Logarithmic Bundles

of Multi-Degree Arrangements in P^{n}

Elena Angelini

Received: October 21, 2014 Revised: March 4, 2015

Communicated by Thomas Peternell

Abstract. Let D = {D_{1}, …, D_{ℓ}} be a multi-degree arrangement with normal crossings on the complex projective space P^{n}, with degrees d_{1}, …, d_{ℓ}; let Ω_{P^{n}}^{1}(log D) be the logarithmic bundle attached to it. First we prove a Torelli type theorem when D has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-d_{i} hypersurfaces of Ω_{P^{n}}^{1}(log D). Then, when n = 2, by describing the moduli spaces containing Ω_{P^{2}}^{1}(log D), we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.

2010 Mathematics Subject Classification: 14J60, 14F05, 14C34, 14C20, 14N05 Keywords and Phrases: Multi-degree arrangement, Hyperplane arrangement, Logarithmic bundle, Torelli theorem

1. Introduction

In the complex projective space P^{n}, let D be a union of ℓ distinct smooth irreducible hypersurfaces with degrees d_{1}, …, d_{ℓ}, i.e. a multi-degree arrangement. We can map D to Ω_{P^{n}}^{1}(log D), the sheaf of differential 1-forms with logarithmic poles on D. This sheaf was originally introduced by Deligne in [7] for arrangements with normal crossings. In this case, for all x ∈ P^{n}, the space of sections of Ω_{P^{n}}^{1}(log D) near x is defined as < d log z_{1}, …, d log z_{k}, dz_{k+1}, …, dz_{n} >_{O_{P^{n},x}}, where z_{1}, …, z_{n} are local coordinates such that D = {z_{1} · . . . · z_{k} = 0}. In particular, Ω_{P^{n}}^{1}(log D) is a locally free sheaf over P^{n} and it is called logarithmic bundle. A natural, interesting question is whether Ω_{P^{n}}^{1}(log D) contains information enough to recover D, which is the so-called Torelli problem for logarithmic bundles. In particular, if the isomorphism class of Ω_{P^{n}}^{1}(log D) determines D, then D is called a Torelli arrangement.

In the mathematical literature, the first situation that has been analyzed is the case of hyperplanes. In [11] Dolgachev, Kapranov proved that, if ℓ ≤ n + 2, then two different arrangements always give the same logarithmic bundle and in [26] Vallès showed that, if ℓ ≥ n+3, then we can reconstruct the hyperplanes from the logarithmic bundle (as its unstable hyperplanes, see Definition 2.4) unless they osculate a rational normal curve C_{n} of degree n in P^{n}, in which case the logarithmic bundle is isomorphic to E_{ℓ−2}(C_{n}^{∨}), the Schwarzenberger bundle ([21], [22]) of degree ℓ − 2 associated to C_{n}^{∨}. Recently, Dolgachev ([10]) and Faenzi, Matei, Vallès ([13]) solved this problem in the case of hyperplanes that do not necessarily satisfy the normal crossings property. Concerning the higher degree case, Ueda and Yoshinaga ([25], [24]) studied the case ℓ = 1, characterizing generically the Torelli arrangements as the ones with d_{1} ≥ 3. In [2] we analyzed hypersurfaces of the same degree d and, by means of the unstable hypersurfaces of Ω_{P^{n}}^{1}(log D) (see Definition 2.4), we proved a Torelli type theorem when ℓ ≥ \binom{n+d}{d} + 3. Pairs of quadrics are also investigated in [2]. Very recently Ballico, Huh, Malaspina ([4]) and Dimca, Sernesi ([9]), generalizing the techniques, respectively, of [2] and [24], answered some Torelli type questions, respectively, in the case of logarithmic bundles over quadrics or products of projective spaces and for plane curves with nodes and cusps. In this paper, after recalling some preliminary tools (§. 2, 3), we consider multi-degree arrangements with normal crossings on P^{n} (§. 4), on P^{2} (§. 5, 6, 7) and conic-arrangements with normal crossings on P^{2} (§. 8). In Theorem 4.2, by generalizing the arguments used in [2] for hypersurfaces of the same degree and by applying a reduction technique, we prove that if the number ℓ_{i} of hypersurfaces of degree d_{i} in D satisfies ℓ_{i} ≥ \binom{n+d_{i}}{d_{i}} + 3, then we can generically recover the components of D. In §. 5, 6, 7 we focus on some line-conic cases on P^{2} and we prove that they are not of Torelli type (Corollaries 5.5, 6.4). In particular, in Theorem 7.1 we show a link between arrangements of three lines and a conic and arrangements with a cubic in the projective plane. Finally, §. 8 is devoted to conics. The cases ℓ ∈ {1, 2} were studied in [2]; here we prove that for ℓ ≥ 4 a Torelli type result holds (Theorem 8.5). ℓ = 3 is still a bit mysterious.

Acknowledgements I am very grateful to Giorgio Ottaviani and Daniele Faenzi for introducing me to this interesting subject and for their help during the preparation of this work. I also thank Jean Vallès for several helpful comments.

2. Preliminary definitions and notations

Let P^{n} be the n-dimensional complex projective space with n ≥ 2 and let D = {D_{1}, …, D_{ℓ}} be an arrangement on P^{n}, i.e. a family of smooth, irreducible, distinct hypersurfaces of P^{n}. Let us assume that D has normal crossings, that is D is locally isomorphic (in the sense of holomorphic local coordinates changes) to a union of coordinate hyperplanes of C^{n}. For all i ∈ {1, …, ℓ}, D_{i} = {f_{i} = 0} with f_{i} ∈ C[x_{0}, …, x_{n}]_{d_{i}} for certain d_{i}; thus D = {f = 0}, where f = f_{1} · . . . · f_{ℓ} has degree d = d_{1} + … + d_{ℓ}. In particular, if all d_{i}’s are equal to 1 we speak of a hyperplane arrangement, if they are equal to 2 we deal with an arrangement of quadrics and so on. If different d_{i}’s appear in D, then we call D a multi-degree arrangement. In order to introduce the notion of sheaf of logarithmic forms on D we refer to Deligne ([8], [7]). Let U be the complement of D in P^{n} and let j be the embedding of U in P^{n}. We denote by Ω_{U}^{1} the sheaf of holomorphic differential 1-forms on U and by j_{∗}Ω_{U}^{1} its direct image sheaf on P^{n}. Since D has normal crossings, then for all x ∈ P^{n} there exists a Euclidean neighbourhood I_{x} ⊂ P^{n} such that I_{x} ∩ D = {z_{1} ⋯ z_{k} = 0}, where {z_{1}, …, z_{k}} is a part of a system of local coordinates. We have the following:

Definition 2.1. The sheaf of differential 1-forms on P^{n} with logarithmic poles on D is the subsheaf Ω_{P^{n}}^{1}(log D) of j_{∗}Ω_{U}^{1}, such that, for all x ∈ P^{n},

Γ(I_{x}, Ω_{P^{n}}^{1}(log D)) = {s ∈ Γ(I_{x}, j_{∗}Ω_{U}^{1}) | s = ∑_{i=1}^{k} u_{i}d log z_{i} + ∑_{i=k+1}^{n} v_{i}dz_{i}}

where u_{i}, v_{i} are locally holomorphic functions and d log z_{i} = \frac{dz_{i}}{z_{i}}.

Another way to describe these sheaves, which is useful for more general divisors and is equivalent to the previous one in the normal crossings case, is the following, ([19], [20]):

Definition 2.2. The sheaf of diff. 1-forms on P^{n} with log. poles on D is

Ω_{P^{n}}^{1}(log D) = T (log D)^{∨}(−1),

where T (log D) is the kernel of the Gauss map O_{P^{n}}^{n+1} ⟶^{(∂_{0}f ,...,∂_{n} f )} O_{P^{n}}(d − 1).

Since D has normal crossings, Ω_{P^{n}}^{1}(log D) is a locally free sheaf of rank n, [7]. It is called the logarithmic bundle attached to D. Definition 2.1 can be used, more generally, to introduce the logarithmic bundle of an arrangement with normal crossings D on a smooth algebraic variety X (see also [2]). Our investigations are mainly based on the following:

Theorem 2.3. Ω_{P^{n}}^{1}(log D) admits the short exact sequences

(1)  0 ⟶ Ω_{P^{n}}^{1} ⟶ Ω_{P^{n}}^{1}(log D) ⟶^{res} ⨁_{i=1}^{ℓ} O_{D_{i}} ⟶ 0,

where res denotes the Poincaré residue morphism ([11]) and

(2) 0 ⟶ Ω_{P^{n}}^{1}(log D)^{∨} ⟶ O_{P^{n}}(1)^{n+1} ⊕ O_{P^{n}}^{ℓ−1} ⟶^{N} ⨁_{i=1}^{ℓ} O_{P^{n}}(d_{i}) ⟶ 0,

where N is a ℓ×(n+ℓ) matrix depending on the f_{j}’s and their partial derivatives ([2]).

