j. differential geometry 66 (2004) 169-209

THE HILBERT COMPACTIFICATION OF THE UNIVERSAL MODULI SPACE OF SEMISTABLE VECTOR BUNDLES OVER SMOOTH CURVES

Alexander Schmitt

Abstract

We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Graßmannian. It has all the properties asked for by Teixidor.

Introduction

For every smooth curve C and integers χ and r > 0, one has the projective moduli space U(C; χ, r) of semistable vector bundles E of rank r and Euler characteristic χ(E) = χ. An automorphism σ of C acts on U(C; χ, r) via [E] ⟼ [σ^{∗}E]. Let 𝔐_{g} be the moduli space of smooth curves of genus g. It is possible to construct a universal moduli space 𝔘(g; χ, r) ⟶ 𝔐_{g}, such that the fibre over [C] is U(C; χ, r)/Aut(C). This leads to the problem of compactifying 𝔘(g; χ, r) over \overline{𝔐}_{g}, the moduli space of stable curves of genus g. There are two natural approaches to this [19]: first, given a stable curve C of genus g, one can look at torsion-free sheaves E of uniform rank r on C with Euler characteristic χ(E) = χ which are semistable w.r.t. the canonical polarization. These objects form again a projective moduli space U(C; χ, r). Pandharipande [14] has constructed a projective moduli space \overline{𝔘}(g; χ, r) ⟶ \overline{𝔐}_{g}, such that the fibre over a stable curve [C] is U(C; χ, r)/Aut(C). Second, for a stable curve C, instead of looking at torsion-free sheaves on C, one can look at vector bundles on semistable models of C. This viewpoint has advantages for certain degeneration arguments. As an approach to the above problem, it has been formalized by Gieseker [6] and further studied by Gieseker and Morrison [7], Nagaraj and Seshadri [12], Teixidor i Bigas [19], Kausz ([8],[9]), and the author [16]. It was also used by Caporaso [2] to solve the problem for r = 1. Without loss of generality, we may assume that, for every smooth curve of genus g and every semistable vector bundle E on C of rank r with χ(E) = χ, E is globally generated, H^{1}(E) = 0, and the evaluation map ev: H^{0}(E) ⊗ O_{C} ⟶ E gives rise to a closed embedding C ↪ Gr(H^{0}(E), r) into the Graßmannian of r-dimensional quotients of H^{0}(E). Thus, we fix a vector space V^{χ} of dimension χ, define 𝔊 := Gr(V^{χ}, r), and look at ℌ(g; χ, r), the closure of the Hilbert scheme of smooth curves in 𝔊 with Hilbert polynomial P (m) = d · m + (1 − g) in the whole Hilbert scheme. Here, χ = d + r(1 − g) and 𝔊 is polarized by O_{𝔊}(1), the determinant of the universal quotient bundle. Note that we have a natural action of SL(V^{χ}) on ℌ(g; χ, r). Our candidate for the Hilbert compactification is, therefore, ℌℭ(g; χ, r) := ℌ(g; χ, r)// SL(V^{χ}). Before we can form the GIT-quotient, we have, however, to find appropriate linearized ample line bundles on ℌ(g; χ, r). First, there are some obvious ones. For this, let ℭ ⊂ 𝔊 × ℌ(g; χ, r) be the universal curve. For every natural number m, we have, on 𝔊 × ℌ(g; χ, r), the exact sequence

During the preparation of the article, the author benefited from support by the DFG through the priority program “Globale Methoden in der komplexen Geometrie — Global Methods in Complex Geometry”.

Received 07/03/2002.

0 ⟶ I_{ℭ} ⊗ π_{𝔊}^{∗}O_{𝔊}(m) ⟶ π_{𝔊}^{∗}O_{𝔊}(m) ⟶ (π_{𝔊}^{∗}O_{𝔊}(m))_{|ℭ} ⟶ 0.

For large m, π_{ℌ(g;χ,r)∗} of this sequence leads to a surjective homomorphism of vector bundles

Ψ_{ℌ(g;χ,r)}: S^{m}V^{χ} ⊗ O_{ℌ(g;χ,r)} ⟶ 𝔈_{m} := π_{ℌ(g;χ,r)∗}((π_{𝔊}^{∗}O_{𝔊}(m))_{|ℭ}).

The rank of 𝔈_{m} is P (m), and ∧^{P (m)}Ψ_{ℌ(g;χ,r)} yields a closed immersion

ℌ(g; χ, r) ↪ ℙ( ⋀^{P (m)} S^{m}V^{χ})

which is equivariant w.r.t. the SL(V^{χ})-actions. Let 𝔏_{m} := 𝔏_{m}(g; χ, r) := det(𝔈_{m}) be the pullback of O(1). Then, we can define ℌℭ_{m}(g; χ, r) := ℌ//_{𝔏_{m}} SL(V^{χ}). This, space will be, however, only useful, if we can settle the following properties:

1) If [C] ∈ ℌ(g; χ, r) is a smooth curve and q_{C}: V^{χ} ⊗O_{C} ⟶ E is the pullback of the universal quotient, then [C] is (semi)stable w.r.t. the linearization in 𝔏_{m}, if and only if E is a (semi)stable vector bundle.

2) If [C] ∈ ℌ(g; χ, r) is semistable w.r.t. the linearization in 𝔏_{m} and q_{C}: V^{χ} ⊗ O_{C} ⟶ E is the pullback of the universal quotient, then C is a semistable curve and V^{χ} ⟶ H^{0}(E) is an isomorphism.

Points (1) and (2) have been settled in the rank two case by Gieseker and Morrison [7], and (1) in general by the author [16]. Unfortunately, nothing is known about (2) in general, and, even if it were true, the computations of the correct notion of semistability would still be extremely difficult (cf. [19]). The way out is to adapt a strategy due to Nagaraj and Seshadri [12]. In our setting, it is described as follows: for every stable curve C, let Q(C; χ, r) be the quot scheme parameterizing quotients V^{χ} ⊗O_{C} ⟶ E where E is a coherent sheaf of uniform rank r with Euler characteristic χ(E) = χ. From Pandharipande’s construction, we get a universal quot scheme \overline{𝔔}(g; χ, r) ⟶ \overline{𝔐}_{g}, such that the fibre over [C] is just Q(C; χ, r)/Aut(C), and a natural SL(V^{χ})-linearized ample line bundle 𝔑 on \overline{𝔔}(g; χ, r). Next, let ℌ^{0}(g; χ, r) ⊂ ℌ(g; χ, r) be the open part corresponding to semistable curves with the following property: if π: C ⟶ C′ is the projection onto the stable model and if q_{C}: V^{χ} ⊗ O_{C} ⟶ E is the pullback of the universal quotient on 𝔊, then π_{∗}(q_{C}): V^{χ} ⊗ O_{C′} ⟶ π_{∗}(E) is surjective and π_{∗}(E) is a torsion-free sheaf which is semistable w.r.t. the canonical polarization. There is a natural map ℌ^{0}(g; χ, r) ⟶ \overline{𝔔} (g; χ, r), landing in the 𝔑-semistable locus. Let ℌ̂(g; χ, r) ⊂ ℌ(g; χ, r) × \overline{𝔔}(g; χ, r) be the closure of the graph of the above morphism. This is an SL(V^{χ})-invariant subscheme. For a ≫ 0, the semistable points w.r.t. the linearization in

𝔏_{a,m} := 𝔏_{a,m}(g; χ, r) := (π_{ℌ(g;χ,r)}^{∗}𝔏_{m}(g; χ, r) ⊗ π_{\overline{𝔔}(g;χ,r)}^{∗}𝔑^{⊗a})_{|ℌ̂(g;χ,r)}

will lie in the set of the preimages of the points in \overline{𝔔}(g; χ, r) which are semistable w.r.t. the linearization in 𝔑. Note that, for every l ≥ 0, we can perform the same constructions w.r.t. g, r, and χ_{l} := χ+l·r·(2g−2). The result of this note is:

Main Theorem. There exist an l_{0} and for every l ≥ l_{0} an m(l), such that for all l ≥ l_{0} and m ≥ m(l) the following properties hold true:

i) All points in ℌ̂(g; χ_{l}, r) which are semistable w.r.t. the linearization in 𝔏_{a,m}(g; χ_{l}, r) for all a ≫ 0 lie in the graph of the morphism ℌ^{0}(g; χ_{l}, r) ⟶ \overline{𝔔}(g; χ_{l}, r).

ii) Let [C] be contained in the graph of the morphism ℌ^{0}(g; χ_{l}, r) ⟶ \overline{𝔔}(g; χ_{l}, r), and let q_{C}: V^{χ_{l}} ⊗ O_{C} ⟶ E be the pullback of the universal quotient. Then, [C] is (semi)stable w.r.t. the linearization in 𝔏_{a,m}(g; χ_{l}, r) for all a ≫ 0, if and only if (C, E ⊗ ω_{C}^{⊗−l}) is H-(semi)stable in the sense of Definition 2.2.10.

Remark.

i) If E is a vector bundle of rank r on the semistable curve C and if π: C ⟶ C′ is the map onto the stable model, then the condition that π_{∗}(E) be torsion-free is a precise condition on the restriction of E to any chain R of rational curves attached at only two points, say p_{1} and p_{2}. Namely, there must not exist a nonzero section of E_{|R} vanishing in both p_{1} and p_{2}. This implies, for example, that R has at most r components and that E_{|R} is strictly standard, i.e., for any component R_{i} ≅ ℙ_{1} of R, E_{|R_{i}} ≅ O_{R_{i}}^{⊕ε_{i}} ⊕O_{R_{i}}(1)^{⊕(r−ε_{i})} with 0 ≤ ε_{i} < r, i = 1, …, s. For the detailed discussion, we refer the reader to [12].

ii) The condition to be semistable with respect to the linearization in 𝔏_{a,m}(g; χ_{l}, r) for all a ≫ 0 is explained as follows: let G be a reductive algebraic group, ρ_{i}: G ⟶ GL(W_{i}), i = 1, 2, two finite-dimensional representations, and ℌ̂ ⊂ ℙ(W_{1}) × ℙ(W_{2}) a Ginvariant closed subscheme. Denote by 𝔏_{a,m} the restriction of O_{ℙ(W_{1})×ℙ(W_{2})}(m, a) to ℌ̂ and by ℌ̂_{a,m}^{(s)s} the set of points which are (semi)stable w.r.t. the linearization in 𝔏_{a,m}. Then, there is an ε_{∞}, such that

π_{2}^{−1}(ℙ(W_{2})^{s}) ⊂ ℌ̂_{a′,m′}^{(s)s} = ℌ̂_{a,m}^{(s)s} ⊂ π_{2}^{−1}(ℙ(W_{2})^{ss})

whenever both m′/a′ and m/a are smaller than ε_{∞}. This follows from the corresponding assertion for ℂ^{∗}-actions and the master space construction of Thaddeus ([20], [13]).

iii) The intrinsically defined concept of H-(semi)stability has the following properties:

• For smooth curves, it agrees with Mumford-(semi)stability.

• If (C, E ⊗ ω_{C}^{⊗−l}) is H-semistable and π: C ⟶ C′ is the morphism to the stable model, then π_{∗}(E ⊗ ω_{C}^{⊗−l}) is semistable w.r.t. the canonical polarization.

• A pair (C, E⊗ω_{C}^{⊗−l}) with C a semistable curve and π_{∗}(E⊗ω_{C}^{⊗−l}) a stable sheaf, π: C ⟶ C′ being the morphism to the stable model, is H-stable.

Therefore, ℌℭ(g; χ, r) := ℌ̂(g; χ_{l}, r)//_{𝔏_{a,m}(g;χ_{l},r)} SL(V^{χ_{l}}) with l ≥ l_{0}, m ≥ m(l), and a ≫ 0 is a well-defined moduli space which compactifies 𝔘(g; χ, r) over \overline{𝔐}_{g} and, furthermore, maps to \overline{𝔘}(g; χ, r).

We remark that the authors of [12] were well aware of the fact that their approach might be used in this generality. The formulation of an intrinsic semistability concept for vector bundles on semistable curves, however, seems to be new.

The final section of this paper is devoted to the study of the geometry of the Hilbert compactification and its map to the moduli space of stable curves.

Notation. Let C be a semistable curve. Then, O_{C}(1) will stand for the canonical sheaf ω_{C}, although this line bundle will be ample if and only if C is stable. Likewise, if E is a coherent sheaf on C, we write P (E) for the polynomial l ⟼ χ(E(l)). A scheme will be a scheme of finite type over the field of complex numbers.

Acknowledgments. The author wishes to thank the referee for some valuable suggestions and Professor Newstead for pointing out reference [1] to him.

1. Preliminaries

1.1. Suitable linearizations. Let ρ_{i}: G ⟶ GL(W_{i}), i = 1, 2, two representations of a reductive group G. This yields an action of G on ℙ(W_{1}) × ℙ(W_{2}). Assume 𝔛 is a G-invariant subscheme, and let π: 𝔛 ⟶ ℙ(W_{2}) be the induced morphism. Finally, let 𝔏_{n_{1},n_{2}} be the G-linearized ample line bundle O_{ℙ(W_{1})×ℙ(W_{2})}(n_{1}, n_{2})_{|𝔛}. Define 𝔛_{n_{1},n_{2}}^{(s)s} as the set of points in 𝔛 which are (semi)stable w.r.t. the linearization in 𝔏_{n_{1},n_{2}}. Likewise, ℙ(W_{2})^{(s)s} is defined. The following is well-known and easy to prove.

Proposition 1.1.1. If n_{2}/n_{1} is large enough, then

π^{−1}(ℙ(W_{2})^{s}) ⊂ 𝔛_{n_{1},n_{2}}^{s} ⊂ 𝔛_{n_{1},n_{2}}^{ss} ⊂ π^{−1}(ℙ(W_{2})^{ss}).

Remark 1.1.2. Let n_{2}/n_{1} be so large that the conclusion of Proposition 1.1.1 holds. Then, a point x ∈ 𝔛 will be (semi)stable w.r.t. the linearization in 𝔏_{n_{1},n_{2}}, if and only if π(x) is semistable and for every one-parameter subgroup λ: ℂ^{∗} ⟶ G with μ_{O_{ℙ(W_{2})}(1)}(λ, π(x)) = 0, one has μ_{O_{ℙ(W_{1})}(1)}(λ, π′(x)) (≥) 0, π′: 𝔛 ⟶ ℙ(W_{1}) being the induced morphism.

