{1}^{[i]}], …, [x_{d}^{(i)}, y_{d}^{[i]}]}, i = 0, 1,

for some [x_{k}^{(i)}, y_{k}^{[i]}] ∈ ℙ^{1}. Define ψ([b]) by

ψ([b, c]) = c^{d^{2}} ∏_{k,l∈[d]} (\frac{x_{k}^{(0)}}{y_{k}^{(0)}} − \frac{x_{l}^{(1)}}{y_{l}^{(1)}} ) . (5.52)

While this section could be infinite, it is well-defined, i.e., independent of the choice of a representative b ∈ B_{T^{∗}} for [b]. With an appropriate coordinate change on ℂ^{n+1}, it can be assumed that H_{i} = {X_{i} = 0}. The map uˆ0 corresponds to (n + 1) homogeneous polynomials of degree d: p_{0}, …, p_{n}. Since the right-hand side of (5.52) is symmetric in the roots of p_{0} and separately in the roots of p_{1}, ψ is a rational function in the coefficients of p_{0} and p_{1}. Thus, ψ extends over all of U_{T^{∗}}(μ). Furthermore, this section extends by zero over \overline{U}_{T^{∗}}(μ) − U_{T^{∗}}(μ).

(2) We now identify the zero set of the section ψ. From Equation (5.52), it is clear that ψ vanishes with multiplicity one if p_{0} and p_{1} have a common root, i.e., if uˆ0 passes through H_{0} ∩ H_{1}. The section ψ also has a pole of order d along the sets of maps

X_{0} = {b : y_{k}^{(0)}(b) = 0 for a unique k ∈ [d], p_{1}(1, 0) ≠ 0},
X_{1} = {b : y_{k}^{(1)}(b) = 0 for a unique k ∈ [d], p_{0}(1, 0) ≠ 0}.

Note that \overline{X}_{i} = ev^{−1}(H_{i}). Finally, while ψ vanishes outside of U_{T^{∗}}(μ), \overline{U}_{T} (μ) has (complex) codimension one in \overline{U}_{T^{∗}}(μ) if and only if T < T^{∗} is a two-bubble strata, i.e., as described just before the statement of the lemma. Let dˆ0 and dˆ1 be the corresponding degrees. It follows from Equation (5.52) that ψ has a zero of order d2 ˆ1 along an open subset of U_{T} (μ). Thus, we obtain

c_{1}(L^{∗⊗d^{2}}) = H − 2da + ∑_{dˆ0+dˆ1=d}^{≥} d2 ˆ1\overline{M}_{(dˆ0,dˆ1)}(μ).

Corollary 5.16. With notation as in Lemma 5.15,

c_{1}(L^{∗}) = \frac{1}{d^{2}} (H − 2da + ∑_{dˆ0+dˆ1=d}^{>} d2 ˆ1\overline{M}_{(dˆ0,dˆ1)}(μ)) .

Proof. This is immediate from Lemma 5.15 and (5.3).

If T = (S^{2}, [N], I; j, \underline{d}) is any bubble type, let Tˆ0 = (S^{2}, Mˆ0T ⊔ Hˆ0T , {ˆ0}; ˆ0, dˆ0). Denote by T_{k} for k ∈ Hˆ0T the simple bubble types corresponding to \overline{T} . Then,

(5.53) \overline{U}_{T} (μ) = \overline{U}_{Tˆ0}(μ) ×_{∏_{k∈Hˆ0T} (ev_{k}×ev)} ∏_{k∈Hˆ0T} \overline{U}_{T_{k}}(μ)

≡ \{(bˆ0, (b_{k})_{k∈Hˆ0T} ) ∈ \overline{U}_{Tˆ0}(μ)

× ∏_{k∈Hˆ0T} \overline{U}_{T_{k}}(μ) : ev_{k}(bˆ0) = ev(b_{h}) ∀k ∈ Hˆ0T \}.

Lemma 5.17. With notation as above, if T < T^{∗} and dˆ0 ≠ 0,

c_{1}(L^{∗}T^{∗})|_{\overline{U}_{T} (μ)} = \{c_{1}(L^{∗}Tˆ0)|_{\overline{U}_{Tˆ0}(μ)} +  ∑_{∅≠M_{0}⊂Mˆ0Tˆ0,M_{0}∩Hˆ0T ≠∅}\overline{U}_{Tˆ0(M_{0})}(μ)\}

×_{∏_{k∈Hˆ0T} (ev_{k}×ev)} ∏_{k∈Hˆ0T} \overline{U}_{T_{k}}(μ).

Proof. Since Lˆ0T^{∗}|_{\overline{U}_{T} (μ)} = Lˆ0T and \overline{U}_{T} (μ) ∩ \overline{U}_{T^{∗}(M_{0})}(μ) = ∅ unless M_{0} ⊂ Mˆ0T , by (5.3)

c_{1}(L^{∗}T^{∗})|_{\overline{U}_{T} (μ)} = c_{1}(L^{∗}T )|_{\overline{U}_{T} (μ)} − ∑_{∅≠M_{0}⊂Mˆ0T} \overline{U}_{T} (μ) · \overline{U}_{T^{∗}(M_{0})}(μ)

= c_{1}(L∗ ; ˆ0Tˆ0)|_{\overline{U}_{T} (μ)} − ∑_{∅≠M_{0}⊂Mˆ0T} \overline{U}_{T (M_{0})}(μ).