Our aim is to study the injectivity of the correspondence

(3)  D ⟼ Ω_{P^{n}}^{1}(log D)

where D is a multi-degree arrangement with normal crossings with fixed degrees d_{1}, …, d_{ℓ}, that is the Torelli problem for logarithmic bundles. In the case of 1 : 1 correspondence we call D an arrangement of Torelli type or, simply, a Torelli arrangement. In the next section we recall the main results concerning this problem in the case of hyperplanes ([11], [26], [1]), of one smooth hypersurface ([25], [24], [2]), of many smooth hypersurfaces of degree d ≥ 2 and of two smooth quadrics ([2]). In some of them, the components of D are recovered by looking at the set of unstable objects of Ω_{P^{n}}^{1}(log D) of a given degree; to that end we make the following:

Definition 2.4. Let D ⊂ P^{n} be a hypersurface. We call D unstable for Ω_{P^{n}}^{1}(log D) if the following condition holds:

(4)  H^{0}(D, Ω_{P^{n}}^{1}(log D)_{|_{D}}^{∨}) ≠ {0}.

Remark 2.5. Let us suppose that D has ℓ = ℓ_{1} + … + ℓ_{m} components such that ℓ_{i} have degree d_{i}, i ∈ {1, …, m}. We are interested in Definition 2.4 when h^{0}(P^{n}, Ω_{P^{n}}^{1}(log D)^{∨}) = {0}, that is, by using the same arguments of Remark 5.3 of [2], when

(5)  ∑_{i=1}^{m}(ℓ_{i} · d_{i}) > n + 1.

Remark 2.6. In Lemma 5.4 of [2], by means of (1) we prove that each component D_{i} of D is an unstable hypersurface of degree d_{i} for Ω_{P^{n}}^{1}(log D). As in Definition 2.4, we can introduce the notion of unstable hypersurface for Ω_{X}^{1}(log D) when X is a smooth algebraic variety and D is an arrangement with normal crossings on it. In a similar way we can prove that each element of D is unstable for Ω_{X}^{1}(log D).

3. Some known Torelli type results

Let H = {H_{1}, …, H_{ℓ}} be a hyperplane arrangement with normal crossings on P^{n}. If ℓ ≤ n + 2, then H isn’t of Torelli type ([11]); otherwise we have the following result ([26], Theorem 3.1):

Theorem 3.1. If ℓ ≥ n + 3 then H is the set of unstable hyperplanes of Ω_{P^{n}}^{1}(log H), unless H_{1}, …, H_{ℓ} osculate a rational normal curve C_{n} ⊂ P^{n} of degree n, in which case all the hyperplanes lying on C_{n}^{∨} ⊂ (P^{n})^{∨} are unstable and Ω_{P^{n}}^{1}(log H) ≅ E_{ℓ−2}(C_{n}^{∨}), the Schwarzenberger bundle of degree ℓ − 2 associated to C_{n}^{∨}.

If D = {D_{1}}, where D_{1} ⊂ P^{n} is a general hypersurface of degree d_{1}, then D is of Torelli type if and only if d_{1} ≥ 3 ([24], Theorem 1; [2], Proposition 6.1). Now, let D = {D_{1}, …, D_{ℓ}} be an arrangement with normal crossings on P^{n}, with ℓ ≥ 2 and d_{i} = d ≥ 2 for all i ∈ {1, …, ℓ}. By associating to D a hyperplane arrangement H in P^{\binom{n+d}{d}−1} through the d-uple Veronese embedding and by applying Theorem 3.1, we get the following result ([2], Theorem 5.5):

Theorem 3.2. If ℓ ≥ \binom{n+d}{d} + 3 and H is a hyperplane arrangement with normal crossings whose components don’t osculate a rational normal curve of degree \binom{n+d}{d} − 1 in P^{\binom{n+d}{d}−1}, then D is the set of smooth, irreducible, degree-d hypersurfaces of P^{n} unstable for Ω_{P^{n}}^{1}(log D).

In ([2], Theorem 7.5) we prove also that if ℓ = d = 2 then D is not a Torelli arrangement. Indeed, by using the simultaneous diagonalization of the matrices of the smooth quadrics and a duality argument, we get that two such arrangements have isomorphic logarithmic bundles if and only if they have the same tangent hyperplanes. In the next sections we present some recent results concerning multi-degree arrangements (§. 4, 5, 6, 7) and an almost complete description of the conic-case (§. 8).

4. Many multi-degree hypersurfaces

Let D = {D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}, D_{1}^{d_{2}}, …, D_{ℓ_{2}}^{d_{2}}, . . . . . . , D_{1}^{d_{m}}, …, D_{ℓ_{m}}^{d_{m}}} be a multidegree arrangement with normal crossings in P^{n} such that the components D_{1}^{d_{i}}, …, D_{ℓ_{i}}^{d_{i}} have degree d_{i}, with i ∈ {1, …, m} and d_{m} > d_{m−1} > ⋯ > d_{1}; let us denote by Ω_{P^{n}}^{1}(log D) the corresponding logarithmic bundle. When the number of components in D is sufficiently large, the Torelli problem can be solved by generalizing the method used in [2] and by applying a reduction technique inspired by the one adopted in [26]. So, let H_{d_{i}} be the arrangement with ℓ_{i} hyperplanes on P^{N_{i}}, with i ∈ \{1, …, m\} and N_{i} = \binom{n+d_{i}}{d_{i}} − 1, associated to {D_{1}^{d_{i}}, …, D_{ℓ_{i}}^{d_{i}}} by means of the d_{i}-uple Veronese embedding, i.e. ν_{d_{i}} : P^{n} ⟶ P^{N_{i}} and ν_{d_{i}}([x_{0}, …, x_{n}]) = [… x^{I} …], where x^{I} ranges over all monomials of degree d_{i} in x_{0}, …, x_{n}. Let us assume that each H_{d_{i}} has normal crossings on P^{N_{i}} and let Ω_{P^{N_{i}}}^{1} (log H_{d_{i}}) be the associated logarithmic bundle. With the previous notation, let us consider the diagonal embedding:

ν : P^{n} ⟶ P = ∏_{i=1}^{m} P^{N_{i}}

ν([x_{0}, …, x_{n}]) = [ν_{d_{1}}([x_{0}, …, x_{n}]), …, ν_{d_{m}}([x_{0}, …, x_{n}])].

Let p_{i} : P ⟶ P^{N_{i}} be the i-th projection and let h_{i} = c_{1}(p_{i}^{∗}(O_{P^{N_{i}}} (1))). By means of ν, we can associate to the multi-degree arrangement D an arrangement A = A_{1} ∪ … ∪ A_{m} on P such that A_{i} is an irreducible divisor of class h_{i} which is the pull-back via p_{i} of H_{d_{i}}.

Let us assume that A has normal crossings and let Ω_{P}^{1}(log A) be its logarithmic bundle (see also [4] for some results concerning logarithmic bundles over products of projective spaces).

Remark 4.1. The following property holds:

(6)  Ω_{P}^{1}(log A) ≅ ⨁_{i=1}^{m} p_{i}^{∗}(Ω_{P^{N_{i}}}^{1} (log H_{d_{i}})).

Moreover, if ℓ_{i} ≥ N_{i} + 2, H_{d_{i}} having normal crossings, Ω_{P^{N_{i}}}^{1} (log H_{d_{i}}) is a Steiner bundle over P^{N_{i}}, [11]. So, because of (6), Ω_{P}^{1}(log A) admits the short exact sequence

(7)  0 ⟶ ⨁_{i=1}^{m}O_{P}(−h_{i})^{ℓ_{i}−N_{i}−1} ⟶ ⨁_{i=1}^{m}O_{P}^{ℓ_{i}−1} ⟶ Ω_{P}^{1}(log A) ⟶ 0.

Now we can state and prove the main result concerning the Torelli problem for multi-degree arrangements with many components.

Theorem 4.2. Let D be a multi-degree arrangement with normal crossings on P^{n} and let H_{d_{1}}, …, H_{d_{m}}, A be the corresponding arrangements, respectively, on P^{N_{1}}, …, P^{N_{m}} and P, in the sense of Veronese maps. Assume that, for all i ∈ {1, …, m}:

1. ℓ_{i} ≥ N_{i} + 4

2. A has normal crossings on P

3. H_{d_{i}} has normal crossings on P^{N_{i}} and its hyperplanes don’t osculate a rational normal curve of degree N_{i} in P^{N_{i}}. Then D = {D ⊂ P^{n} smooth irred. hypers. of degree d_{i}, ∃ i | D satisfies (4)}.

Proof. We perform a double inclusion argument between the two sets in the last line of the statement of Theorem 4.2. We observe that the inclusion ⊂ follows from Remark 2.6. Thus, let us assume that D ⊂ P^{n} is a smooth irreducible hypersurface of degree d_{i} which is unstable for Ω_{P^{n}}^{1}(log D), we want to prove that D ∈ D. First let us suppose that the degree of D is the highest one, i.e. d_{m}. Our aim is to show that, denoting by H_{d_{m}} ⊂ P^{N_{m}} the hyperplane associated to D by means of ν_{d_{m}}, then H = p_{m}^{∗}(H_{d_{m}}) ⊂ P satisfies

(8)  H^{0}(H, Ω_{P}^{1}(log A)_{|_{H}}^{∨}) ≠ {0}.