1.2. Pandharipande’s moduli space; the universal quot scheme. Let C be a stable curve with irreducible components C_{1}, ..., C_{c}, and F a coherent sheaf on C. The tuple \underline{r}(F) := (rk F_{|C_{1}}, …, rk F_{|C_{c}}) is called the multirank of F. We say that F has uniform rank r on C, if \underline{r}(F) = (r, …, r). Finally, set e_{ι} := deg ω_{C|C_{ι}}, ι = 1, …, c. Then, the total rank of F is the quantity trk F := ∑_{ι=1}^{c} e_{ι} · rk F_{|C_{ι}}. Now, a coherent O_{C}-module E is called (semi)stable, if it is torsion-free and for every subsheaf 0 ⊊ F ⊊ E, the inequality

\frac{χ(F)}{trk F} (≤)\frac{χ(E)}{trk E}

is satisfied. One also introduces the notions of S-equivalence and polystability. Pandharipande studies the functor \underline{U}(g; χ, r) which associates to every scheme S the set of equivalence classes of pairs (C_{S}, E_{S}), consisting of a flat family π: C_{S} ⟶ S of stable curves and an S-flat coherent sheaf E_{S} on C_{S} such that E_{S|π^{−1}(s)} is a semistable sheaf of uniform rank r and Euler characteristic χ on π^{−1}(s) for all closed points s ∈ S. Here, (C_{S}, E_{S}) and (C′_{S}, E′_{S}) are considered equivalent, if there are an S-isomorphism ψ: C_{S} ⟶ C′_{S} and a line bundle L_{S} on S such that E_{S} ≅ ψ^{∗}E′_{S} ⊗ π^{∗}L_{S}. In [14], a coarse moduli space \overline{𝔘}(g; χ, r) for the functor \underline{U}(g; χ, r) is constructed.

We now briefly review the construction, because we need some of the details. If C is a stable curve, then O_{C}(10) defines a closed embedding C ↪ ℙ_{N}, N := 10(2g−2)−g. Let ℌ_{g} be the Hilbert scheme of curves in ℙ_{N} with the respective Hilbert polynomial. There is a natural left action of SL(N +1) on ℌ_{g} together with a linearization in an ample line bundle 𝔏_{ℌ_{g}}. As Gieseker [5] has shown, the GIT quotient ℌ_{g}//_{𝔏_{ℌg}} SL(N + 1) yields \overline{𝔐}_{g}.

There is a constant l_{0}, such that for every l ≥ l_{0}, every stable curve C of genus g, and every (semi)stable sheaf E of uniform rank r and Euler characteristic χ on C, one has:

• E(10 · l) is globally generated.

• H^{1}(E(10 · l)) = 0.

• After identification of H^{0}(E(10 · l)) with a previously fixed vector space V^{χ_{10·l}} of dimension χ_{10·l} = χ + 10 · l · r · (2g − 2), the point ev: V^{χ_{10·l}} ⊗ O_{C} ⟶ E(10 · l) in the quot scheme Q(C; χ_{10·l}, r) is (semi)stable.

Let C_{g} ↪ ℌ_{g} × ℙ_{N} be the universal curve. We then have a relative quot scheme ρ: Q = Q(g; χ_{10·l}, r) ⟶ ℌ_{g}, such that the fibre over C ∈ ℌ_{g} is the quot scheme Q(C; χ_{10·l}, r) of quotients E of V^{χ_{10·l}} ⊗ O_{C} with χ(E ⊗_{O_{C}} O_{ℙ_{N}}(l)) = χ + 10 · l · r · (2g − 2) for all l. The natural action of SL(V^{χ_{10·l}}) × SL(N + 1) on Q is linearized in a suitable ρ-ample line bundle 𝔏_{Q}. For any a > 0, define 𝔏_{a} := 𝔏_{Q} ⊗ ρ^{∗}𝔏_{ℌ_{g}}^{⊗a}. For large a, we have, by Proposition 1.1.1, Q_{𝔏_{a}}^{ss} ⊂ ρ^{−1}(ℌ_{g,𝔏_{ℌg}}^{ss} ). Then, \overline{𝔘}(g; χ, r) :=

Q//_{𝔏_{a}}(SL(V^{χ_{10·l}}) × SL(N + 1)).

Now, we can form this GIT quotient in two steps [13]: first, we divide by the SL(N + 1)-action and then by the SL(V^{χ_{10·l}})-action. As ℌ_{g,𝔏_{ℌg}}^{s} = ℌ_{g,𝔏_{ℌg}}^{ss} (more precisely, this holds on an appropriate closed subscheme, see [5], Proposition 2.0.0), 1.1.1 shows that a point (C, q_{C}) ∈ Q is SL(N + 1)-semistable w.r.t. the linearization in 𝔏_{ℌ_{g}}, if and only if C is stable, i.e., there is no condition on the quotient q_{C}. Define \overline{𝔔}(g; χ_{10·l}, r) := Q//_{𝔏_{a}} SL(N + 1). Let \underline{Q}(g; χ_{10·l}, r) be the functor which assigns to a scheme S the set of equivalence classes of pairs (C_{S}, q_{S}: V^{χ_{10·l}} ⊗ O_{C_{s}} ⟶ E_{S}) consisting of a flat family π: C_{S} ⟶ S of stable curves and a quotient q_{S} onto an S-flat sheaf E_{S}, such that χ(E_{S|π^{−1}(s)}) = χ_{10·l} for every closed point s ∈ S. Two families (C_{S}, q_{S}) and (C′_{S}, q′_{S}) will be considered equivalent, if there are an S-isomorphism ψ: C_{S} ⟶ C′_{S} and an isomorphism φ_{S}: E_{S} ⟶ ψ^{∗}E′_{S} with ψ^{∗}q′_{S} = φ_{S}∘q_{S}. The space \overline{𝔔}(g; χ_{10·l}, r) obviously is the coarse moduli scheme for the functor \underline{Q}(g; χ_{10·l}, r). In particular, the fibre over [C] ∈ \overline{𝔐}_{g} identifies with Q(C; χ_{10·l}, r)/Aut(C). As explained in the first section of [13], there is an induced SL(V^{χ_{10·l}})-action on \overline{𝔔}(g; χ_{10·l}, r) and some multiple of 𝔏_{a} descends to an SL(V^{χ_{10·l}})-linearized ample line bundle 𝔑. Moreover, the points in \overline{𝔔}(g; χ_{10·l}, r) which are SL(V^{χ_{10·l}})-(semi)stable for the given linearization are just the images of the (SL(V^{χ_{10·l}}) × SL(N + 1))- (semi)stable points. Let us note the following elementary fact:

Lemma 1.2.1. Let C be a stable curve and q_{C}: V^{χ_{10·l}} ⊗ O_{C} ⟶ E a quotient with χ(E) = χ_{10·l}. Let h ∈ ℌ_{g} be a point such that the fibre of the universal curve over h is isomorphic to C. Then, for any one-parameter subgroup λ: ℂ^{∗} ⟶ SL(V^{χ_{10·l}}),

μ_{𝔏_{a}}(λ, (h, q_{C})) (≥) 0 ⟺ μ_{𝔑}(λ, [C, q_{C}]) (≥) 0.

Proof. The quotient morphism Q_{𝔏_{a}}^{SL(N+1)−ss} ⟶ \overline{𝔔}(g; χ_{10·l}, r) yields the SL(V^{χ_{10·l}})-equivariant and finite morphism

Q_{h} := ρ^{−1}(h) ⟶ Q(C; χ_{10·l}, r)/Aut(C).

This immediately implies the assertion. q.e.d.

1.3. Auxiliary results from Nagaraj and Seshadri. We here recall some results of the paper [12]. Additional information may be found there. We also point out that [12] works with semistable curves the stable model of which is an irreducible curve with exactly one node. As one easily check, this assumption is not essential.

Proposition 1.3.1.

i) Let R be a chain of projective lines, and F a globally generated torsion-free sheaf on R. Then, for any component R_{ι} ≅ ℙ_{1}, the restriction map H^{0}(R, F) ⟶ H^{0}(R_{ι}, F_{|R_{ι}}) is surjective. Moreover, H^{1}(R, F) = 0.

ii) Let π: C′ ⟶ C be a morphism between semistable curves which contracts only some chains of projective lines. Suppose E is a vector bundle on C′ the restriction of which to every projective line contracted by π has nonnegative degree. In that situation:

1) π_{∗}O_{C′} = O_{C}.

2) R^{i}π_{∗}(E) = 0 for i > 0. In particular, H^{j}(C′, E) = H^{j}(C, π_{∗}(E)) for all j.

3) Let R^{j} be a chain of projective lines which is contracted by π and attached at the points p_{1} and p_{2}. Let C̃ be the closure of C′ \ R^{j}. If H^{1}(C̃, I_{p_{1},p_{2}}E_{|C̃}) = H^{1}(C′, I_{R^{j}}E) = 0, then the restriction map H^{0}(C′, E) ⟶ H^{0}(R^{j}, E_{|R^{j}}) is surjective, so that, by i), the restriction map H^{0}(C′, E) ⟶ H^{0}(R_{ι}^{j}, E_{|R_{ι}^{j}}) to

any component R_{ι}^{j} of R^{j} is surjective, too.

Proof. Part ii) is Proposition 3 in [12].

Ad i): The restriction of F to a component R_{ι} is of the form

O_{ℙ_{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ_{1}}(a_{rk F_{|Rι}}) ⊕ Torsion

with a_{i} ≥ 0, i = 1, …, rk F_{|R_{ι}}. Therefore, H^{1}(R_{ι}, F_{|R_{ι}}(−1)) = 0. By successively removing components which are attached at one point only, the result becomes an easy induction on the length of R. q.e.d.

Remark 1.3.2. In the situation of Proposition 1.3.1 ii), there are precise conditions for π_{∗}(E) to be torsion-free ([12], Proposition 5). In particular, any chain of rational curves contracted by π can have length at most rk E. This already bounds the family of semistable curves which might appear in our investigations.

Proposition 1.3.3. Let C be a semistable curve containing the disjoint chains R^{1}, …, R^{c} of projective lines which are attached at two points only, define C̃_{j} as the closure of C \ R^{j}, and let p_{1}^{j}, p_{2}^{j} be the points where R^{j} is attached, j = 1, …, c. Also set R := ⋃_{j=1}^{c} R^{j} and define C̃ as the closure of C \ R. Suppose E is a vector bundle on C which satisfies the following properties:

1) The restriction E_{|R_{ι}^{j}} of E to any component of R^{j} has positive degree, j = 1, …, c.

2) H^{1} (C̃_{j}, I_{p_{1}^{j},p_{2}^{j}}E_{|C̃_{j}}) = 0, j = 1, …, c.

3) The homomorphism

H^{0} (C̃_{j}, I_{p_{1}^{j},p_{2}^{j}}E_{|C̃_{j}}) ⟶ (I_{p_{1}^{j},p_{2}^{j}}E_{|C̃_{j}}) / (I_{p_{1}^{j},p_{2}^{j}}^{2}E_{|C̃_{j}})

is surjective, j = 1, …, c.

4) For any point x ∈ C̃ \ {p_{1}^{j}, p_{2}^{j}, j = 1, …, c}, the homomorphism

H^{0}(C, I_{R}E) ⟶ E_{|C̃}/(I_{x}^{2}E_{|C̃})

is surjective.

5) For any two points x_{1} ≠ x_{2} ∈ C̃ \ {p_{1}^{j}, p_{2}^{j}, j = 1, …, c}, the homomorphism

H^{0}(C, I_{R}E) ⟶ E_{|{x_{1}}} ⊕ E_{|{x_{2}}}

is surjective. Then, the evaluation map ev: H^{0}(C, E) ⊗ O_{C} ⟶ E yields a closed embedding

C ↪ Gr(H^{0}(C, E), rk E).

Proof. This is proved like Proposition 4 in [12]. q.e.d.

Remark 1.3.4. Note that the conditions (2)–(5) will be satisfied for E(l), l ≫ 0.

Proposition 1.3.5 ([12], Lemma 4). Suppose we are given a commutative diagram

Z ⟶^{π} W
p↓  ↓^{q}
T  T

in which p and q are projective, p is flat, and π_{∗}O_{Z} = O_{W}. Assume that E_{Z} is a vector bundle on Z, such that, for every point t ∈ T, we have R^{i}π_{t∗}(E_{Z|p^{−1}(t)}) = 0 for all i > 0, π_{t}: p^{−1}(t) ⟶ q^{−1}(t) being the induced morphism. Then,

π_{∗}(E_{Z})_{|q^{−1}(t)} = π_{t∗}(E_{Z|p^{−1}(t)}), for all closed points t ∈ T .

Remark 1.3.6. Let O_{W} (1) be a q-ample line bundle. Assume

H^{i}(p^{−1}(t), (E_{Z} ⊗π^{∗}O_{W} (n))_{|p^{−1}(t)}) = 0, for n ≫ 0, i > 0, and all t ∈ T .

Then, q_{∗}((π_{∗}E_{Z})(n)) = p_{∗}(E_{Z} ⊗ π^{∗}O_{W} (n)) is locally free for all n ≫ 0, whence π_{∗}E_{Z} is T -flat.

1.4. Modules over Discrete Valuation Rings. Let (R, v: R ⟶ ℤ) be a discrete valuation ring with uniformizing parameter t, i.e., v(t) = 1. A finitely generated module M over R will be called almost torsion-free, if its torsion submodule is annihilated by t. Likewise, a submodule M′ of a free module M of finite rank is called almost saturated, if the module M/M′ is almost torsion-free.

Proposition 1.4.1. Let M = R^{⊕r} be a free module of rank r and

(∗)  0 ⊊ M_{1} ⊊ M_{2} ⊊ ⋯ ⊊ M_{n} ⊊ M

a chain of almost saturated submodules. Then, there is a basis e_{1}, …, e_{r} for M, such that M_{i} is generated by ρ_{1}^{i}e_{1}, …, ρ_{rk M_{i}}^{i}e_{rk M_{i}} for appropriate elements ρ_{j}^{i} ∈ R with v(ρ_{j}^{i}) ∈ {0, 1}, j = 1, …, rk M_{i}, i = 1, …, n.

Note that we do not require rk(M_{i}) > rk(M_{i−1}), i = 2, …, n.