The claim follows by using Equation (5.3) again.

Corollary 5.18. All intersection numbers on \overline{V}_{k}(μ) involving only the powers of a and c_{1}(L_{k}^{∗}) are computable.

Proof. Corollary 5.16 and Lemma 5.17 reduce the computation of such numbers to understanding the restrictions c_{1}(L^{∗}Tˆ0)|_{\overline{U}_{Tˆ0(M_{0})}}, where M_{0} is a subset of Mˆ0Tˆ0 intersecting Hˆ0T . By (5.2),

\overline{U}_{Tˆ0(M_{0})} ≈ \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}} × \overline{U}_{Tˆ0/M_{0}}.

We express c_{1} (L^{∗} Tˆ0) |_{\overline{U}_{Tˆ0 (M_{0})}} in terms of cohomology classes on \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}}. By definition, LTˆ0|_{\overline{U}_{Tˆ0(M_{0})}} comes from a line bundle over \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}}. In fact,

c_{1}(L^{∗}Tˆ0)|_{\overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}}×\overline{U}_{Tˆ0/M_{0}}} = ψˆ0 × 1,

where ψˆ0 is the ψ-class of \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}} corresponding to the marked point ˆ0. Since L^{∗}Tˆ0|_{\overline{U}_{Tˆ0(M_{0})}} is L^{∗}Tˆ0(M_{0}),

c_{1}(L^{∗}Tˆ0)|_{\overline{U}_{Tˆ0(M_{0})}} = c_{1}(L^{∗}Tˆ0)|_{\overline{U}_{Tˆ0(M_{0})}} − ∑_{∅≠M′_{0}⊂Mˆ0T} \overline{U}_{Tˆ0(M′_{0})} · \overline{U}_{Tˆ0(M_{0})}

= ψˆ0 × 1 − ∑_{∅≠M′_{0}⊂(M_{0}−Hˆ0T )} \overline{U}_{Tˆ0(M′_{0};M_{0}−M′_{0})}

= ψ̃_{M_{0}−Hˆ0T} × 1|_{\overline{M}_{\{ˆ0,ˆ1\}⊔M_{0}}×\overline{U}_{Tˆ0/M_{0}}} , where Tˆ0(M′_{0}; M_{0} − M′_{0}) ≡ {Tˆ0(M_{0})}(M′_{0}) and for any proper subset J̃ of J we define the cohomology class ψ̃J̃ on \overline{M}_{0,\{ˆ0,ˆ1\}⊔J} by

ψ̃J̃ = ψˆ0 − ∑_{∅≠J′⊂J̃}\overline{M}_{0,(\{ˆ0\}⊔J′,\{ˆ1\}⊔(J−J′))}.

Here \overline{M}_{0,(\{ˆ0\}⊔J′,\{ˆ1\}⊔(J−J′))} is the closure in \overline{M}_{0,\{ˆ0,ˆ1\}⊔J} of the two-com-ponent strata such that the marked points on one of the components are {ˆ0} ⊔ J′. The numbers

χ(|J|, |J̃|) ≡ ⟨ψ̃|J|−1 ; J̃, [\overline{M}_{0,\{ˆ0,ˆ1\}⊔J}]⟩

are given in Corollary 5.20, which is a consequence of the following well-known lemma; see [9] for example.

Lemma 5.19.

(1) For any j^{∗} ∈ J, ψ̃_{J−\{j^{∗}\}} = 0 in H^{∗} (\overline{M}_{0,\{ˆ0,ˆ1\}⊔J}).

(2) If N \overline{M}_{0,(\{ˆ0\}⊔J′,\{ˆ1\}⊔(J−J′))} is the normal bundle of

\overline{M}_{0,(\{ˆ0\}⊔J′,\{ˆ1\}⊔(J−J′))} ≈ \overline{M}_{0,\{ˆ0,ˆ1\}⊔J′} × \overline{M}_{0,\{ˆ0,ˆ1\}⊔(J−J′)}

in \overline{M}_{0,\{ˆ0,ˆ1\}⊔J},

c_{1} (N \overline{M}_{0,(\{ˆ0\}⊔J′,\{ˆ1\}⊔(J−J′))}) = −ψˆ1 × 1 − 1 × ψˆ0.

Corollary 5.20. If m > 0, χ(m, 0) = 1. If m > k > 0, χ(m, k) = 0.

For our purposes, we can assume that the constraints μ_{1}, …, μ_{N} are disjoint. In the case of ℙ^{2}, the dimension of the space \overline{V}_{k}(μ) is at most 2. Thus, by a dimension count, if \overline{U}_{Tˆ0(M_{0})}(μ) is nonempty and appears in the computation of the intersection numbers of Corollary 5.18 via Lemma 5.17, then Hˆ0T consists of a single element and M_{0} = Hˆ0T . The corresponding moduli space \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}} is a single point and thus

⟨ψ̃_{M_{0}−Hˆ0T}^{|M_{0}|−1} , [\overline{M}_{\{ˆ0,ˆ1\}⊔M_{0}}]⟩ = 1.