Indeed, if this is the case, H_{d_{m}} is an unstable hyperplane for Ω_{P^{Nm}}^{1} (log H_{d_{m}}) and so hypothesis 1. and 3. allow us to apply Theorem 3.1, which implies that H_{d_{m}} ∈ H_{d_{m}}. In particular, we get that D = D_{j}^{d_{m}} ∈ D for certain j ∈ {1, …, ℓ_{m}}. Let us denote by V the image of the map ν; since V is a non singular subvariety of P which intersects transversally A, from Proposition 2.11 of [10] we get the following exact sequence

(9)  0 ⟶ N_{V, P}^{∨} ⟶ Ω_{P}^{1}(log A)_{|_{V}} ⟶ Ω_{V}^{1} (log A ∩ V) ⟶ 0

where N_{V, P}^{∨} denotes the conormal sheaf of V in P. We remark that V ≅ P^{n} and D = A ∩ V, so if we restrict (9) to D, we apply Hom(·, O_{D}) and then we pass to cohomology we get

0 ⟶ H^{0}(D, Ω_{P^{n}}^{1}(log D)_{|_{D}}^{∨}) ⟶ H^{0}(D, Ω_{P}^{1}(log A)_{|_{D}}^{∨}).

Since D is unstable for Ω_{P^{n}}^{1}(log D), necessarily it has to be

(10)  H^{0}(D, Ω_{P}^{1}(log A)_{|_{D}}^{∨}) ≠ {0}.

Now, let us tensor the ideal sheaf sequence of V in P with Ω_{P}^{1}(log A)_{|_{H}}^{∨}; we have the exact sequence

0 ⟶ I_{V ∩H, H} ⊗ Ω_{P}^{1}(log A)_{|_{H}}^{∨} ⟶ Ω_{P}^{1}(log A)_{|_{H}}^{∨} ⟶ Ω_{P}^{1}(log A)_{|_{D}}^{∨} ⟶ 0.

Passing to cohomology we get

0 ⟶ H^{0}(H, I_{V ∩H, H} ⊗ Ω_{P}^{1}(log A)_{|_{H}}^{∨}) ⟶ H^{0}(H, Ω_{P}^{1}(log A)_{|_{H}}^{∨}) ⟶
⟶ H^{0}(D, Ω_{P}^{1}(log A)_{|_{D}}^{∨}) ⟶ H^{1}(H, I_{V ∩H, H} ⊗ Ω_{P}^{1}(log A)_{|_{H}}^{∨}).

We remark that to conclude the proof it suffices to show that

(11)  H^{1}(H, I_{V ∩H, H} ⊗ Ω_{P}^{1}(log A)_{|_{H}}^{∨}) = {0}.

Since hypothesis 1. and 3. hold, we are allowed to use (7), which, by applying Hom(·, P) turns out to be

0 ⟶ Ω_{P}^{1}(log A)^{∨} ⟶ ⨁_{i=1}^{m}O_{P}^{ℓ_{i}−1} ⟶ ⨁_{i=1}^{m}O_{P}(h_{i})^{ℓ_{i}−N_{i}−1} ⟶ 0.

If we tensor with I_{V, P |_{H}} and then we pass to cohomology, the previous sequence becomes

(12)  ⋯ ⟶ ⨁_{i=1}^{m}H^{0}(H, I_{V ∩H, H} ⊗ O_{P}(h_{i})_{|_{H}}^{ℓ_{i}−N_{i}−1}) ⟶

⟶ H^{1}(H, I_{V ∩H, H} ⊗ Ω_{P}^{1}(log A)^{∨}_{|_{H}}) ⟶ ⨁_{i=1}^{m}H^{1}(H, I_{V ∩H, H} ⊗ O_{P}^{ℓ_{i}−1}_{|_{H}}).

In order to prove (11) it suffices to show that

(13)  H^{1−k}(H, I_{V ∩H, H}(kh_{i})) = {0}

for k = {0, 1} and for all i ∈ {1, …, m}. V ∩ H being connected, from the induced cohomology sequence of the ideal sheaf sequence of V in P, restricted to H, we immediately get (13) for k = 0.

So, let us consider the exact commutative diagram:

0  0  0
↓  ↓  ↓
0 → I_{V, P}(h_{i} − h_{m}) → O_{P}(h_{i} − h_{m}) → O_{P^{n}}(d_{i} − d_{m}) → 0
↓  ↓  ↓
0 →  I_{V, P}(h_{i})  →  O_{P}(h_{i})  →  O_{P^{n}}(d_{i})  → 0
↓  ↓  ↓
0 → I_{V ∩H, H}(h_{i}) →  O_{H}(h_{i})  → O_{V ∩H}(d_{i}) → 0
↓  ↓  ↓
0  0  0

Since H^{0}(P, O_{P}(h_{i})) → H^{0}(P^{n}, O_{P^{n}}(d_{i})) is an isomorphism, we always get

(14)  H^{0}(P, I_{V, P}(h_{i})) = H^{1}(P, I_{V, P}(h_{i})) = {0}.

Moreover, looking at the first row of the diagram, we obtain, for all i,

(15)  H^{1}(P, I_{V, P}(h_{i} − h_{m})) = {0}

By using (14) and (15), the first column of the diagram implies (13) for k = 1, as desired. Now, let us suppose that D has degree d_{i} with i ∈ {m − 1, m − 2, …, 1}. In order to prove that D ∈ D, we apply a reduction technique to Ω_{P^{n}}^{1}(log D) and to the hypersurfaces of D of highest degree d_{m}. Let’s start with D_{ℓ_{m}}^{d_{m}}: since for this hypersurface (4) holds, there exists a non-zero surjective homomorphism

Ω_{P^{n}}^{1}(log D)_{|_{D{dm \atop ℓm}}} ⟶ O_{D_{ℓm}^{dm}} ,

which induces a surjective composed homomorphism g_{ℓ_{m}}

Ω_{P^{n}}^{1}(log D) ⟶ Ω_{P^{n}}^{1}(log D)_{|_{D{dm \atop ℓm}}} ⟶ O_{D_{ℓm}^{dm}} .

Its kernel, denoted by K_{ℓ_{m}}^{d_{m}}, turns out to be a rank-n vector bundle over P^{n}. If we apply the snake lemma to the commutative diagram

0 ⟶ ⨁_{i=1}^{m} O_{P^{n}}(−d_{i})^{ℓ_{i}} ⟶ O_{P^{n}}(−1)^{n+1} ⊕ O_{P^{n}}^{(∑_{i=1}^{m} ℓ_{i})−1} ⟶ Ω_{P^{n}}^{1}(log D) ⟶ 0

↓  ↓  ↓ g_{ℓ_{m}}
0 ⟶ O_{P^{n}}(−d_{m}) ⟶  O_{P^{n}}  ⟶ O_{D_{ℓm}^{dm}} ⟶ 0

we get that K_{ℓ_{m}}^{d_{m}} admits the short exact sequence

0 ⟶ ⨁_{i=1}^{m−1} O_{P^{n}}(−d_{i})^{ℓ_{i}} ⊕ O_{P^{n}}(−d_{m})^{ℓ_{m}−1} ⟶^{M_{ℓm}}

⟶^{M_{ℓm}} O_{P^{n}}(−1)^{n+1} ⊕ O_{P^{n}}^{(∑_{i=1}^{m−1} ℓ_{i})+(ℓ_{m}−1)−1} ⟶ K_{ℓ_{m}}^{d_{m}} ⟶ 0

where M_{ℓ_{m}} is the [n + (∑_{i=1}^{m−1} ℓ_{i}) + (ℓ_{m} − 1)]×[(∑_{i=1}^{m−1} ℓ_{i}) + (ℓ_{m} − 1)] matrix obtained from the transpose of the matrix in (2) by removing the last column and row. So we have that

K_{ℓ_{m}}^{d_{m}} ≅ Ω_{P^{n}}^{1}(log{D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}, D_{1}^{d_{2}}, …, D_{ℓ_{2}}^{d_{2}}, . . . . . . , D_{1}^{d_{m}}, …, D_{ℓ_{m}−1}^{d_{m}}}),

i.e. K_{ℓ_{m}}^{d_{m}} is the logarithmic bundle associated to D − {D_{ℓ_{m}}^{d_{m}}}. In particular, D satisfies the condition

(16)  H^{0}(D, K_{ℓ_{m}}^{d_{m}}^{∨}_{|_{D}}) ≠ {0},

that is D is unstable for K_{ℓ_{m}}^{d_{m}}. Indeed, if we apply Hom(·, O_{P^{n}}) to the short exact sequence

0 ⟶ K_{ℓ_{m}}^{d_{m}} ⟶ Ω_{P^{n}}^{1}(log D) ⟶^{g_{ℓm}} O_{D_{ℓm}^{dm}} ⟶ 0

we get

(17)  0 ⟶ Ω_{P^{n}}^{1}(log D)^{∨} ⟶ K_{ℓ_{m}}^{d_{m}}^{∨} ⟶ O_{D_{ℓm}^{dm}} (d_{m}) ⟶ 0.