Proof. We carry out an induction over r, the case r = 1 being clear. For the first submodule M_{1}, we may find a basis e_{1}^{∗}, …, e_{r}^{∗} with the asserted property [18], 10.5. Take e_{1} := e_{1}^{∗}. Then,

(∗∗) 0 ⊊ M̃_{1} := M_{1}/(M_{1} ∩ ⟨e_{1}⟩) ⊊ ⋯ ⊊ M̃_{n} := M_{n}/(M_{n} ∩ ⟨e_{1}⟩) ⊊

⊊ M̃:= M/⟨e_{1}⟩

is a filtration of the free module M/⟨e_{1}⟩ of rank r −1. We claim that M̃_{i} is an almost saturated submodule of M̃, i = 1, …, n. If e_{1} ∈ M_{i}, this is just the isomorphism theorem. Otherwise, te_{1} ∈ M_{i} and M′_{i} := M̃/M̃_{i} is a quotient of M′′_{i} := M/M_{i} = (M/⟨te_{1}⟩)/M̃_{i}. As M′_{i} and M′′_{i} have the same rank, no element of the free part of M′′_{i} can map to the torsion of M′_{i}, i.e., the torsion of M′_{i} is a quotient of the torsion of M′′_{i} which implies the claim.

Therefore, we can apply the induction hypothesis to (∗∗). Let e′_{2}, ..., e′_{r} be any lift of the appropriate basis \overline{e}_{2}, …, \overline{e}_{r} for M̃. Suppose that we have already found a basis of the form e_{1}, …, e_{rk M_{i}}, e′_{rk M_{i}+1}, …, e′_{r}, so that the assertion holds for M_{1}, …, M_{i} and e_{2}, …, e_{rk M_{i}} also lift \overline{e}_{2}, …, \overline{e}_{rk M_{i}}. Then, M_{i+1} is spanned w.r.t. that basis by vectors of the form

v_{j} = ( ∗_{j} ; 0 ; ⋮ ; 0 ; q_{j} ; 0 ; ⋮ ; 0 )

with q_{j}, ∗_{j} ∈ R, v(q_{j}) ∈ {0, 1}, and q_{j} at the j-th place, j = 1, …, rk M_{i+1}. If v(q_{1}) ≤ v(∗_{j}), we may clearly set ∗_{j} = 0. If v(q_{1}) > v(∗_{j}), i.e., v(q_{1}) = 1 and v(∗_{j}) = 0, and also v(q_{j}) = 0, we set e_{j} := v_{j}. In fact, this case can only occur, if j > rk M_{i} or in all the M_{ι} with ι ≤ i, t · e_{j} was the respective basis vector. But then, the result still holds for M_{1}, …, M_{i} and the basis e_{1}, …, e_{j−1}, v_{j}, e_{j+1}, …, e_{rk M_{i}}, e′_{rk M_{i}+1}, ..., e′_{r}. Finally, the case v(q_{1}) = 1, v(∗_{j}) = 0, and v(q_{j}) = 1 cannot occur. Indeed, in that case, the class of the vector (0, …, t, …, 0)^{T}, t at the j-th place, in M/M_{i+1} is nonzero, i.e., the class of (0, …, 1, …, 0)^{T} in M/M_{i+1} is not annihilated by t, a contradiction. q.e.d.

2. Proof of the main theorem

2.1. Construction of the Hilbert compactification. The set of pairs (ν: C̃ ⟶ C, ν^{∗}E), ν being a partial normalization of the stable curve C and E a semistable sheaf of rank r with Euler characteristic χ is clearly bounded. Therefore, we may find an l_{1}, such that, for every l ≥ l_{1}, the following assumptions are met:

Assumption 2.1.1. For every stable curve C, every partial normalization ν: C̃ ⟶ C, resolving the nodes, say, N_{1}, …, N_{κ}, every semistable sheaf E on C, any two points p_{1}, p_{2} ∈ C̃ mapping to a node of C, and Z := ν^{−1}({N_{1}, …, N_{κ}}), one has:

1) H^{1}(C̃, I_{p_{1},p_{2}}ν^{∗}E(l)) = 0.

2) The homomorphism

H^{0}(C̃, I_{p_{1},p_{2}}ν^{∗}E(l)) ⟶ (I_{p_{1},p_{2}}ν^{∗}E(l))/(I_{p_{1},p_{2}}^{2}ν^{∗}E(l))

is surjective.

3) For every x ∈ ν^{−1}(C \ {N_{1}, …, N_{κ}}), the homomorphism

H^{0}(C̃, I_{Z}ν^{∗}E(l)) ⟶ ν^{∗}E(l)/(I_{x}^{2}ν^{∗}E(l))

is surjective.

4) For any two points x_{1} ≠ x_{2} ∈ ν^{−1}(C \ {N_{1}, …, N_{κ}}), the homomorphism

H^{0}(C̃, I_{Z}ν^{∗}E(l)) ⟶ ν^{∗}E(l)_{|{x_{1}}} ⊕ ν^{∗}E(l)_{|{x_{2}}}

is surjective.

In the following, l is assumed to be at least l_{1}. Let ℌ := ℌ(g; χ_{l}, r) be as in the introduction, and let C_{ℌ} ↪ ℌ × 𝔊 be the universal curve. Let ℌ̃ be the open subset of points h for which C_{h} := C_{ℌ} ×_{ℌ} {h} is semistable. Let qℌ̃: V^{χ_{l}} ⊗ O_{Cℌ̃} ⟶ Eℌ̃ be the pullback of the universal quotient. Then, there is a flat family q: C∗ ; ℌ̃ ⟶ ℌ̃ of stable curves

together with an ℌ̃-morphism π: Cℌ̃ ⟶ C∗ ; ℌ̃, such that π_{∗}O_{Cℌ̃} = O_{C∗ ; ℌ̃} and π is fibrewise the contraction onto the stable model. By 1.3.1 ii) (2), we are in the position to apply Proposition 1.3.5. Moreover, we see that the assumptions of Remark 1.3.6 are satisfied. We get the homomorphism

π_{∗}(qℌ̃): V^{χ_{l}} ⊗ O_{C∗ ; ℌ̃} ⟶ π_{∗}Eℌ̃

of ℌ̃-flat sheaves. Let ℌ^{0} be the open set of points h for which: • (π_{∗}Eℌ̃)_{|q^{−1}(h)} is a semistable sheaf.

• π_{∗}(qℌ̃)_{|q^{−1}(h)} is surjective.

• H^{0}(π_{∗}(qℌ̃)_{|q^{−1}(h)}) is an isomorphism.

Set C_{ℌ^{0}}^{∗} := C∗ ; ℌ̃ ×ℌ̃ ℌ^{0}. Then, the quotient family

π_{ℌ^{0}} := π_{∗}(qℌ̃)_{|C_{ℌ^{0}}^{∗}} : V^{χ_{l}} ⊗ O_{C_{ℌ^{0}}^{∗}} ⟶ E_{ℌ^{0}} := (π_{∗}Eℌ̃)_{|C_{ℌ^{0}}^{∗}}

defines a morphism ℌ^{0} ⟶ \overline{𝔔}(g; χ_{l}, r). Let 𝔛 ⊂ ℌ × \overline{𝔔}(g; χ_{l}, r) be the closure of the graph of the above morphism. Let 𝔏_{a,m} be as in the introduction with a so large, that the semistable points in 𝔛 lie in the preimage of the points in \overline{𝔔}(g; χ_{l}, r) which are semistable w.r.t. the linearization in 𝔑.

Theorem 2.1.2. Let x ∈ 𝔛 be a point which is semistable w.r.t. the linearization in 𝔏_{a,m}, a ≫ 0. Then, x is contained in the graph Γ of the morphism ℌ^{0} ⟶ \overline{𝔔}(g; χ_{l}, r).

Proof. By construction of ℌ and 𝔛, the set of (semi)stable points y ∈ 𝔛 corresponding to smooth curves is dense. Let y represent the smooth curve C. By the result of [16], y is (semi)stable (w.r.t. the linearization in 𝔏_{a,m} for all a ≫ 0), if and only if V^{χ_{l}} ⟶ H^{0}(E(l)) is an isomorphism and E is a (semi)stable bundle of rank r with Euler characteristic χ, E := E_{ℌ^{0}|C}. Hence, we may find a smooth curve K, a point k ∈ K, and a morphism κ: K ⟶ 𝔛 with κ(k) = x, such that κ(K \ {k}) is contained in the locus of pairs (C, q_{C}: V^{χ_{l}} ⊗ O_{C} ⟶ E(l)) ∈ Γ for which C is a smooth curve and E is a stable vector bundle. We have an induced morphism \overline{κ}: K ⟶ \overline{𝔔}(g; χ_{l}, r) which lands, by assumption, in the semistable locus. Without loss of generality, we may assume that this morphism is induced by a family (C_{K}^{∗}, q_{K}: V^{χ_{l}} ⊗ O_{C_{K}^{∗}} ⟶ E_{C_{K}^{∗}}). The surface σ: S := C_{K}^{∗} ⟶ K is smooth outside the nodes of σ^{−1}(k) and has singularities of type A_{n} in these nodes. We may resolve these singularities in the usual way in order to get a flat family σ̃: S̃ = C̃_{K} ⟶ K of semistable curves with S̃ smooth. Let q̃_{K} be the pullback of q_{K} to S̃. Then, q̃_{K} defines a rational map S̃ ⟶ 𝔊 which is defined outside some nodes of σ^{−1}(k). By blowing up these nodes and points which are infinitely near to them, we get a new flat family σ′: S′ = C′_{K} ⟶ K and q′_{K}: V^{χ_{l}} ⊗ O_{S′} ⟶ E′_{K} where E′_{K} is locally free. Set C′_{k} := σ′^{−1}(k). Let C̃ be the closure of C′_{k} with all chains of rational curves attached at only two points removed. Then, ν: C̃ ⟶ C is a partial normalization of C. By construction, the morphisms C̃ ⟶ 𝔊 induced by q′_{K|C̃} and ν^{∗}(q_{K|σ^{−1}(k)}) agree, whence these quotients are equivalent. Now, our Assumptions 2.1.1 and Proposition 1.3.3 imply that the image of C′_{k} under the map S′ ⟶ 𝔊 is a semistable curve C′′_{k}, and π: C′_{k} ⟶ C′′_{k} just contracts all rational chains on which E′_{K} is trivial. Next, look at the commutative diagram

C̃  C̃ ⟶^{ν} C
∥  ^{⊂}↓  ∥
C̃ ⟶^{⊂} C′_{k} ⟶^{ν′} C
∥  ^{π}↓  ∥
C̃ ⟶^{⊂} C′′_{k} ⟶^{ν′′} C.

The quotients π_{∗} (q′_{K|σ′^{−1}(k)})_{|C̃} and ν^{∗}(q_{K|σ^{−1}(k)}) are also equivalent.

Thus, there is an isomorphism α: (π_{∗}E′_{K|σ′^{−1}(k)})_{|C̃} ⟶ ν^{∗}(E_{K|σ^{−1}(k)}),

making the following diagram commute:

V^{χ_{l}} ⊗ O_{C′′_{k}} ⟶^{π_{∗}(q′_{K|σ′^{−1}(k)})}  π_{∗}E′_{K|σ′^{−1}(k)}
restriction to C̃↓  ↓restriction to C̃

V^{χ_{l}} ⊗ OC̃ ⟶^{π_{∗}(q′_{K|σ′^{−1}(k)})_{|C̃}} (π_{∗}E′_{K|σ′^{−1}(k)})_{|C̃}
∥  ↓^{α}
V^{χ_{l}} ⊗ OC̃ ⟶^{ν^{∗}(q_{K|σ^{−1}(k)})}  ν^{∗}E_{K|σ^{−1}(k)}.

This latter diagram finally provides us, via projection onto C, with the following commutative diagram:

V^{χ_{l}} ⊗ O_{C} ⟶^{ν′′_{∗}π_{∗}(q′_{K|σ′^{−1}(k)})} ν′′_{∗}π_{∗}E′_{K|σ′^{−1}(k)}
↓  ↓
V^{χ_{l}} ⊗ ν_{∗}OC̃ ⟶^{ν_{∗}ν^{∗}(q_{K|σ^{−1}(k)})} E_{K|σ^{−1}(k)} ⊗ ν_{∗}OC̃.

Furthermore, we have the commutative diagram

V^{χ_{l}} ⊗ O_{C} ⟶^{q_{K|σ^{−1}(k)}}  E_{K|σ^{−1}(k)}
↓  ↓
V^{χ_{l}} ⊗ ν_{∗}OC̃ ⟶^{ν_{∗}ν^{∗}(q_{K|σ^{−1}(k)})} E_{K|σ^{−1}(k)} ⊗ ν_{∗}OC̃

in which the vertical arrows are injective. Therefore, the image of ν′′_{∗}π_{∗}E′_{K|σ′^{−1}(k)} in the sheaf E_{K|σ^{−1}(k)} ⊗ ν_{∗}OC̃ is E_{K|σ^{−1}(k)}. The kernel of the surjection

ν′′_{∗}π_{∗}E′_{K|σ′^{−1}(k)} ⟶ E_{K|σ^{−1}(k)}

must be zero, because both sheaves have the same Hilbert polynomial w.r.t. O_{C}(1).

To conclude, the quotient q′_{K} defines a K-morphism ϑ: C′_{K} ⟶ K ×𝔊 and we have seen:

(a) The image is a flat family of curves σ′′: C′′_{K} ⟶ K with C′′ as the fibre over k.

(b) The family q′′_{K} := ϑ_{∗}(q′_{K}): V^{χ_{l}} ⊗O_{C′′_{K}} ⟶ E_{C′′_{K}} := ϑ_{∗}E_{C′_{K}} is a flat quotient (by Proposition 1.3.5), such that σ′′_{∗}(q′′_{K}) is a quotient onto a family of semistable torsion-free coherent sheaves.

Therefore, the family C′′_{K} defines a morphism κ′: K ⟶ Γ ⊂ 𝔛. Since we have not altered the original family outside the point k, κ′ agrees with the original morphism κ outside k, and, thus, everywhere, because 𝔛 is separated. This shows x = κ′(k) ∈ Γ. q.e.d.

Remark 2.1.3. One could also use the arguments presented in the paper [12]. We have chosen the alternative way, also suggested in [12], because it reflects more of the moduli problem we are dealing with.

Therefore, the GIT-quotient ℌ^{0}//_{𝔏_{a,m}} SL(V^{χ_{l}}) exists as a projective scheme over \overline{𝔐}_{g}. It also comes with an \overline{𝔐}_{g}-morphism to \overline{𝔘}(g; χ, r). The hard task will be to give it a modular interpretation.