In the case of ℙ^{3}, \overline{V}_{1}(μ) is four-dimensional, and we encounter two cases when \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}} is positive-dimensional. One possibility is that Hˆ0T is still a single-element set, but M_{0} contains one of the N marked points. In this case, by Corollary 5.20 or simply by the first statement of Lemma 5.19,

⟨ψ̃_{M_{0}−Hˆ0T}^{|M_{0}|−1} , [\overline{M}_{\{ˆ0,ˆ1\}⊔M_{0}}]⟩ = χ(2, 1) = 0.

In fact, we can replace the first statement of Lemma 5.19 with the direct computation of the degree ψˆ0 on \overline{M}_{0,4} given by Lemma 5.21 below. The other case when \overline{M}_{0,\{ˆ0,ˆ1\}⊔M_{0}} is positive-dimensional is M_{0} = Hˆ0T is a two-element set. Then

⟨ψ̃_{M_{0}−Hˆ0T}^{|M_{0}|−1} , [\overline{M}_{\{ˆ0,ˆ1\}⊔M_{0}}]⟩ = χ(2, 0) = 1.

Lemma 5.21. Let

\overline{M}_{0,4}^{(0)} = \{(y_{1}, y_{2}, y_{3}) ∈ ℂ^{3} : y_{1} + y_{2} + y_{3} = 0,

β(|y_{1}|) + β(|y_{2}|) + β(|y_{3}|) = \frac{1}{2}\}.

Then the action of S^{1} on \overline{M}_{0,4}^{(0)} induced from the standard action on ℂ is free,

\overline{M}_{0,4} = \overline{M}_{0,4}^{(0)}/S^{1} ≈ ℙ^{1},

and the line bundle associated to this quotient is the tautological line bundle over ℙ^{1}.

Proof. Identify \overline{M}_{0,4}^{(0)} with S^{3} ⊂ ℂ^{2} S^{1}-equivariantly by the map

(y_{1}, y_{2}, y_{3}) ⟶ \frac{(y_{1}, y_{2})}{|y_{1}| + |y_{2}|}.

Our assumptions on β imply that this map is a diffeomorphism; see Subsection 1.3.

Corollary 5.22. If T = (S^{2}, [3], \{ˆ0\}; ˆ0, 0), ⟨c_{1}(L^{∗}), [\overline{U}_{T}]⟩ = 1.

Remark. In [13], we extend the definition of \overline{M}_{0,4}^{(0)} of Corollary 5.21

to construct spaces \overline{M}_{T}^{(0)} for all bubble types T .

5.8 The final formulas

We finally put everything together to arrive at formulas for the numbers n_{2,d}(μ) for ℙ^{2} and ℙ^{3}. It can be assumed that μ is a tuple of (3d − 2) points in the case of ℙ^{2} and of p points and q lines, with 2p+q = 4d−3, in the case of ℙ^{3}. In the former case, we write n_{2,d} for n_{2,d}(μ) and n_{d} for the number of rational plane degree d curves passing through 3d − 1 points.

If ν ∈ Γ(Σ × ℙ^{n}; Λ^{0,1}π_{Σ}^{∗}T^{∗}Σ ⊗ π_{ℙ^{n}}^{∗}T^{∗}ℙ^{n}) is generic, for all t ∈ (0, 1), the signed cardinality of the set M_{Σ,tν,d}(μ) is the symplectic invariant RT_{2,d}(; μ). If t > 0 is sufficiently small, every element of M_{Σ,tν,d}(μ) lies either in a small neighborhood U of the set H_{Σ,d}(μ) or in a small neighborhood W of the space of all bubble map with singular domains. Furthermore,

±|M_{Σ,tν,d}(μ) ∩ U| = |H_{Σ,d}(μ)| = 2n_{2,d}(μ).

On the other hand, by Subsection 4.9,

|M_{Σ,tν,d}(μ) ∩ W | (5.54)

= \begin{cases}n_{1}^{(1)}(μ) + 2n_{1}^{(2)}(μ) + 18n_{1}^{(3)}(μ) + n_{2}^{(1)}(μ), & if n = 2; \\ n_{1}^{(1)}(μ) + 2n_{1}^{(2)}(μ) + 18n_{1}^{(3)}(μ) \\ +n_{2}^{(1)}(μ) + 2n_{2}^{(2)}(μ) + n_{3}^{(1)}(μ), & if n = 3. \end{cases}

Thus, n_{2,d}(μ) is one-half of the difference between RT_{2,d}(; μ) and the number in (5.54). We write CR(μ) for the number given by (5.54).

We first consider to the n = 2 case. We abbreviate M_{(d_{1},d_{2})}(μ) as M_{d_{1},d_{2}}. Let

Z_{2;d} = ( ⋃_{d_{1}+d_{2}=0^{d_{1},d_{2}>0}} Z_{d_{1},d_{2}} ) /ℤ_{2}, where

Z_{d_{1},d_{2}} = ⋃_{j_{l}=1,2} \overline{U}_{(S^{2},[N],I;j,\{0,d_{1},d_{2}\})}(μ),

and the partial ordering on I = {ˆ0, 1, 2} is 0 < 1, 2. The set Z_{2;d} is the zero-dimensional space of three-bubble maps passing through the (3d-2) points μ, such that the map is trivial on the principal component. Note that

|Z_{d;2}| = τ_{2}(μ) = \frac{1}{2} ∑_{d_{1}+d_{2}=d} \binom{3d − 2}{3d_{1} − 1}d_{1}d_{2}n_{d_{1}}n_{d_{2}}. (5.55)

The binomial coefficient counts the number of possible ways of distributing the constraints between the two nontrivial bubbles. Without the factor d_{1}d_{2}, the above number would have been precisely the number of two-component rational curves passing through (3d − 2) generic points in ℙ^{2}. However, we have to account for the image of the evaluation map at ˆ0, which must be one of the d_{1}d_{2} points of intersection of two rational curves of degrees d_{1} and d_{2}.