So, if we restrict (17) to D and then consider the induced cohomology sequence, we obtain an injective map

H^{0}(D, Ω_{P^{n}}^{1}(log D)_{|_{D}}^{∨}) ⟶ H^{0}(D, K_{ℓ_{m}}^{d_{m}}^{∨}_{|_{D}}),

which implies (16). Now, starting from K_{ℓ_{m}}^{d_{m}}, we iterate this technique for D_{ℓ_{m}−1}^{d_{m}}, D_{ℓ_{m}−2}^{d_{m}}, …, D_{1}^{d_{m}} and we get a sequence of rank-n vector bundles K_{ℓ_{m}−1}^{d_{m}}, K_{ℓ_{m}−2}^{d_{m}}, …, K_{1}^{d_{m}} over P^{n} such that, for all s ∈ {1, …, ℓ_{m} − 1},

0 ⟶ K_{ℓ_{m}−s}^{d_{m}} ⟶ K_{ℓ_{m}−(s−1)}^{d_{m}} ⟶^{g_{ℓm−s}} O_{D_{ℓm−s}^{dm}} ⟶ 0

is a short exact sequence and

K_{ℓ_{m}−s}^{d_{m}} ≅ Ω_{P^{n}}^{1}(log{D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}, . . . . . . , D_{1}^{d_{m}}, …, D_{ℓ_{m}−(s+1)}^{d_{m}}}).

In particular

K_{1}^{d_{m}} ≅ Ω_{P^{n}}^{1}(log{D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}, . . . . . . , D_{1}^{d_{m−1}}, …, D_{ℓ_{m−1}}^{d_{m−1}}})

and the smooth irreducible hypersurface D of degree d_{i} is unstable for K_{1}^{d_{m}}. If i = m − 1, then D is a hypersurface of highest degree in the arrangement D − {D_{1}^{d_{m}}, …, D_{ℓ_{m}}^{d_{m}}} and so, by repeating the computations of the first case of this proof, we get that there exists j ∈ {1, …, ℓ_{m−1}} such that D = D_{j}^{d_{m−1}}. If i = m−2, we apply the reduction technique to K_{1}^{d_{m}} and to the hypersurfaces {D_{ℓ_{m−1}}^{d_{m−1}}, …, D_{1}^{d_{m−1}}} and so on. If i = 1, with this method D turns out to be unstable for the logarithmic bundle Ω_{P^{n}}^{1}(log{D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}}) and so, from Theorem 3.2 it follows that there exists j ∈ {1, …, ℓ_{1}} such that D = D_{j}^{d_{1}}, which concludes the proof. □

We have the following:

Corollary 4.3. If ℓ_{i} ≥ \binom{n+d_{i}}{d_{i}} + 3, for all i ∈ \{1, …, m\}, then the map

D = {D_{1}^{d_{1}}, …, D_{ℓ_{1}}^{d_{1}}, . . . . . . , D_{1}^{d_{m}}, …, D_{ℓ_{m}}^{d_{m}}} ⟶ Ω_{P^{n}}^{1}(log D)

is generically injective.

Remark 4.4. Hypothesis 1. of Theorem 4.2 implies (5).

Remark 4.5. We don’t know if we can state a Torelli type theorem like 4.2 without assuming 2. and 3.

In the case of arrangements with lines and conics in the projective plane, that is d_{1} = 1 and d_{2} = n = 2, hypothesis 1. of Theorem 4.2 translates to ℓ_{1} ≥ 6 and ℓ_{2} ≥ 9. In the next three sections we describe this kind of arrangements when ℓ_{1} ∈ {1, 2, 3} and ℓ_{2} = 1.

5. A conic and a line

Let D = {L, C} be an arrangement with normal crossings in P^{2} consisting of a line L and a conic C. Without loss of generality, we can assume L = {f_{1} = 0} and C = {f_{2} = 0}, with f_{1} = x_{0} and f_{2} = ∑_{i,j=0}^{2} a_{ij}x_{i}x_{j}, (a_{ij})_{0≤i,j≤2} ∈ GL(2, C), so that, by applying Gaussian elimination to the matrix of (2), we can get the minimal resolution for Ω_{P^{2}}^{1}(log D)

(18)  0 ⟶ O_{P^{2}}(−2) ⟶^{M} O_{P^{2}}(−1)^{2} ⊕ O_{P^{2}} ⟶ Ω_{P^{2}}^{1}(log D) ⟶ 0

with

M = (\begin{matrix}2 ∂_{1}f_{2} \\ 2 ∂_{2}f_{2} \\ −2 x_{0} ∂_{0}f_{2} \end{matrix}) .

As a consequence we get that c_{1}(Ω_{P^{2}}^{1}(log D)) = 0, c_{2}(Ω_{P^{2}}^{1}(log D)) = 1 and, according to the Bohnhorst-Spindler criterion ([5]), that Ω_{P^{2}}^{1}(log D) is a semistable vector bundle over P^{2}.

Theorem 5.1. Let M_{P^{2}}^{ss}(0, 1) be the family of semistable rank 2 vector bundles E over P^{2} with minimal resolution

0 ⟶ O_{P^{2}}(−2) \underrightarrow{^{t}(ℓ_{1} ℓ_{2} q)} O_{P^{2}}(−1)^{2} ⊕ O_{P^{2}} ⟶ E ⟶ 0

where ℓ_{1}, ℓ_{2} ∈ H^{0}(P^{2}, O_{P^{2}}(1)) and q ∈ H^{0}(P^{2}, O_{P^{n}}(2)). Then the map

M_{P^{2}}^{ss}(0, 1) ⟶^{π_{2}} P^{2}

E ⟼ {ℓ_{1} = 0} ∩ {ℓ_{2} = 0}

is an isomorphism.

Proof. Let E and E′ be two elements of M_{P^{2}}^{ss}(0, 1), defined, respectively, by ℓ_{1}, ℓ_{2}, q and ℓ′_{1}, ℓ′_{2}, q′, as in the statement of Theorem 5.1. We have to prove that the intersection point of ℓ_{1} and ℓ_{2} coincides with the one of ℓ′_{1} and ℓ′_{2} if and only if E ≅ E′. If the intersection point is the same, without loss of generality we can assume that ℓ_{1} = ℓ′_{1} = x_{0} and ℓ_{2} = ℓ′_{2} = x_{1}. We remark that, for all x ∈ P^{2}, E_{x} and E′_{x} are the cokernels of two rank 1 maps, in particular if x = [0, 0, 1] then q and q′ have to contain the term x_{2}^{2}. Thus, E ≅ E′ if and only if there exist g_{1}, g_{2} ∈ H^{0}(P^{2}, O_{P^{2}}(1)) such that the following diagram commutes

O_{P^{2}}(−2) \underrightarrow{^{t}(x_{0} x_{1} q)} O_{P^{2}}(−1)^{2} ⊕ O_{P^{2}}
(1) ↓  ↓ (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ g_{1} & g_{2} & 1 \end{matrix})

O_{P^{2}}(−2) \underrightarrow{^{t}(x_{0} x_{1} q′)} O_{P^{2}}(−1)^{2} ⊕ O_{P^{2}}

which is equivalent to say that

(19)  q′ − q = g_{1}x_{0} + g_{2}x_{1}.

Assume that

q = b_{00}x_{0}^{2} + b_{01}x_{0}x_{1} + b_{02}x_{0}x_{2} + b_{11}x_{1}^{2} + b_{12}x_{1}x_{2} + x_{2}^{2}
q′ = b′_{00}x_{0}^{2} + b′_{01}x_{0}x_{1} + b′_{02}x_{0}x_{2} + b′_{11}x_{1}^{2} + b′_{12}x_{1}x_{2} + x_{2}^{2}.

By using the identity principle for polynomials, we immediately get that

g_{1} = (b′_{00} − b_{00})x_{0} + (b′_{01} − b_{01} − 1)x_{1} + (b′_{02} − b_{02})x_{2}
g_{2} = x_{0} + (b′_{11} − b_{11} − 1)x_{1} + (b′_{12} − b_{12})x_{2}

solve (19), which concludes the proof. □

Remark 5.2. Each E ∈ M_{P^{2}}^{ss}(0, 1) is logarithmic for a line and a conic.

Remark 5.3. Theorem 5.1 asserts that Ω_{P^{2}}^{1}(log D) lives in 2-dimensional space, while the number of parameters associated to a line and a conic with normal crossings is 7. So we can immediately conclude that arrangements like these are not of Torelli type.

With the aid of the description given in Theorem 5.1 and with the same notation as in the beginning of this section, we get the following result.

Proposition 5.4. The point in P^{2} corresponding to Ω_{P^{2}}^{1}(log D) by means of π_{2} is the pole of the line L with respect to the conic C.

Proof. By applying Cramer’s rule we get that the point in ℙ^{2} satisfying

∑_{j=0}^{2} a_{1j}x_{j} = ∑_{j=0}^{2} a_{2j}x_{j} = 0

is P = [a_{12}^{2} − a_{11}a_{22}, a_{22}a_{01} − a_{02}a_{12}, a_{02}a_{11} − a_{12}a_{01}]. The polar line of P with respect to C is given by

(a_{12}^{2} − a_{11}a_{22}, a_{22}a_{01} − a_{02}a_{12}, a_{02}a_{11} − a_{12}a_{01}) (\begin{matrix}a_{00} & a_{01} & a_{02} \\ a_{01} & a_{11} & a_{12} \\ a_{02} & a_{12} & a_{22} \end{matrix}) (\begin{matrix}x_{0} \\ x_{1} \\ x_{2} \end{matrix}) = 0

which reduces to x_{0} = 0, that is to L, as desired. □

We immediately get the following:

Corollary 5.5. Let D = {L, C} and D′ = {L′, C′} be arrangements with normal crossings in P^{2} given by a line and a conic. Then

Ω_{P^{2}}^{1}(log D) ≅ Ω_{P^{2}}^{1}(log D′)

if and only if the pole of L with respect to C coincides with the pole of L′ with respect to C′.