2.2. Analysis of semistability. Fix the data g, χ, and r, and set χ_{l} := χ + l · r · (2g − 2). Let C be a semistable curve of genus g. Denote by π: C ⟶ C′ the contraction onto the stable model of C. Let E be a vector bundle of uniform rank r on C with Euler characteristic χ(E) = χ, such that E has positive degree on each rational component and π_{∗}(E) is torsion-free. From Proposition 1.3.3, we infer that, for sufficiently large l, H^{0}(E(l)) ⊗ O_{C} ⟶ E(l) will give rise to a closed embedding C ↪ 𝔊(H^{0}(E(l)), r) and H^{1}(E(l)) = {0}. Identifying H^{0}(E(l)) with some fixed vector space V^{χ_{l}} of dimension χ_{l}, we may ask whether [C] ∈ ℌ^{0}(g; χ_{l}, r) is SL(V^{χ_{l}})-(semi)stable w.r.t. the linearization in 𝔏_{a,m} for large a. We already know from Pandharipande’s construction and Proposition 1.1.1 that:

(a) If π_{∗}(E) is stable w.r.t. the canonical polarization, then [C] will be stable.

(b) If [C] is semistable, then π_{∗}(E) is semistable w.r.t. the canonical polarization.

If π_{∗}(E) is properly semistable, then there will be additional conditions for [C] to be (semi)stable. We will have to analyze those conditions. Abstractly, by Remark 1.1.2, they can be described as follows: suppose we are given q_{C}: V^{χ_{l}}⊗O_{C} ⟶ E(l), such that H^{0}(q_{C}) is an isomorphism and π_{∗}(E) is semistable w.r.t. the canonical polarization. Then, [C] will be SL(V^{χ_{l}})-(semi)stable, if and only if for every one-parameter subgroup λ: ℂ^{∗} ⟶ SL(V^{χ_{l}}) with μ_{𝔑}(λ, [π_{∗}(q_{C})]) = 0, one has

(1) μ_{𝔏_{m}}(λ, ∧^{P (m)}Ψ_{ℌ(g;χ_{l},r)})(≥)0, Ψ_{ℌ(g;χ_{l},r)} as in the introduction.

Let us first recall when μ_{𝔑}(λ, [π_{∗}(q_{C})]) = 0 happens. For this, suppose λ is given with respect to the basis v_{1}, …, v_{χ_{l}} by the weight vector

(2) \underline{γ} = ∑_{i=1}^{χ_{l}−1} α_{i}(\underbrace{i − χ_{l}, …, i − χ_{l}}_{i×}, \underbrace{i, …, i}_{χ_{l}×}),

α_{i} ∈ ℚ_{≥0}, i = 1, …, χ_{l} − 1,

and let i_{1} < i_{2} < ⋯ < i_{k} be the indices with α_{i} > 0. Then, we get a filtration V^{•} : {0} =: V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{k} ⊂ V_{k+1} := V^{χ_{l}} with V_{j} := ⟨ v_{1}, …, v_{i_{j}} ⟩, j = 1, …, k. Recall that μ_{𝔑}(λ, [π_{∗}(q_{C})]) depends only on the pair (V^{•}, \underline{α}), \underline{α} := (α_{i_{1}}, …, α_{i_{k}}) ([11], Section 2.2). Set

F̃_{j} := π_{∗}(q_{C}(V_{j} ⊗ O_{C})(−l)), j = 1, …, k.

Now, from the construction of \overline{𝔘}(g; χ_{l}, r) and Lemma 1.2.1, one knows that the equality μ_{𝔑}(λ, [π_{∗}(q_{C})]) = 0 will occur, if and only if:

• ∑_{j=1}^{k} α_{i_{j}}(P (π_{∗}(E)) trk(F̃_{j}) − P (F̃_{j}) trk(E)) = 0, i.e., each F̃_{j} destabilizes E.

• H^{0}(π_{∗}(q_{C}))(V_{j}) = H^{0}(F̃_{j}(l)), j = 1, …, k. Recall that the family of pairs (C, F) with C a stable curve and F a destabilizing subsheaf of a semistable torsion-free sheaf E of uniform rank r on C with χ(E) = r is bounded. In view of Proposition 1.3.1, we can find an l_{1}, such that for all l ≥ l_{1}, the following assumptions are verified:

Assumption 2.2.1. Let [C] ∈ ℌ^{0}(g; χ_{l}, r) be semistable w.r.t. the linearization in 𝔏_{a,m}(g; χ_{l}, r) for a ≫ 0. Denote by π: C ⟶ C′ the contraction onto the stable model, and by q_{C}: V^{χ_{l}} ⊗ O_{C} ⟶ E the pullback of the universal quotient. Then, π_{∗}(E(−l)) is, by definition, semistable and H^{0}(π_{∗}(q_{C})) is an isomorphism. For every destabilizing subsheaf F′ ⊂ E(−l), define

F := q_{C}((H^{0}(π_{∗}(q_{C}))^{−1}(H^{0}(F′(l)))) ⊗ O_{C}).

Note that π_{∗}(F(−l)) = F′. Then, we assume:

1) For every irreducible component C_{ι}, the restriction homomorphism H^{0}(C, F) ⟶ H^{0}(C_{ι}, F_{|C_{ι}}) is surjective.

2) For every irreducible component C_{ι} which is not a rational curve attached at only two points and on which F has positive rank, and any two points p_{1} ≠ p_{2} ∈ C_{ι}, the evaluation map

H^{0}(C_{ι}, F_{|C_{ι}}) ⟶ F_{|C_{ι}} ⊗_{O_{C}} O_{{p_{1},p_{2}}}

is surjective.

3) For every irreducible component C_{ι} which is not a rational curve attached at only two points and on which F has positive rank and any point p ∈ C_{ι}, the evaluation map

H^{0}(C_{ι}, F_{|C_{ι}}) ⟶ F_{|C_{ι}} ⊗_{O_{C}} (𝔪_{p,C_{ι}}/𝔪_{p,C_{ι}}^{2})

is surjective.

Next, we fix a maximal filtration F_{•}: 0 =: F′_{0} ⊊ F′_{1} ⊊ F′_{2} ⊊ ⋯ ⊊ F′_{k} ⊊ F′_{k+1} := π_{∗}(E) of π_{∗}(E) by destabilizing subsheaves and a vector \underline{α} := (α_{1}, …, α_{k}) of nonnegative rational numbers. Let us also fix some l and define V_{j} := H^{0}(F′_{j}(l)) under the identification of V^{χ_{l}} with H^{0}(π_{∗}(E)(l)) and F_{j} := q_{C}(V_{j} ⊗ O_{C}), j = 1, …, k.

Remark 2.2.2. The F_{j} are saturated subsheaves of E, j = 1, …, k. For, if F̃_{j} is the saturation of F_{j}, then F̃_{j}(l′) will be globally generated for some l′ large enough. This is clear for points which lie on a component on which ω_{C} is ample. Now, let R be the disjoint union of all maximal chains of rational curves attached at only two points, set C^{∗} := C \ R, let \overline{C}^{∗} be the closure of C^{∗}, and define \underline{x} := \overline{C}^{∗} \ C^{∗}. Then, H^{1}(\overline{C}^{∗}, I\underline{x}F̃_{j}(l′)) is zero for l′ ≫ 0, i.e., H^{0}(C, F̃_{j}(l′)) ⟶ H^{0}(R, F̃_{j|R}(l′)) is surjective. As F̃_{j|R}(l′) is globally generated, the claim is settled. Now, if F̃_{j} strictly contains F_{j}, then h^{0}(C, F̃_{j}(l′)) > h^{0}(C, F_{j}(l′)) for all l′ ≫ 0. This is, however, not possible, because the inclusion F′_{j}(l + l′) = π_{∗}F_{j}(l′) ⊂ π_{∗}F̃_{j}(l′) must be an equality as both sheaves have the same multi-rank and F′_{j} is saturated.

Note that, given a basis v_{1}, …, v_{χ_{l}} of V^{χ_{l}} such that ⟨ v_{1}, …, v_{dim V_{j}} ⟩ = V_{j}, j = 1, …, k, the vector \underline{α} defines a one-parameter subgroup λ of SL(V^{χ_{l}}), by Formula (2). By [11], 2.2, again, the values μ_{𝔑}(λ, [π_{∗}(q_{C})]) and μ_{𝔏_{m}}(λ, [∧^{P (m)}Ψ_{ℌ(g;χ_{l},r)}]) do not depend on the choice of such a basis. It will be our task to compute the value of μ_{𝔏_{m}}(λ, [∧^{P (m)}Ψ_{ℌ(g;χ_{l},r)}]). In other words, we will have to find a basis for H^{0}(C, det(E(l)^{⊗m})) for which the weight of the associated element in Hom(∧^{P (m)}S^{m}V, ℂ) becomes minimal. Then, μ_{𝔏_{m}}(λ, ∧^{P (m)}ψ) will be minus that weight.

Case A). The irreducible components of C are smooth. Write C = ⋃_{ι=1}^{c} C_{ι} as the union of its irreducible components and set L_{m} := det(E(l))^{⊗m}. For large m, the restriction map H^{0}(C, L_{m}) ⟶ H^{0}(C_{ι}, L_{m}^{ι}), L_{m}^{ι} := L_{m|C_{ι}}, will be surjective for ι = 1, …, c. We may, therefore, compute first the weights of a basis for H^{0}(C_{ι}, L_{m}^{ι}), ι = 1, …, c. Set E_{ι} := E(l)_{|C_{ι}} and F_{j}^{ι} := Im(F_{j} ⟶ E_{ι}), j = 1, …, k, ι = 1, …, c. We also define Ṽ_{j} := ⟨ v_{dim V_{j−1}+1}, …, v_{dim V_{j}} ⟩, j = 2, …, k +1, and Ṽ_{1} := V_{1}. Then

⋀^{r} V = ⨁ ; (ρ_{1},...,ρ_{k+1}):∑ ρ_{j}=r W_{ρ_{1},...,ρ_{k+1}},

W_{ρ_{1},...,ρ_{k+1}} := ⋀^{ρ_{1}} Ṽ_{1} ⊗ ⋀^{ρ_{2}} Ṽ_{2} ⊗ ⋯ ⊗ ⋀^{ρ_{k+1}} Ṽ_{k+1}.

The spaces W\underline{ρ} are weight spaces for λ for the weight

w\underline{α} ; ι,\underline{ρ}(l) := ρ_{1} · γ_{1}(l) + ⋯ + ρ_{k+1} · γ_{k+1}(l)

where γ_{j}(l) is the weight of a section in Ṽ_{j}, j = 1, …, k +1. Formula (2) shows that the γ_{j}(l) are, in fact, polynomials in l.

Remark 2.2.3. Let N ∈ C be a node of C, i.e., a point where two components C_{1} and C_{2} of C meet. Then, the stalk of a torsion-free sheaf of Q at N is of the form O_{C,N}^{⊕a_{1}} ⊕ O_{C_{1},N}^{⊕a_{2}} ⊕ O_{C_{2},N}^{⊕a_{3}} ([17], Huitiême Partie, Proposition 3). Thus, if E is a torsion-free sheaf and F is a saturated subsheaf, then the image F_{1}of F in E_{|C_{1}} is an almost saturated subsheaf, because Tors(E_{|C_{1}}/F_{1}) ≅ ℂ^{⊕a_{3}} as O_{C_{1},N}-module. Therefore, Proposition 1.4.1 allows one to determine the vanishing orders of sections coming from a weight space W\underline{ρ}. This observation will be crucial for the following subtle analysis of weights and vanishing orders.

Let us look at some specific ι ∈ {1, …, c}. Then, the space of minimal weight which produces sections which do not vanish on C_{ι} is W_{min}^{ι} := W_{rk F_{1}^{ι},rk F_{2}^{ι}−rk F_{1}^{ι},...,rk E_{ι}−rk F_{k}^{ι}}. The associated weight is

w\underline{α} ; ι,min(l) := ∑_{j=1}^{k} α_{j}(dim V_{j} · rk E_{ι} − χ_{l} · rk F_{k}^{ι})

= ∑_{j=1}^{k} α_{j}(P (F′_{j})(l) · rk E_{ι} − P (π_{∗}(E))(l) · rk F_{j}^{ι}).

Let N_{1}^{ι}, …, N_{ν_{ι}}^{ι} be the nodes of C located on C_{ι}, i.e., the points where C_{ι} meets other components of C. Note that each F_{j}^{ι} is a subbundle of E_{ι} outside the above points. This is because F_{j} is a saturated subsheaf of E, by Remark 2.2.2, j = 1, …, k. Let us look at a specific node N ∈ {N_{1}^{ι}, …, N_{ν_{ι}}^{ι}}. Let 𝔼 be the fibre of E at N, and 𝔽_{j} the image of F_{j}^{ι} in 𝔼, j = 1, …, k. Set a_{j} := dim 𝔽_{j}, r_{j} := rk F_{j}^{ι}, and b_{j} := min{a_{j} − a_{j−1}, r_{j} − r_{j−1}}, j = 1, …, k + 1. A general section of W_{min}^{ι} will vanish of order o_{N} := r−b_{N} at N. This is an immediate consequence of Proposition 1.4.1.

Lemma 2.2.4.

i) The sections in the subspace W_{min}^{ι} generate the line bundle det(E_{|C_{ι}}(l)(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s})).

ii) The image of W_{min}^{ι} in H^{0}(det(E_{|C_{ι}}(l)(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s}))) is a very ample linear system, unless C_{ι} is rational and attached at only two points and det(E_{|C_{ι}}(l)(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s})) is trivial.

Proof. The assertion i) about global generation results immediately from Assumption 2.2.1 (1). Likewise, 2.2.1 (2) and (3) settle the very ampleness in ii) when C_{ι} is not a rational component which is attached at only two points. In the remaining case, E_{ι} is of the form O_{ℙ_{1}}^{⊕s} ⊕ O_{ℙ_{1}}(1)^{⊕t}. The F_{j}^{ι}, as globally generated subsheaves of E_{ι}, are also of the form O_{ℙ_{1}}^{⊕s_{j}} ⊕ O_{ℙ_{1}}(1)^{⊕t_{j}}, j = 1, …, k. Let r_{1}, …, r_{b+1} be the ranks occurring among the F_{j}^{ι}, and for β = 1, …, b + 1, let G_{β} be the first among the F_{j}^{ι} to attain that rank. There are now two possibilities: either H^{0}(G_{β+1})/H^{0}(G_{β}) contains a subspace H^{0}(O_{ℙ_{1}}(1)) for some β ∈ {0, …, b} or not. In the first case, by 2.2.1 (1), it is easy to explicitly construct sections separating points and tangent vectors. The second case, however, can only occur, if all the G_{β} are trivial, again by 2.2.1. But then, det(E_{|C_{ι}}(l)(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s})) is obviously trivial. q.e.d.