Lemma 5.23. In the n = 2 case, the total correction is given by

CR(μ) = ⟨78a^{2} + 72ac_{1}(L^{∗}) + 22c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ − 18τ_{2}(μ).

Proof. The four numbers of (5.54) are given by Lemmas 5.6 and 5.3 and by Corollary 5.2. The cardinality of S_{1}(μ) is given by Lemma 5.4.

Lemma 5.24. With notation as above,

⟨ac_{1}(L^{∗}), [\overline{V}_{1}(μ)]⟩ = \frac{1}{d} (−n_{d} + \frac{1}{2} ∑_{d_{1}+d_{2}=d} d_{1}^{2}d_{2}^{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}} ) .

Proof. By Corollary 5.16,

ac_{1}(L^{∗}) = \frac{1}{d^{2}} a (H − 2da + ∑_{d_{1}+d_{2}=d}^{>} d_{2}^{2}M_{d_{1},d_{2}} ) . (5.56)

Note that

(5.57) ∑_{d_{1}+d_{2}=d} d_{2}^{2}⟨a, [M_{d_{1},d_{2}}]⟩ = ∑_{d_{1}+d_{2}=d} d_{1}(d_{1}d_{2})d_{2}^{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}

= \frac{1}{2}d ∑_{d_{1}+d_{2}=d} d_{1}^{2}d_{2}^{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

The reason for the appearance of the factor d_{1}d_{2} in (5.57) is the same one as in (5.55). On the other hand, the factor d_{1} appears because we need to count the number of times the first rational component intersects a line in ℙ^{2}. Since

⟨aH, [\overline{V}_{1}(μ)]⟩ = dn_{d} and ⟨a^{2}, [\overline{V}_{1}(μ)]⟩ = n_{d},

the claim follows by plugging (5.57) into (5.56).

Lemma 5.25. With notation as above,

⟨c_{1}^{2}(L^{∗}), [\overline{V}_{1}(μ)]⟩ = −\frac{1}{2} ∑_{d_{1}+d_{2}=d} \binom{3d − 2}{3d_{1} − 1}d_{1}d_{2}n_{d_{1}}n_{d_{2}}.

Proof. By Corollary 5.16,

c_{1}^{2}(L^{∗}) = \frac{1}{d^{2}} c_{1}(L^{∗}) (H − 2da + ∑_{d_{1}+d_{2}=d} d_{2}^{2}M_{d_{1},d_{2}} ) . (5.58)

Since there are no two-component rational curves of total degree d passing through (3d − 1) generic points in ℙ^{2} and there are no threecomponent rational curves of total degree d passing through (3d − 2) generic points in ℙ^{2}, by Corollary 5.16

⟨Hc_{1}(L^{∗}), [\overline{V}_{1}(μ)]⟩ = \frac{1}{d^{2}} ⟨−2daH, [\overline{V}_{1}(μ)]⟩ = −2n_{d}. (5.59)

Similarly by Corollary 5.16 and Lemma 5.17,

⟨c_{1}(L^{∗}), [M_{d_{1},d_{2}}]⟩ = \frac{1}{d_{1}^{2}} ⟨−2d_{1}aH, [M_{d_{1},d_{2}}]⟩ + |Z_{d_{1},d_{2}} | = −|Z_{d_{1},d_{2}} | (5.60)

= −d_{1}d_{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

Note that by symmetry

∑_{d_{1}+d_{2}=d} d_{1}d_{2}^{3}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}} (5.61)

= \frac{1}{2} ∑_{d_{1}+d_{2}=d} d_{1}d_{2} (d^{2} − 2d_{1}d_{2}) \binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

The claim now follows from Equations (5.58)–(5.61) and Lemma 5.24.

Corollary 5.26. The total correction term is given by

CR(μ) = 78n_{d} + 72\frac{1}{d} (−n_{d} + \frac{1}{2} ∑_{d_{1}+d_{2}=d} d_{1}^{2}d_{2}^{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}} )

− 20 ∑_{d_{1}+d_{2}=d} d_{1}d_{2}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

Proof. This claim is immediate from Lemmas 5.23–5.25 and Equation (5.55).

Lemma 5.27. The genus-two degree-d RT-invariant of ℙ^{2} is given by

RT_{2,d}(; μ) ≡ RT_{2,d}(; p_{[3d−2]}) = 6d^{2}n_{d} + ∑_{d_{1}+d_{2}=d} d_{1}^{3}d_{2}^{3}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

Proof. Applying the genus-reducing composition law of [10] twice, we obtain

RT_{2,d}(; p_{[3d−2]}) = 2RT_{1,d}(p, ℙ^{2}; p_{[3d−2]}) + RT_{1,d}(ℓ, ℓ; p_{[3d−2]}) (5.62)
= 4RT_{0,d}(p, ℙ^{2}, p, ℙ^{2}; p_{[3d−2]})
+ 4RT_{0,d}(p, ℙ^{2}, ℓ, ℓ; p_{[3d−2]}) + RT_{0,d}(ℓ, ℓ, ℓ, ℓ; p_{[3d−2]})
= 0 + 4RT_{0,d}(p, ℓ, ℓ; p_{[3d−2]}) + RT_{0,d}(ℓ, ℓ, ℓ, ℓ; p_{[3d−2]}).