Figure 1. L is the polar line of P with respect to C

Remark 5.6. These results can be extended in a natural way to the case of a multi-degree arrangement D with normal crossings in P^{n}, n ≥ 3, consisting of a hyperplane H and a smooth quadric Q. In this setting Ω_{P^{n}}^{1}(log D) is no more semistable over P^{n}, but its isomorphism class is still described by the pole of H with respect to Q, [3].

6. A conic and two lines

Let D = {L_{1}, L_{2}, C} be an arrangement with normal crossings in P^{2}, where L_{i}, for i ∈ {1, 2}, is a line and C is a conic. We can assume that L_{1} = {f_{1} = 0}, L_{2} = {f_{2} = 0} and C = {f_{3} = 0} where f_{1} = x_{0}, f_{2} = x_{1} and f_{3} = ∑_{i,j=0}^{2} a_{ij}x_{i}x_{j}, (a_{ij})_{0≤i,j≤2} ∈ GL(2, C), so that, by means of (2), Ω_{P^{2}}^{1}(log D) fits in the minimal resolution

(20)  0 ⟶ O_{P^{2}}(−2) ⟶^{M} O_{P^{2}}(−1) ⊕ O_{P^{2}}^{2} ⟶ Ω_{P^{2}}^{1}(log D) ⟶ 0

where

M = (\begin{matrix}2 ∂_{2}f_{3} \\ −2 x_{0} ∂_{0}f_{3} \\ −2 x_{1} ∂_{1}f_{3} \end{matrix}) .

In particular, (20) implies that the normalized bundle Ω_{P^{2}}^{1}(log D)(−1) belongs to M_{P^{2}}(−1, 2), the moduli space of rank 2 stable vector bundles over P^{2} with Chern classes c_{1} = −1 and c_{2} = 2. In the following result, which is likely to be known to experts, we give an interesting description of M_{P^{2}}(−1, 2); in order to state it, we denote by σ_{2}(ν_{2}(P^{2})) the 2-secant variety of the image of the quadratic Veronese map ν_{2} : P^{2} ⟶ P^{5}.

Theorem 6.1. M_{P^{2}}(−1, 2) is isomorphic to σ_{2}(ν_{2}(P^{2}))−ν_{2}(P^{2}), the projective space of symmetric matrices of order 3 and rank 2.

Proof. A vector bundle E lives in M_{P^{2}}(−1, 2) if and only if it is endowed with a short exact sequence like

0 ⟶ O_{P^{2}}(−3) \underrightarrow{^{t}(ℓ_{1} q_{1} q_{2})} O_{P^{2}}(−2) ⊕ O_{P^{2}}^{2}(−1) ⟶ E ⟶ 0

where ℓ_{1} ∈ H^{0}(P^{2}, O_{P^{2}}(1)) and q_{1}, q_{2} ∈ H^{0}(P^{2}, O_{P^{2}}(2)). We note that E has a unique line L ⊂ P^{2} such that H^{0}(L, E_{|_{L}}(−1)) ≠ {0}, known as jumping line of E, which is {ℓ_{1} = 0}. On this line, the linear series given by q_{1} and q_{2} has two distinct double points, which we denote by P_{1} and P_{2}. Then the map given by

M_{P^{2}}(−1, 2) ⟶ σ_{2}(ν_{2}(P^{2})) − ν_{2}(P^{2})
E ⟼ {P_{1}, P_{2}}

is an isomorphism, which concludes the proof. □

Remark 6.2. Theorem 6.1 implies that Ω_{P^{2}}^{1}(log D)(−1) is characterized by 4 parameters, while D needs 9 parameters to be described. So in this case D is not a Torelli arrangement.

Remark 6.3. The jumping line of Ω_{P^{2}}^{1}(log D) is {∂_{2}f_{3} = 0} and it is the polar line with respect to C of L_{1} ∩ L_{2} = [0, 0, 1]. Moreover, the linear series on this line is given by L_{1} ∪ s_{2} and L_{2} ∪ s_{1}, where s_{2} is the polar line with respect to C of {∂_{2}f_{3} = 0} ∩ L_{2} = [a_{22}, 0, −a_{02}] and s_{1} is the polar line with respect to C of {∂_{2}f_{3} = 0} ∩ L_{1} = [0, a_{22}, −a_{12}], that is s_{2} = {a_{22}∂_{0}f_{3} − a_{02}∂_{2}f_{3} = 0} and s_{1} = {a_{22}∂_{1}f_{3} − a_{12}∂_{2}f_{3} = 0}. The logarithmic bundle Ω_{P^{2}}^{1}(log D) corresponds to the two intersection points {P_{1}, P_{2}} of C and {∂_{2}f_{3} = 0}.

Corollary 6.4. Let D = {L_{1}, L_{2}, C} and D′ = {L′_{1}, L′_{2}, C′} be arrangements with normal crossings in P^{2} consisting of two lines and a conic. Let {P_{1}, P_{2}}, resp. {P′_{1}, P′_{2}}, be the points in P^{2} associated to Ω_{P^{2}}^{1}(log D), resp. to Ω_{P^{2}}^{1}(log D′), in the sense of Remark 6.3. Then

Ω_{P^{2}}^{1}(log D) ≅ Ω_{P^{2}}^{1}(log D′) ⟺ {P_{1}, P_{2}} = {P′_{1}, P′_{2}}.

7. A conic and three lines

Let D = {L_{1}, L_{2}, L_{3}, C} be an arrangement with normal crossings in P^{2} consisting of three lines and a conic, let us say that L_{1} = {x_{0} = 0}, L_{2} = {x_{1} = 0}, L_{3} = {x_{2} = 0} and C = {f_{4} = 0} where

(21)  f_{4} = ∑_{i,j=0}^{2} d_{ij}x_{i}x_{j}, (d_{ij})_{0≤i,j≤2} ∈ GL(2, C).

Figure 2. The points {P_{1}, P_{2}} associated to Ω_{P^{2}}^{1}(log D)

In this case, starting from (2), the minimal resolution for the logarithmic bundle turns out to be

(22)  0 ⟶ O_{P^{2}}(−2) ⟶^{M} O_{P^{2}}^{3} ⟶ Ω_{P^{2}}^{1}(log D) ⟶ 0

where

M = (\begin{matrix}−x_{0}∂_{0}f_{4} \\ −x_{1}∂_{1}f_{4} \\ −x_{2}∂_{2}f_{4} \end{matrix}) .

From (22) we get that Ω_{P^{2}}^{1}(log D) is stable and that its normalized bundle Ω_{P^{2}}^{1}(log D)(−1) lives in the moduli space M_{P^{2}}(0, 3), which has dimension 9, as we can see in [17]. Since the number of parameters associated to three lines and a conic is 11, also in this case we can’t get a Torelli type theorem. By using the second part of Theorem 2.3, we note that Ω_{P^{2}}^{1}(log D)(−1) admits an exact sequence like the one for the logarithmic bundle of a smooth plane cubic curve. The link between these two vector bundles is explained in the following result:

Theorem 7.1. Let D be the multi-degree arrangement with normal crossings on P^{2} given by {x_{0}x_{1}x_{2}f_{4} = 0}, where f_{4} is as in (21). Then there exists D′ = {D}, where D ⊂ P^{2} is a smooth cubic curve, such that

Ω_{P^{2}}^{1}(log D) ≅ Ω_{P^{2}}^{1}(log D′)(1).

Proof. Our aim is to find g ∈ H^{0}(P^{2}, O_{P^{2}}(3)) such that, for all i ∈ {0, 1, 2},

(23)  ∂_{i}g = e_{0}^{i}(−x_{0}∂_{0}f_{4}) + e_{1}^{i}(−x_{1}∂_{1}f_{4}) + e_{2}^{i}(−x_{2}∂_{2}f_{4})

for certain e_{j}^{i} ∈ C. By using Schwarz’s theorem, from (23) we get, for all i, h ∈ {0, 1, 2}, i ≠ h,

∑_{j=0}^{2} e_{j}^{h}∂_{i}(x_{j}∂_{j}f_{4}) = ∑_{j=0}^{2} e_{j}^{i}∂_{h}(x_{j}∂_{j}f_{4}).

Let us denote by {a_{uv}^{j}} the coefficients of x_{j}∂_{j}f_{4} for j ∈ {0, 1, 2}; by using the identity principle for polynomials we get the following linear system of 9 equations with variables e_{j}^{i}: for all i, u, v ∈ {0, 1, 2}, i ≠ u,

(24)  ∑_{j=0}^{2} a_{uv}^{j}e_{j}^{i} = ∑_{j=0}^{2} a_{iv}^{j}e_{j}^{u}.

Since a_{uv}^{j} depend on the coefficients of f_{4}, the matrix of (24) is:

H = (\begin{matrix}d_{01} & d_{01} & 0 & −2d_{00} & 0 & 0 & 0 & 0 & 0 \\ 0 & 2d_{11} & 0 & −d_{01} & −d_{01} & 0 & 0 & 0 & 0 \\ 0 & d_{12} & d_{12} & −d_{02} & 0 & −d_{02} & 0 & 0 & 0 \\ d_{02} & 0 & d_{02} & 0 & 0 & 0 & −2d_{00} & 0 & 0 \\ 0 & d_{12} & d_{12} & 0 & 0 & 0 & −d_{01} & −d_{01} & 0 \\ 0 & 0 & 2d_{22} & 0 & 0 & 0 & −d_{02} & 0 & −d_{02} \\ 0 & 0 & 0 & d_{02} & 0 & d_{02} & −d_{01} & −d_{01} & 0 \\ 0 & 0 & 0 & 0 & d_{12} & d_{12} & 0 & −2d_{11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 2d_{22} & 0 & −d_{12} & −d_{12} \end{matrix}) .