Corollary 2.2.5. The space S^{m}W_{min}^{ι} generates

H^{0} ((det(E(l))_{|C_{ι}} (− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s}))^{⊗m})

for all m ≫ 0.

Let d′_{ι} be the degree of the line bundle det(E_{|C_{ι}}(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s})) and e_{ι} := deg(ω_{C|C_{ι}}). Because of Corollary 2.2.5, the elements in S^{m}W_{min}^{ι} will contribute the weight

K\underline{α} ; ι(l, m) := m · (m · (d′_{ι} + l · r · e_{ι}) + 1 − g(C_{j})) · w\underline{α} ; ι,min(l)

to a basis of H^{0}(C_{ι}, L_{m}^{ι}). Thus, we only have to worry about sections vanishing of lower order than m · o_{N} at N. Note that W_{ρ_{1},...,ρ_{k+1}} will produce sections which do not vanish on C_{ι} if and only if the condition

∑_{i=1}^{j} ρ_{i} ≤ r_{j}, j = 1, …, k, (3)

is satisfied. A tuple \underline{ρ} = (ρ_{1}, …, ρ_{k+1}) satisfying (3) will be called admissible. Note that there are only finitely many admissible tuples.

Next, we let κ_{1} < ⋯ < κ_{s} be the elements in {1, …, k + 1} with a_{κ} − a_{κ−1} > r_{κ} − r_{κ−1}. Set

K := {κ_{1}, …, κ_{s}} and K^{∗} := {1, …, k + 1} \ K.

Lemma 2.2.6. Fix a vanishing order o < o_{N}, and let w\underline{α} ; ι,\underline{ρ}(l) be the minimal weight of a section with vanishing order o. Then, \underline{ρ} may be chosen to satisfy

\begin{aligned}r_{κ} − r_{κ−1} & ≤ & ρ_{κ} & ≤ & a_{κ} − a_{κ−1} & for κ ∈ K \\ a_{κ} − a_{κ−1} & ≤ & ρ_{κ} & ≤ & r_{κ} − r_{κ−1} & for κ ∈ K^{∗}. \end{aligned}

Proof. We begin with the right-hand side inequalities. Suppose ρ_{κ} violates the right-hand inequality. In particular, ρ_{κ} > r_{κ} −r_{κ−1}, whence ∑_{j=1}^{κ−1} ρ_{j} < r_{κ−1}. Define \underline{ρ}′ = (ρ′_{1}, …, ρ′_{k+1}) by ρ′_{κ−1} := ρ_{κ−1} + 1, ρ′_{κ} := ρ_{κ} − 1, and ρ′_{j} := ρ_{j} for j ≠ κ − 1, κ. Then, \underline{ρ}′ is obviously admissible, w\underline{α} ; ι,\underline{ρ}′(l) ≤ w\underline{α} ; ι,\underline{ρ}(l), and W\underline{ρ′} will still produce sections of vanishing order o. The latter property results from the fact that ρ_{κ} was, by assumption, strictly bigger than the maximal number of sections in Ṽ_{κ} with linearly independent images in 𝔼.

The other inequality asserts ρ_{j} ≥ b_{j} for j = 1, …, k + 1. For k + 1, we have ∑_{j=1}^{k} ρ_{j} ≤ r_{k}, so ρ_{k+1} = r − ∑_{j=1}^{k} ρ ≥ r_{k+1} − r_{k}. Suppose ρ_{j_{0}} < b_{j_{0}}. Then, there is an index j′ > j_{0} with ρ_{j′} ≠ 0. Otherwise, ∑_{j=1}^{j_{0}} ρ_{j} = r and then ρ_{j_{0}} ≥ r_{j_{0}} − r_{j_{0}−1} as before, a contradiction. Let j′ be minimal with the above properties. Define \underline{ρ}′ = (ρ′_{1}, …, ρ′_{k+1}) with ρ′_{j_{0}} := ρ_{j_{0}} + 1, ρ′_{j′} := ρ_{j′} − 1, and ρ′_{j} = ρ_{j} for j ≠ j_{0}, j′. As ∑_{j=1}^{j_{0}−1} ρ_{j} ≤ r_{j_{0}−1} and ρ_{j_{0}} < r_{j_{0}} −r_{j_{0}−1}, \underline{ρ}′ is again admissible. Moreover, w\underline{α} ; ι,\underline{ρ}′(l) ≤ w\underline{α} ; ι,\underline{ρ}(l) and W\underline{ρ′} will still lead to sections of vanishing order o. This time, the last assertion is the consequence of the assumption that ρ_{κ} was strictly smaller than the maximal number of sections in Ṽ_{κ} with linearly independent images in 𝔼. q.e.d.

Let I ⊂ K^{∗} × K be the set of all (i, j) with i < j. Fix an order “⪯” on I, such that (i′, j′) ⪯ (i, j) implies γ_{j′}(l) − γ_{i′}(l) ≤ γ_{j}(l) − γ_{i}(l). The idea for the following investigations is the following: suppose we are given an admissible tuple \underline{ρ}, satisfying the inequalities of Lemma 2.2.6, and (i, j) ∈ I. Then, we define a new tuple \underline{ρ}′ with ρ′_{i} := ρ_{i} − 1 and ρ′_{j} = ρ_{j} + 1 and let all other entries of \underline{ρ}′ agree with those of \underline{ρ}. As i < j, \underline{ρ}′ is still admissible. However, we will perform this operation only if \underline{ρ}′ still satisfies the inequalities of Lemma 2.2.6. In that case, the generic vanishing order of sections in W\underline{ρ}′ will be one less than the generic vanishing order in W\underline{ρ}. Thus, if we are given a specific vanishing order o, we carry out s := o_{N} − o operations of the above type as follows: we start with (i, j) which is minimal w.r.t. “⪯” (because the corresponding process will increase the weight the least), perform the operation on (i, j) as many times as possible, say s_{(i,j)} times, then pass to the next pair (i′, j′) ∈ I w.r.t. the order “⪯” and so on, until we have performed s such processes in total. Then, we arrive at a tuple \underline{ρ}′, such that the generic vanishing order of sections from W\underline{ρ}′ is precisely o. The difficult part is to show that the corresponding weight will be, in fact, minimal.

Fix a vanishing order o < o_{N}, let w\underline{α} ; ι,\underline{ρ}(l) be the minimal weight of a section which vanishes of order o, and assume that \underline{ρ} satisfies the inequalities of Lemma 2.2.6. Then, we define natural numbers s_{(i,j)} for (i, j) ∈ I inductively w.r.t. “⪯” as follows: for (i, j) ∈ I, set

c_{i} = ∑_{(i,j′)≺(i,j)} s_{(i,j′)}

c_{j} = ∑_{(i′,j)≺(i,j)} s_{(i′,j)},

where empty sums are by definition zero. Then,

s_{(i,j)} := min{r_{i} − r_{i−1} − ρ_{i} − c_{i}, ρ_{j} − r_{j} + r_{j−1} − c_{j} }.

Observation 2.2.7.

i) For every index κ ∈ K^{∗} and every admissible tuple \underline{ρ}, satisfying the conditions of Lemma 2.2.6, we have

∑_{i∈K:i<κ}(ρ_{i} − r_{i} + r_{i−1}) ≤ ∑_{j∈K^{∗}:j≤κ}(r_{j} − r_{j−1} − ρ_{j}).

From this, one easily infers that

ρ_{j} = r_{j} − r_{j−1} + ∑_{(i,j)∈I} s_{(i,j)}  for j ∈ K

ρ_{i} = r_{i} − r_{i−1} − ∑_{(i,j)∈I} s_{(i,j)}  for i ∈ K^{∗},

so that the s_{(i,j)} determine \underline{ρ}. ii) Suppose we are given a tuple \underline{s} = (s′_{(i,j)}, (i, j) ∈ I) with 0 ≤ s′_{(i,j)} ≤

s_{(i,j)} for all (i, j) ∈ I. Define \underline{ρ}\underline{s} by

ρ\underline{s} ; j = r_{j} − r_{j−1} + ∑_{(i,j)∈I} s′_{(i,j)}  for j ∈ K

ρ\underline{s} ; i = r_{i} − r_{i−1} − ∑_{(i,j)∈I} s′_{(i,j)}  for i ∈ K^{∗}.

The tuple \underline{ρ}\underline{s} is clearly admissible.

Lemma 2.2.8. Fix a vanishing order o < o_{N}, and let w\underline{α} ; ι,\underline{ρ}(l) be the minimal weight of a section with vanishing order o where \underline{ρ} fulfills the conditions of Lemma 2.2.6. Then, \underline{ρ} may be chosen in such a way that the s_{(i,j)} satisfy

s_{(i,j)} = min{s − ∑_{(i′,j′)≺(i,j)} s_{(i′,j′)}, r_{i} − r_{i−1} − a_{i} + a_{i−1} − c_{i},

a_{j} − a_{j−1} − r_{j} + r_{j−1} − c_{j} }.

Here, s := ∑ s_{(i,j)}, and c_{i} and c_{j} are as before.

Proof. Assume that the assertion were wrong for s_{(i,j)}, i.e., s_{(i,j)} is strictly smaller than the right-hand side. Then, there are three cases. In the first case, s_{(i,j)} = r_{i} − r_{i−1} − ρ_{i} − c_{i} = ρ_{j} − r_{j} + r_{j−1} − c_{j}. In that case

ρ_{i} > a_{i} − a_{i−1} and ρ_{j} < a_{j} − a_{j−1}. (4)

Then, there is an (i′, j′) ≻ (i, j) with s_{(i′,j′)} > 0. Set \underline{ρ}′ = (ρ′_{1}, …, ρ′_{k+1}) with ρ′_{i′} := ρ_{i′} + 1, ρ′_{j′} := ρ_{j′} − 1, and ρ′_{j} = ρ_{j} for j ≠ i′, j′. The tuple \underline{ρ}′ is still admissible, by Observation 2.2.7. It is, in fact, defined w.r.t. \underline{s} with s′_{(i,j)} = s_{(i,j)}, (i, j) ≠ (i′, j′), and s′_{(i′,j′)} = s_{(i′,j′)} − 1. Introduce \underline{ρ}′′ = (ρ′′_{1}, …, ρ′′_{k+1}) by ρ′′_{i} := ρ′_{i} − 1, ρ′′_{j} := ρ′_{j} + 1, and ρ′′_{j′} := ρ′_{j′} for j′ ≠ i, j. This is again admissible, w\underline{α} ; ι,\underline{ρ}′′(l) ≤ w\underline{α} ; ι,\underline{ρ}(l), and W\underline{ρ}′′ still contains sections of vanishing order o, by (4). In other words, we set s′_{(i,j)} := s_{(i,j)} + 1.

In the second case, s_{(i,j)} = r_{i} − r_{i−1} − ρ_{i} − c_{i} < ρ_{j} − r_{j} + r_{j−1} − c_{j}. Then, as ρ_{j} − r_{j} + r_{j−1} − c_{j} > 0, there is an index (i′, j) ≻ (i, j) with s_{(i′,j)} > 0. One may now proceed as before. The last case, s_{(i,j)} = ρ_{j} − r_{j} + r_{j−1} − c_{j} < r_{i} − r_{i−1} − ρ_{i} − c_{i}, is handled the same way. q.e.d.

Given s, the condition in Lemma 2.2.8 uniquely determines a tuple \underline{ρ} with ∑ s_{(i,j)} = s for which w\underline{α} ; ι,\underline{ρ}(l) becomes minimal. Note that W\underline{ρ} yields sections with vanishing order ≥ o_{N} − s where “=” is achieved. An immediate consequence is:

Corollary 2.2.9.

i) Fix a vanishing order o < o_{N}, and let w\underline{α} ; ι,\underline{ρ}(l) be the minimal weight of a section with vanishing order o where \underline{ρ} fulfills the conditions of Lemmas 2.2.6 and 2.2.8. Then, ∑ s_{(i,j)} = o_{N} − o.

ii) Denote by w_{ι,\underline{α}}^{N,o}(l) the minimal weight of a section with vanishing order o at N. Then,

w_{ι,\underline{α}}^{N,o−1}(l) − w_{ι,\underline{α}}^{N,o}(l) ≥ w_{ι,\underline{α}}^{N,o}(l) − w_{ι,\underline{α}}^{N,o+1}(l).

Define 𝕆_{o} := W\underline{ρ}, where the tuple \underline{ρ} is determined by the conditions of Lemmas 2.2.6 and 2.2.8 and ∑ s_{(i,j)} = o_{N} − o, and 𝕆 := ⨁_{o=0}^{o_{N}} 𝕆_{o}. We have to find the minimal weights of sections in H^{0}(C_{ι}, L_{m}^{ι}) vanishing of order 0 ≤ o′ ≤ m · o_{N} − 1. We clearly have to look only at sections in

S^{m}𝕆 = ⨁ ; m_{0},...,m_{o_{N}}:∑ m_{ν}=m S^{m_{0}}𝕆_{0} ⊗ ⋯ ⊗ S^{m_{o_{N}}} 𝕆_{o_{N}} .

Now, the sections in S^{m_{0}}𝕆_{0} ⊗ ⋯ ⊗ S^{m_{o_{N}}} 𝕆_{o_{N}} vanish of order at least m_{1} + 2 · m_{2} + ⋯ + o_{N} · m_{o_{N}}, and we can find some with exactly that vanishing order. On the other hand, the weight of sections in that space is

m_{0} · w_{ι,\underline{α}}^{N,0}(l) + ⋯ + m_{o_{N}} · w_{ι,\underline{α}}^{N,o_{N}} (l)
= m · w_{ι,\underline{α}}^{N,0}(l) − m_{0}(w_{ι,\underline{α}}^{N,0}(l) − w_{ι,\underline{α}}^{N,1}(l)) − ⋯ −
− m_{o_{N}} (w_{ι,\underline{α}}^{N,0}(l) − w_{ι,\underline{α}}^{N,o_{N}} (l))
= m · w_{ι,\underline{α}}^{N,0}(l) − m_{0}(w_{ι,\underline{α}}^{N,0}(l) − w_{ι,\underline{α}}^{N,1}(l)) − ⋯ −
− m_{o_{N}} ((w_{ι,\underline{α}}^{N,0}(l) − w_{ι,\underline{α}}^{N,1}(l)) + ⋯ + (w_{ι,\underline{α}}^{N,o_{N} −1}(l) − w_{ι,\underline{α}}^{N,o_{N}} (l))).