Since the genus-zero three-point RT-invariant is the usual enumerative invariant, the middle term above is simply 4d^{2}n_{d}. On the other hand, by the component-splitting composition law of [10],

RT_{0,d}(ℓ, ℓ, ℓ, ℓ; p_{[3d−2]}) (5.63)

= 2RT_{0,0}(ℓ, ℓ, ℙ^{2}; )RT_{0,d}(ℓ, ℓ, p; p_{[3d−2]})

+ ∑_{d_{1}+d_{2}=d} ∑_{J_{1}+J_{2}=[3d−2]} RT_{0,d_{1}}(ℓ, ℓ, ℓ; p_{J_{1}})RT_{0,d_{2}}(ℓ, ℓ, ℓ; p_{J_{2}})

= 2d^{2}n_{d} + ∑_{d_{1}+d_{2}=d} d_{1}^{3}d_{2}^{3}\binom{3d − 2}{3d_{1} − 1}n_{d_{1}}n_{d_{2}}.

The lemma follows from Equations (5.62) and (5.63).

Theorem 1.1 is nearly proved. We can simplify the expression in Corollary 5.26 by using a recursive relation for the numbers n_{d}; see [10, p. 363]. The expression of Theorem 1.1 is half of the difference between the quantity of Lemma 5.27 and Corollary 5.26. Note that the numbers n_{d} with d = 1, 2, 3 have long been known to be zero; see [1]. Strictly speaking, our computation does not apply to the cases d = 1, 2. However, these two cases do provide a consistency check.

The case of ℙ^{3} is significantly harder than the n = 2 case. An explicit recursive formula as in Theorem 1.1 would be rather long, so we do not provide one. Instead we express n_{2,d}(μ) in terms of the corresponding symplectic invariant and intersection numbers of the spaces \overline{V}_{1}(μ), \overline{V}_{2}(μ), and \overline{V}_{3}(μ).

Theorem 5.28. If d is a positive integer and μ is a tuple of p points and q lines in general position in ℙ^{3} with 2p + q = 4d − 3,

2n_{2,d}(μ) = RT_{2,d}(·; μ) − CR(μ), where
\frac{1}{2}CR(μ) = ⟨480a^{3}c_{1}(L^{∗}) + 476a^{2}c_{1}^{2}(L^{∗}) + 240ac_{1}^{3}(L^{∗})
+ 49c_{1}^{4}(L^{∗}), [\overline{V}_{1}(μ)]⟩ + 36τ_{3}(μ)
− ⟨324a^{2} + 144a(c_{1}(L_{1}^{∗}) + c_{1}(L_{2}^{∗}))
+ 27(c_{1}^{2}(L_{1}^{∗}) + c_{1}^{2}(L_{2}^{∗})) + 25c_{1}(L_{1}^{∗})c_{1}(L_{2}^{∗}), [\overline{V}_{2}(μ)]⟩.

Furthermore, RT(·; μ) and all intersection numbers above are computable.

Proof. The six numbers of (5.54) in the n = 3 case are given by Lemmas 5.14, 5.11, 5.12, 5.7, 5.3, and Corollary 5.2, respectively. The numbers ⟨a, [\overline{S}_{1}(μ)]⟩, ⟨c_{1}(L^{∗}), [\overline{S}_{1}(μ)]⟩, and |\overline{S}_{2}(μ)| are given by Lemmas 5.5 and 5.13. The symplectic invariant RT_{2,d}(·; μ) is well-known to be computable; see [10]. The above intersection numbers are computable by Corollary 5.18.

As in the case of ℙ^{2}, we recover the well-known fact that all degreeone, -two, and -three numbers are zero. The only degree-one number, the number of genus-two degree-one curves through a line, is zero because there are no holomorphic degree-one maps from a positive-genus curve into ℙ^{n}; see [1]. The eight degree-two and -three numbers are zero because the image of any holomorphic map of degree two or three from a genus-two curve into ℙ^{n} is a line, see [1], while no line passes through the required constraints. The first three degree-four numbers given in the table below have also been known to be zero, since the image of any holomorphic map of degree four from a genus-two curve into ℙ^{n} must lie in a plane. Finally, observe that the fourth degree-four number is the number n_{2,4} given by Theorem 1.1, as should be the case.

degree  4  5
(p,q) (6,1) (5,3) (4,5) (3,7)  (0,13)  (5,7)
RT_{2,d}(·; μ) 7,872 64,960 548,608 4,906,304 5,130,826,752 290,439,680
CR(μ) 7,872 64,960 548,608 4,877,504 4,998,465,792 258,287,360
n_{2,d}(μ) 0  0  0  14,400 66,180,480 16,076,160

6. Appendix

6.1 A short exact sequence on ℙ^{n}

If M is a Kahler manifold and E ⟶ M is a holomorphic vector bundle, let O(E) denote the sheaf of holomorphic sections of E. If E ⟶ M is the trivial holomorphic line bundle, we write O for O(E). Let H ⟶ ℙ^{n} be the hyperplane bundle.