By using Gaussian elimination, we get that rank H ≤ 8. So, let us assume that \overline{e}_{j}^{i} is a solution of our system, we need a cubic polynomial g such that conditions in (23) are satisfied. Let us integrate with respect to x_{0} the equation (23) with i = 0, we get

(25)  g(x_{0}, x_{1}, x_{2}) = −2\overline{e}_{0}^{0}x_{0}^{2} (\frac{x_{0}}{3} a_{00}^{0} + \frac{x_{1}}{2} a_{01}^{0} + \frac{x_{2}}{2} a_{02}^{0}) + h(x_{1}, x_{2})+

−2x_{0} [\overline{e}_{1}^{0} (\frac{x_{0}x_{1}}{2} a_{01}^{0} + x_{1}^{2}a_{11}^{0} + x_{1}x_{2}a_{12}^{0}) + \overline{e}_{2}^{0} (\frac{x_{0}x_{2}}{2} a_{02}^{0} + x_{1}x_{2}a_{12}^{0} + x_{2}^{2}a_{22}^{0})]

where h is a function to be determined. If we compute ∂_{1}g from (25), we substitute it in (23) with i = 1 and we integrate with respect to x_{1} we get

(26)  h(x_{1}, x_{2}) = x_{0}x_{1} [\overline{e}_{0}^{0}x_{0}a_{01}^{0} + 2\overline{e}_{1}^{0} (\frac{x_{0}}{2} a_{01}^{0} + x_{1}a_{11}^{0} + x_{2}a_{12}^{0})] +
+x_{0}x_{1} [2\overline{e}_{2}^{0}x_{2}a_{12}^{0} − \overline{e}_{0}^{1} (2x_{0}a_{00}^{0} + x_{1}a_{01}^{0} + 2x_{2}a_{02}^{0})] + i(x_{2})+

−\overline{e}_{1}^{1}x_{1}^{2} (x_{0}a_{01}^{0} + 2\frac{x_{1}}{3} a_{11}^{0} + x_{2}a_{12}^{0}) − x_{1}x_{2}\overline{e}_{2}^{1} (2x_{0}a_{02}^{0} + x_{1}a_{12}^{0} + 2x_{2}a_{22}^{0})

where we have to determine the function i. Finally, if we compare ∂_{2}g from (25) with (23) for i = 2, using also (26) and we integrate with respect to x_{2}, we can find explicitly i, so that the required polynomial is

g(x_{0}, x_{1}, x_{2}) = −\frac{2}{3}\overline{e}_{0}^{0}a_{00}^{0}x_{0}^{3}−2\overline{e}_{0}^{1}a_{00}^{0}x_{0}^{2}x_{1}−2\overline{e}_{1}^{0}a_{11}^{0}x_{0}x_{1}^{2}− \frac{2}{3}\overline{e}_{1}^{1}a_{11}^{0}x_{1}^{3}−2\overline{e}_{2}^{0}a_{22}x_{0}x_{2}^{2}+

−2\overline{e}_{0}^{2}a_{00}^{0}x_{0}^{2}x_{2} − 2(\overline{e}_{1}^{0} + \overline{e}_{2}^{0})a_{12}^{0}x_{0}x_{1}x_{2} − 2\overline{e}_{1}^{2}a_{11}^{0}x_{1}^{2}x_{2} − 2\overline{e}_{2}^{1}a_{22}^{0}x_{1}x_{2}^{2} − \frac{2}{3}\overline{e}_{2}^{2}a_{22}^{0}x_{2}^{3}.

□

Remark 7.2. The proof of Theorem 7.1 implies also Hermite’s Theorem (1868), which asserts that a net of conics can be regarded as the net of the polar conics with respect to a given cubic curve (see [12], book III, chapter III, section 29).

Remark 7.3. If we require that ∂_{i}g = x_{i}∂_{i}f_{4}, for all i ∈ {0, 1, 2}, then

(27)  g(x_{0}, x_{1}, x_{2}) = \frac{2}{3} (a_{00}^{0}x_{0}^{3} + a_{11}^{0}x_{1}^{3} + a_{22}^{0}x_{2}^{3}) ,

provided that the conic is given by f_{4}(x_{0}, x_{1}, x_{2}) = d_{00}x_{0}^{2} + d_{11}x_{1}^{2} + d_{22}x_{2}^{2}. So, let D = {x_{0}x_{1}x_{2}f_{4} = 0} and D′ = {x_{0}x_{1}x_{2}f′_{4} = 0} be two arrangements with normal crossings in P^{2} each of which with a conic given by a diagonalized quadratic form. D and D′ correspond to a logarithmic bundle which is isomorphic to the logarithmic bundle of a smooth cubic like the one of (27). Since in [25] it is proved that two smooth cubics which are both Fermat yield isomorphic logarithmic bundles, then Ω_{P^{2}}^{1}(log D) ≅ Ω_{P^{2}}^{1}(log D′).

Remark 7.4. Although we know that a multi-degree arrangement with three lines and a conic, because of parameters computations, isn’t Torelli and that Theorem 7.1 holds, in this case the problem of determining the fiber of (3) is still open.

8. Arrangements with few conics

Let D = {C_{1}, …, C_{ℓ}} be an arrangement of ℓ ∈ {4, …, 8} conics with normal crossings on P^{2}. Let F_{5}^{2} = {(x, y) ∈ P^{2} × P_{5} | x ∈ C_{y}} be the incidence variety point-conic in P^{2} × P_{5}, where C_{y} ⊂ P^{2} denotes the conic defined by the point y ∈ P_{5} with the Veronese correspondence and let \overline{α}, \overline{β} the restrictions to F_{5}^{2} of the usual projections α and β:

F_{5}^{2} ⊂ P^{2} × P_{5}
↙^{\overline{α}}  ↘^{\overline{β}}
P^{2} P_{5}

Remark 8.1. Let UC(Ω_{P^{2}}^{1}(log D)) be the set of unstable conics of Ω_{P^{2}}^{1}(log D), in the sense of Definition 2.4. UC(Ω_{P^{2}}^{1}(log D)) coincides with the support of the first direct image sheaf R^{1}(\overline{β}_{∗}\overline{α}^{∗}Ω_{P^{2}}^{1}(log D)(−1)): indeed, for all y ∈ P_{5},

R^{1}(\overline{β}_{∗}\overline{α}^{∗}Ω_{P^{2}}^{1}(log D)(−1))_{y} = H^{1}(\overline{β}^{−1}(y), \overline{α}^{∗}Ω_{P^{2}}^{1}(log D)(−1)_{|_{\overline{β}^{−1}(y)}}) =
= H^{1}(C_{y}, Ω_{P^{2}}^{1}(log D)(−1)_{|_{Cy}}) = H^{0}(C_{y}, Ω_{P^{2}}^{1}(log D)_{|_{Cy}}^{∨})^{∨},

where the last inequality follows from Serre’s duality.

So, let tensor with O_{P^{2}}(−1) the exact sequence (2) where n = 2, d_{i} = 2 and let apply the functor \overline{β}_{∗}\overline{α}^{∗}, we get:

(28) 0 → R^{0}\overline{β}_{∗}\overline{α}^{∗}(O_{P^{2}}(−3))^{ℓ} → R^{0}\overline{β}_{∗}\overline{α}^{∗}(O_{P^{2}}(−2)^{3} ⊕ O_{P^{2}}(−1)^{ℓ−1}) →
→ R^{0}\overline{β}_{∗}\overline{α}^{∗}(Ω_{P^{2}}^{1}(log D)(−1)) → R^{1}\overline{β}_{∗}\overline{α}^{∗}(O_{P^{2}}(−3))^{ℓ} →

→ R^{1}\overline{β}_{∗}\overline{α}^{∗}(O_{P^{2}}(−2)^{ℓ−1} ⊕ O_{P^{2}}(−1)^{3}) → R^{1}\overline{β}_{∗}\overline{α}^{∗}(Ω_{P^{2}}^{1}(log D)(−1)) → 0.

In order to determine the terms in (28) we consider

0 ⟶ O_{P^{2}×P_{5}}(−2, −1) ⟶ O_{P^{2}×P_{5}} ⟶ O_{F_{5}^{2}} ⟶ 0,

we do the tensor product with α^{∗}(O_{P^{2}}(t)), where t ∈ {−1, −2, −3} and we apply the functor β_{∗}. In this way (28) becomes

(29)  0 ⟶ R^{0}\overline{β}_{∗}\overline{α}^{∗}(Ω_{P^{2}}^{1}(log D)(−1)) ⟶ (Ω_{P_{5}}^{1})^{ℓ} ⟶^{F}

⟶^{F} (O_{P_{5}}(−1)^{3})^{3} ⊕ O_{P_{5}}(−1)^{ℓ−1} ⟶ R^{1}\overline{β}_{∗}\overline{α}^{∗}(Ω_{P^{2}}^{1}(log D)(−1)) ⟶ 0.