It follows easily from Corollary 2.2.9 ii) that the elements in S^{m}𝕆 producing sections of minimal weight vanishing of order o with (t−1)·m ≤ o ≤ t · m − 1 lie in

⨁_{i=1}^{m} S^{i}𝕆_{t−1} ⊗ S^{m−i}𝕆_{t}, t = 1, …, o_{N}.

These contribute the weight

m · w_{ι,\underline{α}}^{N,t}(l) + \frac{m(m − 1)}{2}(w_{ι,\underline{α}}^{N,t−1}(l) − w_{ι,\underline{α}}^{N,t}(l))

to a basis for ℍ := H^{0}(C_{ι}, L_{m}^{ι})/H^{0}((det(E(l))_{|C_{ι}}(− ∑_{s=1}^{ν_{ι}} o_{N_{s}}N_{s}))^{⊗m}). The total contribution to a basis for ℍ, coming from the node N, thus amounts to

C_{ι}^{N,\underline{α}}(l, m) := m(w_{ι,\underline{α}}^{N,0}(l) + ⋯ + w_{ι,\underline{α}}^{N,o_{N}} (l))

+ \frac{m(m − 1)}{2}(w_{ι,\underline{α}}^{N,0}(l) − w_{ι,\underline{α}}^{N,o_{N}} (l)).

All in all, a basis for H^{0}(C_{ι}, L_{m}^{ι}) will have minimal weight

C\underline{α} ; ι(l, m) := K\underline{α} ; ι(l, m) + ∑_{n=1}^{ν_{ι}} C_{ι}^{N_{n},\underline{α}}(l, m).

Let N be an intersection of two components C_{ι} and C_{ι′} of C. Then,

w_{ι,\underline{α}}^{N,0}(l) = w_{ι′,\underline{α}}^{N,0}(l) =: wN ; \underline{α}(l).

Let N be the set of nodes of C. For large m, there is the exact sequence

0 ⟶ H^{0}(C, det E^{⊗m}) ⟶ ⨁_{ι=1}^{c} H^{0}(C_{ι}, L_{m}^{ι}) ⟶ ⨁_{N∈N} ℂ · e_{N} ⟶ 0.

This shows that

H^{0}(C, det E^{⊗m}) = ⨁_{ι=1}^{c} H^{0} (C_{ι}, L_{m}^{ι} (− ∑_{n=1}^{ν_{ι}} N_{n}^{ι})) ⊕ ⨁_{N∈N} ℂ · e_{N}.

Thus, we see that the minimal weight of a basis for H^{0}(C, det E^{⊗m}) is

P \underline{α} ; F_{•}(l, m) := ∑_{ι=1}^{c} C\underline{α} ; ι(l, m) − m · ∑_{N∈N} wN ; \underline{α}(l).

Note that this polynomial is intrinsically defined in terms of the curve C, the filtration F_{•}, and \underline{α}.

Case B). C has nodal irreducible components. In this case, we pass to the semistable curve π: C′ ⟶ C, where we introduce a projective line for every node at which C is irreducible and the filtration by the F_{j} is not a filtration by subbundles, and pullback E to E′ on C′. Now, let C′_{ι} be any irreducible component of C′. Then, ν_{ι} := π_{|C′_{ι}}: C′_{ι} ⟶ C_{ι} is a partial normalization. For any node N ∈ C_{ι} which is resolved by ν_{ι}, W_{min}^{ι} will produce sections of det(E′(l)) which vanish at both points p_{N,1} and p_{N,2} in ν_{ι}^{−1}(N). As the space of sections of det(E(l)) vanishing at N identifies with the space of sections of det(E′(l)) vanishing at both p_{N,1} and p_{N,2}, it is easy to see that the analogs of Lemma 2.2.4 and Corollary 2.2.5 continue to hold. The rest of the considerations clearly go through as before.

H-semistability. We are now ready to define our semistability concept.

Definition 2.2.10. A pair (C, E), consisting of a semistable curve C and a vector bundle E of rank r on C with χ(E) = χ, will be called H-(semi)stable, if it satisfies the following conditions:

1) The push forward π_{∗}(E) to the stable model via π: C ⟶ C′ is a semistable torsion-free sheaf (see the remark in the introduction).

2) For every maximal filtration of F_{•} of π_{∗}(E) by destabilizing subsheaves and every vector \underline{α} of nonnegative rational numbers, there exists an index l^{∗}, such that for all l ≥ l^{∗}

P \underline{α} ; F_{•}(l, m) (⪯) 0 as polynomial in m.

Note that this semistability concept has all the properties that were asserted in the introduction. Part ii) of the main theorem is a direct consequence of:

Theorem 2.2.11. There exist an index l_{0} and for every l ≥ l_{0} an index m(l), such that for every l ≥ l_{0}, m′ ≥ m(l), and every pair (C, E), consisting of a semistable curve C and a vector bundle E of rank r on C with χ(E) = χ, which satisfies (1) and (2) of Definition 2.2.10

P \underline{α} ; F_{•}(l, m′) (≤) 0 ⟺ P \underline{α} ; F_{•}(l, m) (⪯) 0 as polynomial in m

for every filtration F_{•} and every tuple \underline{α}.

Proof. First note that, given \underline{α}, P \underline{α} ; F^{•} depends only on the following data:

• The tuples (rk F_{1}^{ι}, …, rk F_{k}^{ι}), ι = 1, …, c. These determine all the Hilbert polynomials of the F′_{j}, j = 1, …, k, because these are destabilizing sheaves.

• The tuples (a_{1}^{N}, …, a_{k}^{N}), N a node of C, and a_{j}^{N} the dimension of the image of F_{j} in the fibre of E at N, j = 1, …, k.

By boundedness, the sets of data of the above type is in fact finite. Therefore, we will be done, once we have shown that, for a given set of such data, we have to take only finitely many vectors \underline{α} into account.

Given tuples (r_{1}^{ι}, …, r_{k}^{ι}), ι = 1, …, c, and (a_{1}^{N}, …, a_{k}^{N}), N a node of C, we define sets K_{N,ι} and K_{N,ι}^{∗} as before. Note that in our construction before, we had to look at the quantities γ_{j}(l)−γ_{i}(l), (i, j) ∈ K_{N,ι}^{∗}×K_{N,ι}. By Formula (2),

γ_{j}(l) − γ_{i}(l) = ∑_{t=i}^{j−1} α_{t} · χ_{l}.

For every ordering “⪯_{N,ι}” of K_{N,ι}^{∗} × K_{N,ι}, we get the set of inequalities

∑_{t=i′}^{j′−1} α_{t} ≤ ∑_{t=i}^{j−1} α_{t}, (i′, j′) ⪯_{N,ι} (i, j). (∗)_{⪯_{N,ι}}

Let Q ⊂ ℝ^{k} be the quadrant of vectors all the entries of which are nonnegative. This is a rational polyhedral cone. For a given ordering “⪯_{N,ι}”, the inequalities (∗)_{⪯_{N,ι}} define a proper rational polyhedral subcone of Q. Given two distinct orderings, the resulting cones will meet only along faces, i.e., if we let “⪯_{N,ι}” vary over all possible orderings, we get a fan decomposition Q = ⋃_{β=1}^{B_{N,ι}} Q_{β}^{N,ι} of Q.

We have seen that once the ordering “⪯_{N,ι}” is fixed, every given vanishing order o uniquely determines a vector \underline{ρ} with w\underline{α} ; ι,\underline{ρ}(l) = w_{ι,\underline{α}}^{N,o}(l) for all \underline{α} in the cone Q_{β}^{N,ι} cut out by the inequalities (∗)_{⪯_{N,ι}}. In particular, for \underline{α}, \underline{α}′ ∈ Q_{β}^{N,ι}

w_{ι,\underline{α}+\underline{α}′}^{N,o}(l) = w_{ι,\underline{α}}^{N,o}(l) + w_{ι,\underline{α}′}^{N,o}(l) (5)

for all possible vanishing orders.

As the intersection of two rational polyhedral cones is again a rational polyhedral cone, we can form the rational polyhedral cones of the form Q_{β_{1}}^{N_{1},ι_{1}} ∩⋯∩Q_{β_{ν}}^{N_{ν},ι_{ν}}. Here, N_{1}, …, N_{ν} are the nodes of C (or, in Case B, the nodes of the corresponding partial normalization), ι_{i} is an index such that C_{ι_{i}} contains N_{i}, and β_{i} ∈ {1, …, B_{N,ι_{i}}}, i = 1, …, ν. This defines a fan decomposition Q = ⋃_{β=1}^{B} Q_{B}.

Let F_{•} be a maximal filtration, realizing the data (r_{1}^{ι}, …, r_{k}^{ι}), ι = 1, …, c, and (a_{1}^{N}, …, a_{k}^{N}), N a node of Q. Then, for every β ∈ \{1, …, N\}, and any \underline{α}, \underline{α}′ ∈ Q_{β}, we have, by (5),

P_{F^{•}}^{\underline{α}+\underline{α}′}(l, m) = P \underline{α} ; F^{•}(l, m) + P_{F^{•}}^{\underline{α}′}(l, m). (6)

For every edge e of the cone Q_{β}, denote the minimal integral generator by \underline{α}_{e,β}. Then, by (6), we have to verify the inequalities in Definition 2.2.10 only for \underline{α} in the finite set

\{\underline{α}_{e,β} | β = 1, …, B, e an edge of Q_{β} \}.

The theorem is now settled. q.e.d.

2.3. The Hilbert compactification as a moduli space. Introduce the functors

\underline{HC}^{(s)s}(g; χ, r): \underline{Schemes}_{ℂ} ⟶ \underline{Sets}

which assign to every scheme S the equivalence classes of pairs (C_{S}, E_{S}) where π: C_{S} ⟶ S is a flat family of semistable curves, and E_{S} is an S-flat sheaf, such that, for every closed point s ∈ S, the restriction E_{S|π^{−1}(s)} is an H-(semi)stable vector bundle of uniform rank r and Euler characteristic χ. Two families (C_{S}, E_{S}) and (C′_{S}, E′_{S}) are equivalent, if there are an isomorphism φ_{S}: C′_{S} ⟶ C_{S} and a line bundle L_{S} on S, such that

φ_{S}^{∗}(E_{S} ⊗ π^{∗}(L_{S})) ≅ E′_{S}.

Then, our considerations imply:

Theorem 2.3.1.

i) There is a natural transformation

ϑ: \underline{HC}^{ss}(g; χ, r) ⟶ h_{ℌℭ(g;χ,r)},

such that for every other scheme 𝔖 and every other natural transformation ϑ′: \underline{HC}^{ss}(g; χ, r) ⟶ h_{𝔖}, one has a unique morphism

t: ℌℭ(g; χ, r) ⟶ 𝔖

with ϑ′ = h(t) ∘ ϑ. ii) The space ℌ ℭ (g; χ, r) contains an open subscheme ℌℭ(g; χ, r)^{s}

which is a coarse moduli scheme for \underline{HC}^{s}(g; χ, r).

3. Properties of the Hilbert compactification

3.1. Dimension and smooth points. We will call a pair (C, E) with C a semistable curve and E a vector bundle on C strictly H-stable, if it is H-stable and there is no automorphism φ: C ⟶ C with φ^{∗}E ≅ E.

Remark 3.1.1. If (C, E) is strictly H-stable, then E must be a simple bundle, i.e., End(E) ≅ ℂ · id_{E}. In fact, the universal bundle on ℌ̂ := ℌ̂(g; χ_{l}, r) possesses a GL(V^{χ_{l}})-linearization whence the GL(V^{χ_{l}})- stabilizers of a point in ℌ̂ corresponding to a strictly H-stable pair (C, E) identify with the automorphisms of E on C which form a dense set in the space of endomorphisms. Therefore, End(E) can have dimension at most one, because the GL(V^{χ_{l}})-stabilizer may have dimension at most one.

Let ℌℭ(g; χ, r)^{⋆} ⊂ ℌℭ(g; χ, r) be the open subset parameterizing the strictly H-stable curves.

Theorem 3.1.2.

i) The Hilbert compactification is a normal variety of dimension 3g− 3 + r^{2}(g − 1) + 1.

ii) The subset ℌℭ(g; χ, r)^{⋆} is smooth.

Proof. Let κ_{g}: ℌℭ(g; χ, r) ⟶ \overline{𝔐}_{g} be the natural morphism. The irreducibility and the dimension statement in i) are clear, because they are known for the preimage U of the moduli space 𝔐_{g} of smooth curves under κ_{g} (see [14]), and we have seen in Theorem 2.1.2 that U is dense in the Hilbert compactification.

Let ℌ^{0}(g; χ, r) be as in the introduction, and ℌ^{†}(g; χ, r) ⊂ ℌ^{0}(g; χ, r) the open subset of those (C, q_{C}: V^{χ} ⊗ O_{C} ⟶ E) for which H^{1}(E_{C}) vanishes. As our considerations in Chapter 2 show, the Hilbert compactification is a quotient of an open subset of ℌ^{†}(g; χ, r). Please accept for the moment the following statement:

Proposition 3.1.3. The scheme ℌ^{†}(g; χ, r) is smooth.

This proposition settles i). Let ℌ^{⋆}(g; χ, r) be the open part of the Hilbert scheme which parameterizes the strictly H-stable objects. Now, statement ii) follows, because the quotient morphism ℌ^{⋆}(g; χ, r) ⟶ ℌℭ(g; χ, r)^{⋆} is a principal PGL(V^{χ_{l}})-bundle.

We now turn to the proof of Proposition 3.1.3. Let 𝔐_{g}^{⋆} be the moduli space of automorphism free smooth curves, and set

U^{⋆} := κ_{g}^{−1}(𝔐_{g}^{⋆}) ∩ ℌℭ^{s}.

Then, U^{⋆} is a smooth quasi-projective variety of dimension 3g−3+r^{2}(g− 1) + 1. Since the quotient morphism is over U^{⋆} a principal PGL(V^{χ_{l}})- bundle, the preimage of U^{⋆} under the quotient morphism is a smooth quasi-projective variety of dimension 3g − 3 + r^{2}(g − 1) + χ_{l}^{2}. Moreover, by Theorem 2.1.2, it is dense in ℌ^{†}(g; χ, r), whence the latter is an irreducible scheme of the same dimension. To prove smoothness, we have to determine the dimension of the tangent spaces. If x ∈ ℌ^{†}(g; χ, r) corresponds to the curve C_{x} ↪ 𝔊, the tangent space to ℌ^{†}(g; χ, r) at x is given by Hom(I_{C_{x}}/I_{C_{x}}^{2}, O_{C_{x}}). Since C is a local complete intersection (which is an intrinsic property by [10], Prop. 3.2.1 and Cor. 3.2.2), the conormal sheaf I_{C_{x}}/I_{C_{x}}^{2} is locally free, and we have the exact sequence

0 ⟶ I_{C_{x}}/I_{C_{x}}^{2} ⟶ Ω_{𝔊|C_{x}}^{1} ⟶ Ω_{C_{x}}^{1} ⟶ 0.