Lemma 6.1. There exists an exact sequence of sheaves over ℙ^{n}:

0 ⟶ O ⟶ (n + 1)O(H) ⟶ O(T ℙ^{n}) ⟶ 0.

Proof. (1) Let [X_{0} : ⋯: X_{n}] denote the homogeneous coordinates on ℙ^{n}. Denote by \overline{X}_{i} the section of the hyperplane bundle given by

\overline{X}_{i}|_{[X_{0}:···:X_{n}]}(X_{0}, …, X_{n}) = X_{i} ∈ ℂ.

Then we define a sheaf map O ⟶ (n + 1)O(H) by

f ⟶ (f\overline{X}_{0}, …, f\overline{X}_{n}).

Let U_{i} = {[X_{0} : ⋯: X_{n}] : X_{i} ≠ 0}. On U_{i}, we can use the complex coordinates

z_{i,k} = \frac{X_{k}}{X_{i}} , k ∈ \{0, …, n\} − \{i\}.

Using these coordinates, we define a sheaf map (n+1)O(H) ⟶ O(T ℙ^{n}) by

(p_{0}, …, p_{n}) ⟶ ∑_{k≠i} (p_{k}(z_{i,0}, …, z_{i,n}) − z_{i,k}p_{i}(z_{i,0}, …, z_{i,n}))\frac{∂}{∂z_{i,k}} , (6.1)

where z_{i,i} = 1. We need to see that this map is well-defined. Suppose j ≠ i. Then,

(6.2)  z_{j,l} = z_{i,j}^{−1}z_{i,l} ⟹ \frac{∂}{∂z_{i,k}} = ∑_{l≠j} \frac{∂z_{j,l}}{∂z_{i,k}} \frac{∂}{∂z_{j,l}}

= \begin{cases}z_{i,j}^{−1} \frac{∂}{∂z_{j,k}} , & if k ≠ j; \\ −z_{i,j}^{−2} (\frac{∂}{∂z_{j,i}} + ∑_{l≠i,j} z_{i,l} \frac{∂}{∂z_{j,l}} ) , & if k = j. \end{cases}

Since each p_{l} is a linear functional, if k ≠ i, j, we can write the kth summand in (6.1) as

(z_{j,i}^{−1}p_{k}(z_{j,0}, …, z_{j,n}) − z_{j,i}^{−2}z_{j,k}p_{i}(z_{j,0}, …, z_{j,n}))z_{i,j}^{−1} \frac{∂}{∂z_{j,k}} (6.3)

= (p_{k}(z_{j,0}, …, z_{j,n}) − z_{j,i}^{−1}z_{j,k}p_{i}(z_{j,0}, …, z_{j,n}))\frac{∂}{∂z_{j,k}} .

The remaining, k = j, summand in (6.1) is equal to

(z_{j,i}^{−1}p_{j}(z_{j,0}, …, z_{j,n}) − z_{j,i}^{−2}p_{i}(z_{j,0}, …, z_{j,n})) (6.4)

· (−z_{i,j}^{−2}) (\frac{∂}{∂z_{j,i}} + ∑_{k≠i,j} z_{i,k} \frac{∂}{∂z_{j,k}} )

= (p_{i}(z_{j,0}, …, z_{j,n}) − z_{j,i}p_{j}(z_{j,0}, …, z_{j,n})) (\frac{∂}{∂z_{j,i}} + ∑_{k≠i,j} z_{i,k} \frac{∂}{∂z_{j,k}} ) .

Since z_{j,i}z_{i,k} = z_{j,k}, collecting similar terms in (6.3) and (6.4), we obtain Equation (6.1) with i replaced by j.

(2) It is clear that the first map is injective, the second is surjective, and the composite is zero. Finally, if (p_{0}, …, p_{n}) is mapped to zero by the second map, then (6.1) implies that \overline{X}_{j}p_{i} = \overline{X}_{i}p_{j} for all i and j. Thus, the function f, given by

f([X_{0} : ⋯: X_{n}]) = \frac{p_{i}(X_{0}, …, X_{n})}{X_{i}},

is well-defined and holomorphic wherever (p_{0}, …, p_{n}) is.

6.2 On regularity of the kernel of D_{b}

Lemma 6.2. If u : S^{2} ⟶ ℙ^{n} is a holomorphic map, there is a surjection

(n + 1)H^{1}(S^{2}; O(u^{∗}H ⊗ (−(k + 1)p)))

⟶ H^{1}(S^{2}; O(u^{∗}T ℙ^{n} ⊗ (−(k + 1)p))),

where p denotes the divisor corresponding to a point p ∈ S^{2}. If the degree of u is at least k, then both cohomology groups are trivial.

Proof. Pulling back the short exact sequence of sheaves of Lemma 6.1 by u, tensoring it with −(k + 1)p, and taking the corresponding long exact sequence, we obtain:

⟶ (n + 1)H^{1}(S^{2}; O(u^{∗}H ⊗ (−(k + 1)p))) (6.5)

⟶ H^{1}(S^{2}; O(u^{∗}T ℙ^{n} ⊗ (−(k + 1)p)))
⟶ H^{2}(S^{2}; O(−(k + 1)p)) ⟶ · · · .