Remark 8.2. In order to investigate UC(Ω_{P^{2}}^{1}(log D)), it suffices to study the cokernel of the map F appearing in (29).

Remark 8.3. More generally, all the previous arguments can be applied to a vector bundle E fitting in an exact sequence like the one of Ω_{P^{2}}^{1}(log D).

Now, let us assume that ℓ = 4. In what follows, by using Macaulay2 software system, we produce D_{0} = {C_{0, 1}, C_{0, 2}, C_{0, 3}, C_{0, 4}} such that

(30)  UC(Ω_{P^{2}}^{1}(log D_{0})) = {C_{0, 1}, C_{0, 2}, C_{0, 3}, C_{0, 4}}.

Example 1 D_{0} is made of four smooth random conics with normal crossings:

C_{0,1} : 42x_{0}^{2} − 50x_{0}x_{1} + 9x_{1}^{2} + 39x_{0}x_{2} − 15x_{1}x_{2} − 22x_{2}^{2} = 0,
C_{0,2} : 50x_{0}^{2} + 45x_{0}x_{1} − 39x_{1}^{2} − 29x_{0}x_{2} + 30x_{1}x_{2} + 19x_{2}^{2} = 0,
C_{0,3} : −38x_{0}^{2} + 2x_{0}x_{1} − 36x_{1}^{2} − 4x_{0}x_{2} − 16x_{1}x_{2} − 6x_{2}^{2} = 0,
C_{0,4} : −32x_{0}^{2} + 31x_{0}x_{1} − 38x_{1}^{2} − 32x_{0}x_{2} + 31x_{1}x_{2} + 24x_{2}^{2} = 0.

By multiplying the four polynomials defining the conics, we get the polynomial f ∈ k[x_{0}, x_{1}, x_{2}]_{8} = R_{8} associated to D_{0}, where k is the field Z_{101}. According to Definition 2.2, we consider the kernel E of the Gauss map and we construct the matrix M ∈ M_{6,4}(R) associated to the module defining Ω_{P^{2}}^{1}(log D_{0}). Then we determine the elements of UC(Ω_{P^{2}}^{1}(log D_{0})): as we can see in Remark 8.1, UC(Ω_{P^{2}}^{1}(log D_{0})) is the zero locus of the order 4 minors of the matrix Z, whose cokernel is equal to the cokernel of F. In particular, posing T = k[y_{0}, …, y_{5}], Z ∈ M_{4,12}(T) is the product of C ∈ M_{4,24}(T) and B ∈ M_{24,12}(T), where C is the matrix of variables needed to get Ω_{P^{5}}^{1} and B is the syzygy matrix of A ∈ M_{12,24}(T) whose entries are the coefficients of the polynomials in M. The ideal J generated by the 4 × 4 minors of Z has dimension 1 and degree 4, from which (30) follows. This is the script of our algorithm.

k=ZZ/101
R=k[x_0..x_2]
ran=random(R^{1:0},R^{4:-2})
f=1_R; for t from 0 to rank source ran-1 do f=f*(ran_(0,t))
E=ker map(R^{1:-1+(degree f)_0},R^{3:0},diff(vars R,f))
M=(res dual E).dd_1
T=k[y_0..y_5]
coe=(M,k,i,j)->diff(symmetricPower(k,vars R),transpose(symmetricPower

(k-2,vars R))*submatrix(M,{i},{j}))
coe2=(M,i,j)->diff(transpose(vars(R))*submatrix(M,{i},{j}),symmetric
Power(2,vars R))
expa=(M,k)->matrix table(rank target M,rank source M,(i,j)->coe(M,k,
i,j))
expa2=(M)->matrix table(rank target M,rank source M,(i,j)->coe2(M,i,
j))
A=sub(matrix(expa(submatrix(M,{0..2},{0..3}),2), expa2(submatrix(M,
{3..5},{0..3}))),T)
B=syz A
C=(id_(T^{4:0}))**(vars T)
Z=C*B
J=minors(4,Z)
dim J
degree J

Remark 8.4. The previous algorithm can be performed for all ℓ. In particular, if ℓ = 5 then we can get another example such that the unstable conics of the logarithmic bundle coincide with the conics of the arrangement.

Starting from the previous example, we can prove the following:

Theorem 8.5. If ℓ ≥ 4, then the map

D = {C_{1}, …, C_{ℓ}} ⟶ Ω_{P^{2}}^{1}(log D)

is generically injective.

Proof. First, let us assume that ℓ = 4. Let us consider the incidence variety W = {(D, C) ∈ (P_{5} × P_{5} × P_{5} × P_{5}) × P_{5} | C ∈ UC(Ω_{P^{2}}^{1}(log D))} and let \overline{a}, \overline{b} be the restrictions to W of the projection morphisms, respectively, from (P_{5} × P_{5} × P_{5} × P_{5}) and P_{5}. From the previous example we have that \overline{a}^{−1}(D_{0}) = D_{0}. So, for all arrangements D ∈ P_{5} × P_{5} × P_{5} × P_{5}, dim \overline{a}^{−1}(D) ≥ 0 and h^{0}(\overline{a}^{−1}(D), O_{W}) = length \overline{a}^{−1}(D) ≥ 4. To conclude the proof, it suffices to show that there exists V ⊂ P_{5} × P_{5} × P_{5} × P_{5} open such that, for all D ∈ V

(31)  dim \overline{a}^{−1}(D) = 0,
(32)  length \overline{a}^{−1}(D) = 4.

We remark that the dimension d of the fiber given by the morphism \overline{a} has the upper semicontinuity property ([16], chapter 1, section 8, corollary 3), which implies that {w ∈ W | d(w) ≥ 1} is a closed subset in W. So the set V_{1} = \overline{a}(\{w ∈ W | d(w) ≤ 0\}) = \overline{a}(\{w ∈ W | d(w) = 0\}) is open in P_{5}×P_{5}×P_{5}× P_{5}. By using the upper semicontinuity of the length of the fiber given by the morphism \overline{a} (this fact is a consequence of theorem 12.8, chapter 3 of [14]; this theorem holds with the hypothesis of flatness, in our case we have the generic flatness) we get that the set \{D ∈ P_{5} × P_{5} × P_{5} × P_{5} | length \overline{a}^{−1}(D) ≥ 5\} is closed in P_{5} × P_{5} × P_{5} × P_{5}. As above, the set V_{2} = {D ∈ P_{5} × P_{5} × P_{5} × P_{5} | length \overline{a}^{−1}(D) ≤ 4\} = \{D ∈ P_{5} × P_{5} × P_{5} × P_{5} | length \overline{a}^{−1}(D) = 4\} is open in P_{5} × P_{5} × P_{5} × P_{5}. The points of the open set V = V_{1} ∩ V_{2} satisfy the required properties (31) and (32). Now, if ℓ ≥ 5, then we can apply the reduction technique, performed in the proof of Theorem 4.2, to Ω_{P^{2}}^{1}(log D) and to the conics of D: at each step we get a logarithmic bundle of a conic-arrangement with one component less, till we reduce to the case of four conics, studied above. □

Finally we discuss the case of ℓ = 3. Let D = {C_{1}, C_{2}, C_{3}} be an arrangement of conics with normal crossings on P^{2}. Let us start by analyzing UC(Ω_{P^{2}}^{1}(log D)). In order to do that, let us consider the exact sequence (29) with ℓ = 3: UC(Ω_{P^{2}}^{1}(log D)), the support of R^{1}\overline{β}_{∗}\overline{α}^{∗}(Ω_{P^{2}}^{1}(log D)(−1)), is the maximal degeneration locus of the mor-

phism (Ω_{P_{5}}^{1})^{3} ⟶^{F} (O_{P_{5}}(−1)^{3})^{3} ⊕ O_{P_{5}}(−1)^{2}, i.e. it coincides with the scheme D_{10}(F) = {y ∈ P_{5} | rank(F_{y}) ≤ 10}, which, according to [18], has expected codimension 5 in P_{5} (we note that the computation of the expected codimension is meaningless when ℓ ≥ 4). If this is the case, the number of points in D_{10}(F) is determined by Porteous’ formula:

(33)  [D_{10}(F)] = det[c_{1−i+j}((((O_{P_{5}}(−1)^{3})^{3} ⊕ O_{P_{5}}(−1)^{2}) − (Ω_{P_{5}}^{1})^{3})],

where 1 ≤ i, j ≤ 5. The generic entry of the matrix (33) is the coefficient of the term of degree (1 − i + j) in the formal series in one variable coming from the quotient of the Chern polynomials of ((O_{P_{5}}(−1)^{3})^{3} ⊕ O_{P_{5}}(−1)^{2}) and (Ω_{P_{5}}^{1})^{3}. Thus [D_{10}(F)] = 21. More generally, we get the following:

Proposition 8.6. Let E be a vector bundle over P^{2} such that

0 ⟶ O_{P^{2}}(−2)^{3} ⟶ O_{P^{2}}(−1)^{3} ⊕ O_{P^{2}}^{2} ⟶ E ⟶ 0

is exact and let UC(E) be the set of unstable conics of E, in the sense of (4). UC(E) is expected to be a 0-dimensional scheme of P_{5} with 21 points.