Here, the left exactness follows, because (a) the sequence is in any case exact away from the nodes of C_{x} and (b) since I_{C_{x}}/I_{C_{x}}^{2} is torsion-free, it does not contain any subsheaf the support of which has dimension strictly less than one. We derive the exact sequence

0 ⟶ Hom(Ω_{C_{x}}^{1}, O_{C_{x}}) ⟶ H^{0}(T_{𝔊|C_{x}}) ⟶
⟶ Hom(I_{C_{x}}/I_{C_{x}}^{2}, O_{C_{x}}) ⟶ Ext^{1}(Ω_{C_{x}}^{1}, O_{C_{x}}) ⟶ H^{1}(T_{𝔊|C_{x}}).

We claim that H^{1}(T_{𝔊|C_{x}}) vanishes. For this, we use the exact sequence

0 ⟶ End(E_{𝔊}) ⟶ E_{𝔊}^{⊕χ_{l}} ⟶ T_{𝔊} ⟶ 0.

Let E be the restriction of E_{𝔊} to the curve C_{x}, so that we obtain the exact sequence

0 ⟶ End(E) ⟶ E^{⊕χ_{l}} ⟶ T_{𝔊|C_{x}} ⟶ 0.

Now, our assumption is that H^{1}(E)^{⊕χ_{l}} = H^{1}(E^{⊕χ_{l}}) vanishes, and, for dimension reasons, H^{2}(End(E)) = 0, whence also H^{1}(T_{𝔊|C_{x}}) = 0, as asserted. We also see that

h^{0}(T_{𝔊|C_{x}}) = χ_{l} · h^{0}(E) − χ(End(E)) = χ_{l}^{2} + r^{2}(g − 1).

Next, by Serre duality

dim(Hom(Ω_{C_{x}}^{1}, O_{C_{x}}))−dim(Ext^{1}(Ω_{C_{x}}^{1}, O_{C_{x}})) = χ(Ω_{C_{x}}^{1} ⊗ω_{C_{x}}) = 3g−3.

The exact sequence above thus shows

dim(Hom(I_{C_{x}}/I_{C_{x}}^{2}, O_{C_{x}})) = h^{0}(T_{𝔊|C_{x}}) + χ(Ω_{C_{x}}^{1} ⊗ ω_{C_{x}})
= χ_{l}^{2} + r^{2}(g − 1) + 3g − 3.

This proves that x is a smooth point of ℌ^{†}(g; χ, r). q.e.d.

Remark 3.1.4.

i) Deligne and Mumford [3] applied Schlessinger’s deformation theory [15] in order to show that any semistable curve C admits a miniversal deformation over the base scheme M := Specℂ[[t_{1}, …, t_{N}]] with N := dim(Ext^{1}(Ω_{C}, O_{C})). This means that there is a family C_{M} ⟶ M of curves parameterized by M with C as the fibre over the origin, such that for any flat family of curves C_{B} ⟶ B with B the spectrum of a local Artin algebra and C as the fibre over the closed point there is a morphism φ: B ⟶ M with C_{B} ≅ C_{M} ×_{M} B. Moreover, φ is unique in case B = Spec(ℂ[ε]/⟨ε^{2}⟩). The tangent space to 0 ∈ M thus identifies with Ext^{1}(Ω_{C}, O_{C}). Finally, suppose C has M nodes, then the space of local deformations is the deformation space of this set of nodes and, thus, identifies with M_{loc} = Specℂ[[u_{1}, …, u_{M}]]. Here, one can arrange the generators t_{1}, …, t_{N} and u_{1}, …, u_{M} in such a way that the natural morphism M ⟶ M_{loc} comes from the homomorphism ℂ[[u_{1}, …, u_{M}]] ⟶ ℂ[[t_{1}, …, t_{N}]], u_{i} ⟼ t_{i}, i = 1, …, M.

Now, let x ∈ ℌ^{†}(g; χ, r), and let 𝔘 be its formal neighborhood. By the smoothness of M and its versality, the universal curve over the Hilbert scheme ℌ^{†}(g; χ, r) provides us with a morphism φ: 𝔘 ⟶ M the differential of which is the map

Hom(I_{C_{x}}/I_{C_{x}}^{2}, O_{C_{x}}) ⟶ Ext^{1}(Ω_{C_{x}}^{1}, O_{C_{x}}).

As we have seen before, this map is surjective, so that φ is a submersion whence a smooth morphism.

ii) If we work in the setting of Artin and Deligne-Mumford stacks, we can sharpen the second statement of Theorem 3.1.2. We will do this in Section 3.4 below.

3.2. Existence of universal families. The aim of this section is to prove:

Theorem 3.2.1. Suppose χ and r are coprime. Then, every point x ∈ ℌℭ(g; χ, r)^{⋆} possesses an étale neighborhood U, such that there exists a universal family over U.

3.2.1. The universal curve over ℌℭ(g; χ, r)^{s}. Let Cℌ̂ ↪ ℌ̂ × 𝔊 be the universal closed subscheme. It is clearly invariant under the SL(V^{χ_{l}})-action on ℌ̂×𝔊. On ℌ̂, we choose a line bundle Oℌ̂(1) := 𝔏_{a,m} where m and a ≫ 0 are chosen in such a way that the conclusion of the main theorem holds, and on 𝔊 the usual ample line bundle O_{𝔊}(1). Then, for positive integers s and t,

L_{s,t} := (π∗ ; ℌ̂Oℌ̂(s) ⊗ π_{𝔊}^{∗}O_{𝔊}(t))_{|Cℌ̂}

is an ample SL(V^{χ_{l}})-linearized line bundle on Cℌ̂. If we choose s/t large enough, Proposition 1.1.1 grants that points in the preimage Cs ; ℌ̂ of the

points in ℌ̂ which are stable w.r.t. the linearization in 𝔏_{a,m} are stable w.r.t. the linearization in L_{s,t}. Therefore, the geometric quotient

C_{ℌℭ}^{s} := Cs ; ℌ̂// SL(V^{χ_{l}})

exists. Here, we have set ℌℭ^{s} := ℌℭ(g; χ, r)^{s}. Moreover, there is a natural morphism

σ: C_{ℌℭ}^{s} ⟶ ℌℭ^{s}.

We call C_{ℌℭ^{s}} — abusively — the universal curve. For an H-stable pair (C, E), define

Aut(C, E) := {α: C ⟶^{≅} C | α^{∗}E ≅ E }.

From the GIT set up, it follows that the PGL(V^{χ_{l}})-stabilizer of a stable point (C, q: V^{χ_{l}} ⊗ O_{C} ⟶ E) in ℌ̂ identifies with the group Aut(C, E). Thus, we see:

Corollary 3.2.2. For any point [C, E] ∈ ℌℭ^{s}, the fibre σ^{−1}[C, E] is isomorphic to the curve C/Aut(C, E).

In particular, we may hope for a universal family only over the open subset ℌℭ(g; χ, r)^{⋆}.

3.2.2. Proof of Theorem 3.2.1. Now, let x ∈ ℌℭ^{⋆} := ℌℭ(g; χ, r)^{⋆}. Then, x has an étale neighborhood U, such that the family σ_{U}: C_{U} := C_{ℌℭ^{⋆}} ×_{ℌℭ^{⋆}} U ⟶ U possesses a section which meets every fibre in a smooth point. Then, L_{U} := O_{C_{U}} (σ_{U}(U)) is a relative ample invertible sheaf. Let 𝔘^{⋆} ⊂ ℌ̂ be the SL(V^{χ_{l}})-invariant open subset which parameterizes the strictly H-stable points, and let ψ: 𝔘^{⋆} ⟶ ℌℭ^{⋆} be the quotient morphism. Then, since Mumford’s GIT supplies universal geometric quotients, the map ψ_{U}: U := 𝔘^{⋆} ×_{ℌℭ^{⋆}} U ⟶ U is a geometric quotient, too. For the same reason, the vertical maps in the following Cartesian diagram are both geometric quotients:

\xymatrix{C_{U} := C_{ℌℭ^{⋆}} ×_{ℌℭ^{⋆}} U \ar[r]^{σ_{U}} \ar[d]_{ψ_{C_{U}}} & U \ar[d]^{ψ_{U}} \\ C_{U} \ar[r] & U.}

Define L_{U} as the pullback of L_{U} under the quotient morphism ψ_{C_{U}}. This is obviously an SL(V^{χ_{l}})-linearized relative ample invertible sheaf.

We have seen that the SL(V^{χ_{l}})-stabilizer of a point u ∈ U corresponding to a strictly H-stable object consists exactly of the scalar matrices (Remark 3.1.1). The same holds for the points x ∈ C_{U}. Let μ_{χ_{l}} ⊂ ℂ^{∗} be the subgroup generated by a primitive χ_{l}-th root of unity ζ. For any point u ∈ U or x ∈ C_{U}, the SL(V^{χ_{l}})-stabilizer now identifies with μ_{χ_{l}}. If F is an SL(V^{χ_{l}})-linearized sheaf on U or C_{U}, we will say that it is of weight k, if μ_{χ_{l}} acts by ζ^{k} · id_{F⟨x⟩} for all x ∈ U or x ∈ C_{U}, respectively. For example, L_{U} is of weight zero.

By construction, we have the quotient q_{U}: V^{χ_{l}} ⊗ O_{U} ⟶ E_{U}, and the question we have to answer is whether E_{U} descends — possibly after tensorizing it with the pullback of a line bundle on U — to the quotient C_{U}. By Kempf’s descent lemma (see [4]), an SL(V^{χ_{l}})-linearized vector bundle E on C_{U} descends to the quotient, if and only if it is of weight zero. Now, the sheaf E_{U} is of weight one. Thus, our task will be to find an SL(V^{χ_{l}})-linearized invertible sheaf A_{U} of weight one. Then, E_{U} ⊗ σ_{U}^{∗}(A_{U}^{∨}) will descend to C_{U}, and we will be done.

For any m, the sheaf E_{U}[m] := E_{U} ⊗ L_{U}^{⊗m} is an SL(V^{χ_{l}})-linearized vector bundle of weight one, and, if m is sufficiently large,

F_{m} := σ_{U∗}(E_{U}[m])

will be an SL(V^{χ_{l}})-linearized vector bundle of rank χ_{l} +r ·m and weight one. Then, for m ≫ 0,

N_{U} := det(F_{m+1}) ⊗ det(F_{m})^{∨}

is a line bundle of weight r. Since χ and r and thus also c := χ_{l} + r · m and r are coprime, we may find integers α and β with α · c + β · r = 1, so that

A_{U} := det(F_{m})^{⊗α} ⊗ N_{U}^{⊗β}

will indeed have weight one. q.e.d.

3.3. The fibres of the morphism κ_{g}. As before, let κ_{g}: ℌℭ(g; χ, r) ⟶ \overline{𝔐}_{g} be the natural morphism. By \overline{𝔐}_{g}^{⋆}, we denote the quasi-pro-jective moduli space of automorphism free stable curves. In this section, we want to establish:

Theorem 3.3.1. For any stable curve C_{0} without automorphisms, the variety

κ_{g}^{−1}([C_{0}]) ∩ ℌℭ^{s}(g; χ, r)

has only analytical normal crossings as singularities.

We will follow the strategy of Gieseker’s paper [6] in order to prove the result.

3.3.1. A family of semistable curves with fixed stable model. Let C_{0} be a fixed stable curve. Then, by the results of Deligne and Mumford [3], C_{0} has a universal deformation over M := Specℂ[[t_{1}, …, t_{N}]] with N := dim(Ext^{1}(Ω_{C_{0}}, O_{C_{0}})) = 3g − 3. Moreover, let M_{loc} = Specℂ[[u_{1}, …, u_{M}]] be the deformation space of the nodes of C_{0}. Finally, there is the morphism M ⟶ M_{loc} normalized in such a way that ℂ[[u_{1}, …, u_{M}]] ⟶ ℂ[[t_{1}, …, t_{N}]] is given by u_{i} ⟼ t_{i}, i = 1, …, M, and t_{i} = 0 is the equation of the i-th node of C_{0}, i = 1, …, M.

Next, let C be a semistable curve the stable model of which is C_{0}, and let π: C ⟶ C_{0} be the contraction map. Let R_{1}, …, R_{S} be the maximal connected chains of rational curves which are contracted by π. We label the nodes c_{1}, …, c_{M} of C_{0} in such a way that {c_{i}} = π(R_{i}), i = 1, …, S. We then define

N := {t_{i} = 0, i = S + 1, …, N } ⊂ M.

By restriction of the universal family over M to N, we obtain the family σ_{N} : C_{N} ⟶ N which is smooth outside the nodes “which don’t move”, i.e., outside the nodes c_{S+1}, …, c_{M}. For i = 1, …, S, let d_{i,j}, j = 1, …, ι_{i}, be the nodes of C mapping under π to c_{i}. Define

Q := Spec(ℂ[[ x_{i,j}; i = 1, …, S, j = 1, …, ι_{i}]]).

The homomorphism

φ^{∗}: ℂ[[ t_{1}, …, t_{S} ]] ⟶ ℂ[[ x_{i,j}; i = 1, …, S, j = 1, …, ι_{i}]]
t_{i} ⟼ x_{i,1} · · · · · x_{i,ι_{i}}, i = 1, …, S,

defines a morphism φ: Q ⟶ N. The pull back of the family C_{N} provides us with the family σ_{Q}: C_{Q} ⟶ Q of stable curves. Near the i-th node, the family C_{Q} is defined by the equation

y_{i} · z_{i} − x_{i,1} · · · · · x_{i,ι_{i}} = 0,

for appropriate parameters y_{i} and z_{i}, i = 1, …, S.

Now, let

ϑ: Ĉ_{Q} ⟶ Q

be the blow up of the curve C_{Q} along the ideal generated by y_{1} and x_{1,1}. Near the node c_{1}, we may embed C_{Q} into

𝔸 = Spec(ℂ[[ y_{1}, z_{1}, x_{i,j}; i = 1, …, S, j = 1, …, ι_{i}]]).