Since S^{2} is a one-dimensional complex manifold, the last cohomology group in (6.5) must vanish, and the first statement of the lemma follows. On the other hand, by Kodaira-Serre duality,

H^{1}(S^{2}; O(u^{∗}H ⊗ (−(k + 1)p))) = H^{1}(S^{2}; Ω^{1}(u^{∗}H ⊗ (−(k − 1)p)) (6.6)

≈ H^{0}(S^{2}; O((u^{∗}H ⊗ (−(k − 1)p))^{∗}))^{∗}.

The last group in (6.6) is trivial if O(u^{∗}H ⊗(−(k −1)p)) is positive, i.e., if

⟨c_{1}(u^{∗}H ⊗ (−(k − 1)p), [S^{2}]⟩ = d − (k − 1) > 0,

where d is the degree of u.

Corollary 6.3. If u : S^{2} ⟶ ℙ^{n} is holomorphic map of degree d, for any p ∈ S^{2} and nonzero v ∈ T_{p}S^{2}, the map

ϕ_{p,v}^{(k)} : ker D_{u} ⟶ ⨁_{m∈⟨k⟩} T_{u(p)}ℙ^{n}, ϕ_{p,v}^{(k)}ξ = (ξ_{p}, Dξ|_{p,v}, …, D^{(k)}ξ|_{p,v}),

where Dξ|_{p,v} denotes the covariant derivative of ξ along u in the direction of v, is surjective provided d ≥ k.

Remark. If one defines D^{(k)}ξ with respect to the metric g_{ℙ^{n},u(p)} on ℙ^{n}, D^{(k)} ∈ T_{u(p)}ℙ^{n} ⊗ T^{∗}S^{2⊗k}, where T^{∗}S^{2} is viewed as a complex line bundle. However, the statement is independent of the choice of metric on ℙ^{n}.

Proof. Since ξ is holomorphic, if ϕ_{p,v}^{(k)}ξ is zero, ξ has a zero of order k + 1 at p. Thus, ϕ_{p,v}^{(k)} induces a short exact sequence of sheaves on S^{2}:

0 ⟶ O(u^{∗}T ℙ^{n} ⊗ (−(k + 1)p))

⟶ O(u^{∗}T ℙ^{n}) ⟶^{ϕ_{p,v}^{(k)}} (k + 1)O((u^{∗}T ℙ^{n})_{p}) ⟶ 0,

where we view O((u^{∗}T ℙ^{n})_{p}) as a sheaf on S^{2} via extension by 0; see [3, p. 38]. Taking the corresponding long exact sequence in cohomology, we obtain

(6.7) ⋯ ⟶ H^{0}(S^{2}; O(u^{∗}T ℙ^{n})) ⟶^{ϕ_{p,v}^{(k)}} (k + 1)H^{0}(S^{2}; O((u^{∗}T ℙ^{n})_{p}))
⟶ H^{1}(S^{2}; O(u^{∗}T ℙ^{n} ⊗ (−(k + 1)p))) · · · .

By Lemma 6.2, the last cohomology group in (6.7) is zero if d ≥ k. It follows that the map ϕ_{p,v}^{(k)} is surjective.

6.3 Dimension counts

Lemma 6.4. Let Σ be a compact Riemann surface. If u : Σ ⟶ ℙ^{n} is a holomorphic map, there exists a surjection

(n + 1)H^{1}(Σ; O(u^{∗}H)) ⟶ H^{1}(Σ; O(u^{∗}T ℙ^{n})).

Proof. Pulling back the short exact sequence of Lemma 6.1 by u gives a long exact sequence in sheaf cohomology:

⋯ (n + 1)H^{1}(Σ; O(u^{∗}H)) ⟶ H^{1}(Σ; O(u^{∗}T ℙ^{n})) ⟶ H^{2}(Σ; O) · · · . (6.8)

Since the complex dimension of Σ is one, the last group vanishes, and the claim follows.

Corollary 6.5. Let Σ be a compact Riemann surface. If u : Σ ⟶ ℙ^{n} is a holomorphic map, the \overline{∂}-operator for the bundle u^{∗}T ℙ^{n},

D_{u} : Γ(Σ; u^{∗}T ℙ^{n}) ⟶ Γ(Σ; Λ^{0,1}T^{∗}Σ ⊗ u^{∗}T ℙ^{n})

is surjective, provided d + χ(Σ) > 0, where d is the degree of u.

Proof. The cokernel of D_{u} is H1 ; \overline{∂}(Σ; u^{∗}T ℙ^{n}). By Dolbeault Theorem,

H1 ; \overline{∂}(Σ; u^{∗}T ℙ^{n}) = H^{1}(Σ; O(u^{∗}T ℙ^{n})). (6.9)

On the other hand, by Kodaira-Serre duality (see [3, p. 153]),

H^{1}(Σ; O(u^{∗}H)) = H^{1}(Σ; Ω^{1}(T Σ ⊗ u^{∗}H)) (6.10)

= H^{0}(Σ; O((T Σ ⊗ u^{∗}H)^{∗}))^{∗} = H0 ; \overline{∂}(Σ; (T Σ ⊗ u^{∗}H)^{∗})^{∗}.