Remark 8.7. If we apply the algorithm performed in Example 1 of this section in the case of ℓ = 3, we can find some arrangements D such that UC(Ω_{P^{2}}^{1}(logD)) satisfies the expected properties of Proposition 8.6. Indeed, according to the notations introduced in such algorithm, the variety in P_{5} defined by the ideal J has 21 distinct points, which, in terms of the quadratic Veronese embedding of the projective plane, correspond to smooth conics in P^{2}. Between these 21 points, 3 correspond to the component of D and the remaining 18 belong to a net quadrics in P_{5}, whose base locus is a K3-surface with 12 singular points, that don’t seem to be related to the 18 conics we are interested in. The explicit determination of such 18 points or, equivalently, of a primary decomposition of J saturated with the ideals defining the conics of D as points in P_{5}, would be interesting to solve the Torelli problem in this case, but, at the moment, it seems to be hard, also with a computer.

According to Remark 8.7, instead of studying UC(Ω_{P^{2}}^{1}(log D)), we can focus on UL(Ω_{P^{2}}^{1}(log D)), the set of unstable lines of Ω_{P^{2}}^{1}(log D) in the sense of Definition 2.4. Let F_{2}^{2} be the incidence variety point-line in P^{2} × P_{2}, i.e.

(34)  F_{2}^{2} = {(x, y) ∈ P^{2} × P_{2} | x ∈ L_{y}}

where L_{y} ⊂ P^{2} is the line defined by y ∈ P_{2} and let \overline{p}, \overline{q} be, respectively, the restrictions to F_{2}^{2} of the projection maps p, q as in the following diagram:

F_{2}^{2} ⊂ P^{2} × P_{2}
↙^{\overline{p}}  ↘^{\overline{q}}
P^{2} P_{2}

We remark that UL(Ω_{P^{2}}^{1}(log D)), as a subset of P_{2}, is the support of R^{1}(\overline{q}_{∗}\overline{p}^{∗}Ω_{P^{2}}^{1}(log D)(−2)). Namely, if y ∈ P_{2} then we have that

R^{1}(\overline{q}_{∗}\overline{p}^{∗}Ω_{P^{2}}^{1}(log D)(−2))_{y} = H^{1}(\overline{q}^{−1}(y), \overline{p}^{∗}Ω_{P^{2}}^{1}(log D)(−2)_{|_{\overline{q}^{−1}(y)}}) =
= H^{1}(L_{y}, Ω_{P^{2}}^{1}(log D)(−2)_{|_{Ly}}) = H^{0}(L_{y}, Ω_{P^{2}}^{1}(log D)_{|_{Ly}}^{∨})^{∨},

where the last equality follows from Serre’s duality. In order to study this support, we apply the functor \overline{q}_{∗}\overline{p}^{∗} to the exact sequence (2) in the case of three conics twisted by −2 and we get

(35) 0 ⟶ R^{0}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)^{3}) ⟶ R^{0}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−3)^{3} ⊕ O_{P^{2}}(−2)^{2}) ⟶
⟶ R^{0}\overline{q}_{∗}\overline{p}^{∗}(Ω_{P^{2}}^{1}(log D)(−2)) ⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)^{3}) ⟶

⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−3)^{3} ⊕ O_{P^{2}}(−2)^{2}) ⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(Ω_{P^{2}}^{1}(log D)(−2)) ⟶ 0. Our aim is to describe the terms of (35). So we tensor

0 ⟶ O_{P^{2}×P_{2}}(−1, −1) ⟶ O_{P^{2}×P_{2}} ⟶ O_{F_{2}^{2}} ⟶ 0

with p^{∗}O_{P}^{2}(t), where t ∈ {−4, −3, −2} and then we apply q_{∗}. By using Serre’s duality and the Poincaré-Euler sequence, (35) turns out to be

(36)  0 ⟶ R^{0}\overline{q}_{∗}\overline{p}^{∗}(Ω_{P^{2}}^{1}(log D)(−2)) ⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)^{3}) ⟶^{G}
⟶^{G} (Ω_{P_{2}}^{1})^{3} ⊕ O_{P_{2}}(−1)^{2} ⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(Ω_{P^{2}}^{1}(log D)(−2)) ⟶ 0

where R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)) fits in the exact sequence

0 ⟶ R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)) ⟶ O_{P_{2}}(−1)^{6} ⟶ O_{P_{2}}^{3} ⟶ 0,

and it has rank 3 over P^{2}. The support of R^{1}(\overline{q}_{∗}\overline{p}^{∗}Ω_{P^{2}}^{1}(log D)(−2)) is the maximal degeneration locus of the morphism G in (36), i.e. it’s the scheme D_{7}(G) = {y ∈ P_{2} | rank(G_{y}) ≤ 7}. According to [18], the expected codimension over P_{2} of D_{7}(G) is 2, that is we expect a finite number of unstable lines for Ω_{P^{2}}^{1}(log D). Assuming that D_{7}(G) is 0-dimensional, the number of its points is given by Porteous’ formula:

[D_{7}(G)] = det[c_{1−i+j}(((Ω_{P_{2}}^{1})^{3} ⊕ O_{P_{2}}(−1)^{2}) − R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)^{3}))],

where 1 ≤ i, j ≤ 2. The generic entry of [D_{7}(G)] is the coefficient of the degree- (1−i+j) term of the formal series in one variable defined as the quotient of the Chern polynomial of (Ω_{P_{2}}^{1})^{3} ⊕ O_{P_{2}}(−1)^{2} with the one of R^{1}\overline{q}_{∗}\overline{p}^{∗}(O_{P^{2}}(−4)^{3}). So [D_{7}(G)] = 21. These arguments imply the following:

Proposition 8.8. Let D = {C_{1}, C_{2}, C_{3}} be a normal crossing arrangement of conics in P^{2}. UL(Ω_{P^{2}}^{1}(log D)) is expected to be a 0-dimensional scheme of P_{2} with 21 points.

Remark 8.9. The previous proposition holds, more generally, for all vector bundles E over P^{2} admitting the exact sequence

0 ⟶ O_{P^{2}}(−2)^{3} ⟶ O_{P^{2}}(−1)^{3} ⊕ O_{P^{2}}^{2} ⟶ E ⟶ 0.

By using Macaulay2 software system, we can find some examples of arrangements that behave as stated in Proposition 8.8. Example 2 Let us consider the arrangement D_{0} = {C_{0,1}, C_{0,2}, C_{0,3}} of conics with normal crossings such that

C_{0,1} : x_{0}^{2} + x_{1}^{2} − 16x_{2}^{2} = 0,
C_{0,2} : x_{0}^{2} + 9x_{1}^{2} − 36x_{2}^{2} = 0,
C_{0,3} : 25x_{0}^{2} + 100x_{0}x_{2} + x_{1}^{2} − 2x_{1}x_{2} + 76x_{2}^{2} = 0.

In order to determine UL(Ω_{P^{2}}^{1}(log D_{0})), we contruct the 3×5 matrix associated to Ω_{P^{2}}^{1}(log D_{0})^{∨} in the sense of (2) and then we restrict it to a generic line L ⊂ P^{2} parametrized by x_{0} = bx_{1} + cx_{2}. Afterwards we produce the 9 × 8 matrix M of the map

H^{0}(L, O_{P^{2}}(1)_{|_{L}}^{3} ⊕ O_{P^{2}}^{2}_{|_{L}}) ⟶ H^{0}(L, O_{P^{2}}(2)_{|_{L}}^{3})

with respect to the basis given by {{x_{1}, x_{2}}, {x_{1}, x_{2}}, {x_{1}, x_{2}}, {1}, {1}} and {{x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}}, {x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}}, {x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}}}. Since we are interested in the kernel of this linear map, we consider, over the ring k[b, c], where for simplicity k = Q, the ideal J generated by the 8 maximal minors of M: as expected, we get that J has dimension 0 and degree 21, in particular the algebra k[b, c]/J is a C-vector space of dimension 21. Therefore, let B a basis of k[b, c]/J and let compb (resp. compc) be the 21 × 21 matrix associated, with respect to B, to the linear map (companion)

k[b, c]/J ⟶ k[b, c]/J

defined by the multiplication by b (resp. by c). According to [6] (chapter 2, section 4), the eigenvalues of compb (resp. compc) coincide with the bcoordinates (resp. c-coordinates) of the points of the variety associated to J. By using Stickelberger’s Theorem (see for example [23], chapter 2, section 2.3), in order to get the pair of parameters (b_{i}, c_{i}) defining an unstable line it suffices to match the eigenvalue b_{i} of compb corresponding to the same eigenvector (up to a change of sign) of the eigenvalue c_{i} of compc. If D_{0} is as above, then Ω_{P^{2}}^{1}(log D_{0}) has 21 unstable lines such that 11 are real.

Remark 8.10. As we can see in Figure 3, it seems to be hard but interesting to understand what these lines represent for the conic-arrangement and how it is possible to get the conics from them: we observe, for example, that they are

Figure 3. D_{0} and the 11 real unstable lines plotted with [15]

not tangent lines and they don’t cross the conics in special points. So we can say that the three conics case represents still an open problem.

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Elena Angelini Dipartimento di Matematica e Informatica Università di Ferrara Via Machiavelli 30 44121 Ferrara elena.angelini@unife.it 530

Documenta Mathematica 20 (2015)