The blow up 𝔸̂ of 𝔸 along y_{1} and x_{1,1} is the scheme

𝔸̂ := {((y_{1}, z_{1}, x_{i,j}; i = 1, …, S), [w_{0} : w_{1}]) | y_{1}·w_{0} = x_{1,1}·w_{1}} ⊂ 𝔸×ℙ_{1}.

One checks that the strict transform of C_{Q} is given in the chart w_{0} = 1 by the equation

w_{1} · z_{1} − x_{i,2} · · · · · x_{i,ι_{1}} = 0

and in the chart w_{1} = 1 by

w_{0} · y_{1} − x_{i,1} = 0.

We may now iterate the blow up, i.e., blow up Ĉ_{Q} at the ideal generated by w_{1} and x_{i,2} and so on and perform the same procedure at the other nodes, too, in order to construct a flat family

σ̃_{Q}: C̃_{Q} ⟶ Q

with C as the fibre over the origin. By construction, C̃_{Q} is given near the node d_{i,j} by the equation

y_{i,j} · z_{i,j} − x_{i,j} = 0

for suitable local parameters y_{i,j} and z_{i,j}, i = 1, …, S, j = 1, …, ι_{i}. In particular, it is near d_{i,j} isomorphic to the miniversal deformation of that node, and x_{i,j} = 0 is the locus where the node d_{i,j} “is kept”, i = 1, …, S, j = 1, …, ι_{i}.

By X ↪ Q, we denote the subscheme defined by the equations

x_{i,1} · · · · · x_{i,ι_{i}} = 0, i = 1, …, S.

The scheme X obviously has only analytical normal crossing singularities.

3.3.2. The versality property of C̃_{Q}. The family σ̃_{Q}: C̃_{Q} ⟶ Q together with the Q-morphism π_{Q}: C̃_{Q} ⟶ C_{Q} has the following property:

Proposition 3.3.2. Let τ: S := Spec(A) ⟶ N be an N -scheme where A is a local Artin algebra. Suppose that there is a flat family σ_{S}: C_{S} ⟶ S of semistable curves over S together with an S-morphism π_{S}: C_{S} ⟶ τ^{∗}C_{N}. Suppose that the closed point s of S maps to the origin of N and that π_{S|σ_{S}^{−1}(s)} equals the map π.

Then, there is an N -morphism ψ: S ⟶ Q, such that C_{S} is over τ^{∗}C_{N} = ψ^{∗}C_{Q} isomorphic to ψ^{∗}C̃_{Q}.

Proof. First note that the homomorphism

H^{1}(Hom(Ω_{C}^{1}, O_{C})) ⟶ H^{1}(Hom(π^{∗}Ω_{C_{0}}^{1}, O_{C}))

is injective. In fact, as the computations in [6] used for proving the analogous statement (Corollary 4.4) are completely local, they apply to our situation, too. The rest of the proof may now be copied from [6], proof of Proposition 4.5. q.e.d.

3.3.3. Proof of Theorem 3.3.1. Let ℌ^{†}(g; χ, r) be as in Section 3.1. There is a morphism

κ^{†}: ℌ^{†}(g; χ, r) ⟶ \overline{𝔐}_{g},

and we define

ℌ_{C_{0}} := κ^{†−1}([C_{0}]).

Let σ_{ℌ_{C_{0}}} : C_{ℌ_{C_{0}}} ⟶ ℌ_{C_{0}} be the restriction of the universal family. A suitably high power of the relative dualizing sheaf ω_{C_{ℌ_{C_{0}}} /ℌ_{C_{0}}} will yield a morphism

C_{ℌ_{C_{0}}} ⟶^{ι} P
σ_{ℌ_{C_{0}}} ↓  ↓
ℌ_{C_{0}}  ℌ_{C_{0}},

where P is some projective bundle. The image of ι is a flat family of stable curves, all of which are isomorphic to C_{0}. As C_{0} does not have any automorphisms, this family is trivial. Let

φ_{ℌ_{C_{0}}} : C_{ℌ_{C_{0}}} ⟶ C_{0} × ℌ_{C_{0}}

be the induced morphism. Let x ∈ ℌ_{C_{0}} be a point and 𝔘 its formal neighborhood. Denote the fibre of the family C_{ℌ_{C_{0}}} over x by C, and let τ: 𝔘 ⟶ N be the constant map to the origin. Finally, define

σ_{𝔘}: C_{𝔘} ⟶ 𝔘

as the restriction of the family C_{ℌ_{C_{0}}}. By Proposition 3.3.2, there is a morphism

ψ: 𝔘 ⟶ X ↪ Q.

Our observation in Remark 3.1.4 i) implies that the morphism ψ is smooth. Therefore, ℌ_{C_{0}} has only analytic normal crossing singularities at x. Finally, as C_{0} does not have any automorphisms, the quotient morphism

ℌ_{C_{0}} ∩ ℌ^{0}(g; χ, r)^{s} ⟶ κ_{g}^{−1}([C_{0}]) ∩ ℌℭ^{s}(g; χ, r)

is a principal PGL(V^{χ_{l}})-bundle. This proves the theorem. q.e.d.

Remark 3.3.3. If the automorphism group of C_{0} is nontrivial, the same arguments show that the fibre κ_{g}^{−1}([C_{0}]) ∩ ℌℭ^{s}(g; χ, r) is the quotient of a variety with analytical normal crossings by the automorphism group of C_{0}. However, even if C_{0} is a smooth curve, the action of the group Aut(C_{0}) on the moduli space of semistable bundles has not been thoroughly studied, so far. We refer the reader to the paper [1] for information concerning the action of a single automorphism. 3.4. The moduli stacks. In the following, let \underline{Schemes}_{ℂ} be the category of schemes of finite type over ℂ, viewed as a 2-category, and \underline{Groupoids} the 2-category of groupoids, that is the 2-category whose objects are groupoids, i.e., categories in which all morphisms are isomorphisms, the 1-morphisms are functors and the 2-morphisms are natural transformations between functors.

As usual, defining presheaves of groupoids and establishing isomorphisms between them involves many schemes characterized by some universal property, such as fibre products. However, these schemes will be defined only up to canonical isomorphy and there is no equivalence relation which compensates for this. Thus, we have to fix a priori a representative for every such isomorphy class. In the following, we assume to have done this.

Next, we introduce the 2-functors

HC_{g/χ/r}^{(s)s} : \underline{Schemes}_{ℂ} ⟶ \underline{Groupoids}.

For any scheme S of finite type, the objects of HC_{g/χ/r}^{(s)s}(S) are families (C_{S}, E_{S}) of H-(semi)stable vector bundles as before, the morphisms between (C′_{S}, E′_{S}) and (C_{S}, E_{S}) are pairs (φ_{S}, ψ_{S}), consisting of an Sisomorphism φ_{S}: C′_{S} ⟶ C_{S} and an isomorphism ψ_{S}: E′_{S} ⟶ φ_{S}^{∗}E_{S}. For any morphism f: T ⟶ S pullback of families defines a natural transformation

HC_{g/χ/r}^{(s)s}(f): HC_{g/χ/r}^{(s)s}(S) ⟶ HC_{g/χ/r}^{(s)s}(T).

On the other hand, there are the quotient stacks [ℌ^{ss}/ GL(V^{χ_{l}})] and [ℌ^{s}/ GL(V^{χ_{l}})]. Here, ℌ^{(s)s} is the open part of ℌ^{0}(g; χ_{l}, r) which parameterizes the (semi)stable objects. For any scheme S, the objects of [ℌ^{(s)s}/ GL(V^{χ_{l}})] are pairs (ϑ_{S}: P ⟶ S, η_{S}: P ⟶ ℌ^{(s)s}) where ϑ_{S}: P ⟶ S is a principal GL(V^{χ_{l}})-bundle and η_{S} is an equivariant morphism. One has a natural notion of isomorphism and, as before, pullback defines the functor associated with a morphism f: T ⟶ S.

Theorem 3.4.1. The presheaves HC_{g/χ/r}^{(s)s} and [ℌ^{(s)s}/ GL(V^{χ_{l}})] are isomorphic.

Proof. The assertions amount to prove that, for every scheme S, the groupoids HC_{g/χ/r}^{(s)s}(S) and [ℌ^{(s)s}/ GL(V^{χ_{l}})](S) are equivalent.

First, suppose (ϑ_{S}: P ⟶ S, η_{S}: P ⟶ ℌ^{(s)s}) is an object of the groupoid [ℌ^{(s)s}/ GL(V^{χ_{l}})](S). Then, by means of pullback, the morphism η_{S} yields a GL(V^{χ_{l}})-invariant, P-flat family of semistable curves of genus g

C_{P} ↪ P × 𝔊

and a GL(V^{χ_{l}})-linearized vector bundle E_{P} on C_{P}. Now, as S is the geometric quotient of P by the GL(V^{χ_{l}})-action, the same arguments which were used in Section 3.2 show that we have the curve π_{S}: C_{S} ⟶ S. This time, as there are no stabilizers present, every fibre of π_{S} is indeed a semistable curve of genus g. We have to check that the family C_{S} is indeed S-flat. For this, choose a Zariski-open set U ⊂ S over which the principal bundle P is trivial (this is possible, since we are dealing with GL(V^{χ_{l}}).) Set V := U × GL(V^{χ_{l}}). We will show that C_{S|V} is in fact GL(V^{χ_{l}})-equivariantly a product C_{U} × GL(V^{χ_{l}}) for some U-flat family C_{U}. This clearly settles the affair. By the GL(V^{χ_{l}})-equivariance of ϑ_{S}, we have the commutative diagram

\xymatrix{(x, g, h) \ar@{|->}[d] & ∈ & U × GL(V^{χ_{l}}) × GL(V^{χ_{l}}) \ar[d] \ar[r]^{η_{S|V} ×id} & ℌ^{(s)s} × GL(V^{χ_{l}}) \ar[d] \\ (x, g · h) & ∈ & U × GL(V^{χ_{l}}) \ar[r]^{η_{S|V}} & ℌ^{(s)s}.}

Now, consider the map

V = U × GL(V^{χ_{l}}) ⟶ (U × GL(V^{χ_{l}})) × GL(V^{χ_{l}})
(x, g) ⟼ (x, id_{V^{χ_{l}}} , g).

Define η^{0} : U ⟶ ℌ^{(s)s} by η^{0}(x) := η_{S|V} (x, id_{V^{χ_{l}}}). The content of the diagram before may then be summarized by the suggestive formula

η_{S|V} (x, g) = η^{0}(x) · g.

Finally, the morphism η̃: V ⟶ ℌ^{(s)s}, (x, g) ⟼ η^{0}(x)·g, is by definition of the group action obtained in the following manner: let (C_{U}, E_{U}) be the family induced by the morphism η^{0}. Note that we get even a quotient

q_{U}: V^{χ_{l}} ⊗ O_{C_{U}} ⟶ E_{U}.

Let

Γ: V ⊗ O_{GL(V^{χ_{l}})} ⟶ V ⊗ O_{GL(V^{χ_{l}})}

be the tautological automorphism. Then, define the following quotient on C_{U} × GL(V^{χ_{l}}):

q_{V} : V ⊗ O_{C_{U} ×GL(V^{χ_{l}})} ⟶^{π_{GL(V^{χ_{l}})}^{∗}(Γ)} V ⊗ O_{C_{U} ×GL(V^{χ_{l}})} ⟶^{π_{C_{U}}^{∗} (q_{U})} π_{C_{U}}^{∗} E_{U}.

This quotient defines an embedding C_{U} × GL(V^{χ_{l}}) ⟶ V × 𝔊, and the resulting morphism V ⟶ ℌ^{(s)s} is just η̃ = η_{S|V}. In particular, C_{S|V} is GL(V^{χ_{l}})-equivariantly isomorphic to C_{U} × GL(V^{χ_{l}}), as asserted. By Kempf’s descent lemma, the bundle E_{P} descends to C_{S}, so that (C_{S}, E_{S}) is an object of HC_{g/χ/r}^{(s)s}(S). An isomorphism in [ℌ^{(s)s}/ GL(V^{χ_{l}})](S) will

clearly lead to a unique isomorphism in HC_{g/χ/r}^{(s)s}(S).

Now, suppose we are given a scheme S and a family (π_{S}: C_{S} ⟶ S, E_{S}) of H-(semi)stable vector bundles. Then, we know that

ϑ_{S}: Isom(V^{χ_{l}} ⊗ O_{S}, π_{S∗}E_{S}) ⟶ S

is a principal GL(V^{χ_{l}})-bundle. On P := Isom(V^{χ_{l}} ⊗ O_{S}, π_{S∗}E_{S}), there is the tautological isomorphism

τ_{P}: V^{χ_{l}} ⊗ O_{P} ⟶ ϑ_{S}^{∗}π_{S∗}E_{S}.

Now, form the cartesian diagram

C_{P} ⟶^{ψ} C_{S}
π_{P} ↓  ↓^{π_{S}}
P ⟶^{ϑ_{S}} S.

By flat base change

ϑ_{S}^{∗}π_{S∗}E_{S} ≅ π_{P∗}ψ^{∗}E_{S}.

If we set E_{P} := ψ^{∗}E_{S}, then

V^{χ_{l}} ⊗ O_{C_{P}} ⟶^{π_{P}^{∗}τ_{P}} π_{P}^{∗}π_{P∗}E_{P} ⟶^{ev} E_{P}

defines a morphism P ⟶ ℌ^{(s)s} which is by construction GL(V^{χ_{l}})- equivariant. Again, isomorphisms in the category HC_{g/χ/r}^{(s)s}(S) will lead canonically to isomorphisms in the groupoid [ℌ^{(s)s}/ GL(V^{χ_{l}})](S).

The two operations just introduced clearly establish the desired equivalence of categories. q.e.d.

Now, by the results of Section 3.1, we know that the schemes ℌ^{ss} and ℌ^{s} are smooth. Moreover, the quotient map ℌ^{(s)s} ⟶ [ℌ^{(s)s}/ GL(V^{χ_{l}})] is smooth, whence [ℌ^{ss}/ GL(V^{χ_{l}})] and [ℌ^{s}/ GL(V^{χ_{l}})] are a smooth Artin stack and a smooth Deligne-Mumford stack, respectively.

Corollary 3.4.2. The Hilbert compactification HC_{g/χ/r}^{ss} is a smooth Artin stack, and its open substack HC_{g/χ/r}^{s} is a smooth Deligne-Mumford stack.

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