The bundle (T Σ ⊗ u^{∗}H)^{∗} does not admit any holomorphic section if it is negative, i.e., if

⟨c_{1}((T Σ ⊗ u^{∗}H)), [Σ]⟩ = ⟨c_{1}(T Σ) + c_{1}(u^{∗}H), [Σ]⟩ = χ(Σ) + d > 0.

Thus, the claim follows from Equations (6.9) and (6.10) and Lemma 6.4,

Proposition 6.6. Let Σ be a Riemann surface of genus 2 and let d and n be positive integers with n ≤ 4. If n = 4, assume that d ≠ 2. Suppose μ = (μ_{1}, …, μ_{N}) is an N-tuple of proper complex submanifolds of ℙ^{n} of total complex codimension d(n + 1) − n + N in general position. If

T = (Σ, [N], I; j, \underline{d}′) < T^{∗} = (Σ, [N], \{ˆ0\}; ˆ0, d)

is a bubble type such that d′ ; ˆ0 > 0, then H_{T} (μ) = ∅. Furthermore, if

b = (Σ, [N], {ˆ0}; , (ˆ0, y), u) ∈ H_{T^{∗}}(μ),

then the map u is not multiply-covered.

Proof. (1) If d′ ; ˆ0 ≥ 3, by Corollaries 6.3 and 6.5 and standard arguments such as in [7], the space H_{T} is a smooth manifold and the maps ev_{l} are smooth. If b ∈ H_{T}, a neighborhood of b in H_{T} can be modeled

on ker D_{b} ⊕ ⨁_{l=1}^{l=n} T_{y_{l}}Σ_{b,j_{l}}. In particular, by the Index Theorem,

dim_{ℂ} H_{T} = ∑_{i∈I} (d′_{i}(n + 1) + n(1 − g(Σ_{b,i}))) − (n − 1)|Î| + N
= d(n + 1) − n + |Î| + N.

Thus, if the map

ev_{[N]} ≡ ev_{1} × ⋯ × ev_{N} : H_{T} ⟶ ℙ^{n} × ⋯ × ℙ^{n},

is smooth and transversal to μ_{1} × ⋯ × μ_{N}, H_{T} (μ) is a smooth manifold of (complex) dimension |Î|. Since the map ev_{[N]} is invariant under the action of 2|Î|-dimensional group

G_{T} ≡ {g ∈ P SL_{2} : g(∞) = ∞}Î,

G_{T} acts smoothly on H_{T} (μ). Furthermore, the stabilizer at each point is finite. Thus, H_{T} (μ) = ∅.

(2) Suppose d′ ; ˆ0 = 2. If b = (Σ, [N], I; x, (j, y), u) ∈ H_{T}, the map uˆ0 must factor through a degree-one map ũˆ0 : S^{2} ⟶ ℙ^{n}; see [1, p. 116]. Thus, it is enough to show that the space H_{T′}(μ) is empty, where T′ = (S^{2}, [N], I; j, \underline{d}′′), d′′_{h} = d′_{h} if h ∈ Î and d′′ ; ˆ0 = 1. By Corollary 6.3, the space H_{T′} is a smooth manifold of dimension

dim_{ℂ} H_{T′} = (d − 1)(n + 1) + n + |Î| + N.

Similarly to (1) above, it follows that H_{T′}(μ) is a smooth manifold of dimension n−1+|Î| on which the (2|Î|+3)-dimensional group P SL_{2}×G_{T} acts with only finite stabilizers. It follows that H_{T′}(μ) = ∅ if n < |Î|+4. Note that the case Î = ∅ can occur only if d = dˆ0 = 2. Finally, if dˆ0 = 1, the entire space H_{T} is empty, since there are no holomorphic degree-one maps from Σ into ℙ^{n}.

(3) Suppose b = (Σ, [N], {ˆ0}; , (ˆ0, y), u) ∈ H_{T^{∗}}(μ) and u : Σ ⟶ ℙ^{n} factors through a k-fold cover of S^{2}, where k ≥ 2 and k divides d. Then b arises from the space H_{T′}(μ), where

T′ = (S^{2}, [N], {ˆ0}; ˆ0, d/k).

Similarly to the above, this space is a smooth manifold of dimension

((d/k)(n + 1) + n + N) − (d(n + 1) − n + N)

= −\frac{k − 1}{k}d(n + 1) + 2n.

Thus, H_{T′}(μ) = ∅, provided d ≥ 3. In fact, since H_{T′}(μ) has a threedimensional group of symmetry, H_{T′}(μ) = ∅ unless d = 2 and n ≥ 4.

(4) Suppose b is as in (3) and u factors through a k-fold cover of a torus T, where k ≥ 2 and k divides d. Then b arises from the space

H̃_{1,d/k}(μ) ≡ {(E, y_{[N]}, u) : E is smooth elliptic curve,

u : ℂ ⟶ ℙ^{n}, \overline{∂}u = 0, u_{∗}[E] = \frac{d}{k}λ; u(y_{l}) ∈ μ_{l} ∀l ∈ [N]\}.

Similarly to the above, Corollary 6.5 implies that H̃_{1,d/k}(μ) is a smooth space of dimension

((d/k)(n + 1) + 1 + N) − (d(n + 1) − n + N)

= −(\frac{k − 1}{k}d − 1)(n + 1) < 1.

Since H̃_{1,d/k}(μ) has a one-dimensional group of symmetries, H̃_{1,d/k}(μ) = ∅.

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