UNIVERSITÀ DEGLI STUDI DI NAPOLI FEDERICO II

Dottorato di Ricerca in Matematica. XII ciclo, 1996-2000.

CARLA DIONISI Tesi di Dottorato

MULTIDIMENSIONAL MATRICES AND MINIMAL RESOLUTIONS

OF VECTOR BUNDLES

Coordinatore del dottorato: Prof. Salvatore Rionero

Contents

Introduction 3

Acknowledgements 7

1 Stabilizers for nondegenerate matrices of boundary format and Steiner bundles 8 1.1 Hyperdeterminants . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 Definition and basic properties . . . . . . . . . . . . . . . 8 1.1.2 Hyperdeterminant of boundary format and its applications 14 1.1.3 Singularities of hyperdeterminants . . . . . . . . . . . . . 17 1.1.4 Multidimensional diagonal matrices and Vandermonde ma-

trix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Geometric invariant theory and nondegenerate matrices of bound-

ary format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 Jumping hyperplanes, Steiner bundles and stabilizers . . . . . . 29 1.4 Cohomology of Schwarzenberger bundles . . . . . . . . . . . . . 36

2 Minimal resolution of general stable vector bundles on ℙ^{2} 47 2.1 Generalities and notations . . . . . . . . . . . . . . . . . . . . . 47 2.2 Admissible pairs and resolutions . . . . . . . . . . . . . . . . . . 48

2.2.1 Natural pairs and general vector bundles . . . . . . . . . 58 2.3 The sheaves ℱ(r, f_{1}, f_{2}) . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.1 Jumping lines of ℱ(x_{0}, x_{1}^{2}, x_{2}^{2}) . . . . . . . . . . . . . . . 71 2.3.2 The moduli space 𝔐_{Sheaf,ℙ^{2}}(−1, 2) . . . . . . . . . . . . . 72

2.4 Monads of stable rank-2 bundles on ℙ^{2} . . . . . . . . . . . . . . 75 2.4.1 𝔐_{ℙ^{2}}(2; −1, 4) . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4.2 𝔐_{ℙ^{2}}(2; −1, 5) . . . . . . . . . . . . . . . . . . . . . . . . 79 2.4.3 𝔐_{ℙ^{2}}(2; −1, 6) . . . . . . . . . . . . . . . . . . . . . . . . 80 2.4.4 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 86

Introduction

Cayley [Cay48] in 1848 found an explicit invariant of a given 2 × 2 × 2 matrix which generalizes the usual determinant of a square matrix; for that invariant he proposed the name of hyperdeterminant. This notion was forgotten for many years; until recently Gelfand, Kapranov and Zelevinski rediscovered Cayley’s results and gave a modern account of the whole subject (see the book [GKZ94] for a detailed story of this topic).

The hyperdeterminant of a multidimensional matrix A ∈ V_{0} ⊗ … ⊗ V_{p} with dim V_{i} = k_{i} + 1 exists if and only if k_{j} ≤ ∑_{i≠j} k_{i} for all j = 0, …, p (theorem 1.1.2) and A is called of boundary format if k_{0} = ∑ k_{i}.

We are interested in multidimensional matrices because, in the case of boun– dary format and nondegenerate, they correspond to a class of vector bundles on a product of projective spaces which are called Steiner bundles (theorem 1.2.4). This class is interesting even when the base space is given by a single projective space.

Ancona and Ottaviani in [AO99] began a study of multidimensional matrices of boundary format A ∈ V_{0} ⊗ … ⊗ V_{p} under the action of the reductive group SL(V_{0}) × … × SL(V_{p}) from the point of view of Mumford’s Geometric Invariant Theory.

The action of SL(V_{0})×…×SL(V_{p}) on such matrices translates to an action on the moduli space of the corresponding Steiner bundles, the invariants coincide and the stable points of both actions correspond under that translation. By investigating the properties and the invariants of the above actions, they proved that for p ≤ 2 the stabilizer of a multidimensional matrix is contained in SL(U) where U is a 2-dimensional vector space. The main result of the first chapter of the thesis is the proof that the assumption p ≤ 2 can be dropped; it holds for every p (theorem 1.3.13).

We emphasize that SL(V_{0}) × … × SL(V_{p}) is a ”big” group, so it is quite surprising that the stabilizer found lies always in the 3-dimensional group SL(U) without any dependence on p and on dimV_{i}.

The maximal stabilizer is obtained by the ”most symmetric” class of matrices corresponding to the identity matrix. Under the identifications V_{i} = S^{k_{i}}U the identity is given by the natural map

S^{k_{1}}U ⊗ … ⊗ S^{k_{p}}U → S^{k_{0}}U

which is defined under the assumption k_{0} = ∑ k_{i}. This explains again why the condition of boundary format is so important.

When the base is a single projective space ℙ(V_{2}) (p = 2), Steiner bundles S are defined by the following exact sequence

0 → V_{1} ⊗ 𝒪_{ℙ(V_{2})}(−1) → V_{0} ⊗ 𝒪_{ℙ(V_{2})} → S → 0  (1)

They are stable, they give smooth points in the moduli space of all stable bundles with the same rank and Chern classes, and they belong to an irreducible component of dimension (k − 1)(n − 1)(n + k + 1) where k = k_{1} + 1 and n = k_{2} ([AO94]).

This class of bundles was first studied by Schwarzenberger in 1961 [Sch61] who defined a particular class of bundles (called Schwarzenberger bundles) which corresponds, under the above correspondence, to the identity matrix. Since they have a “big” group of symmetry, their cohomological behavior is expected to be quite special.

In the section 1.4 we prove that if 𝒮 is a Schwarzenberger bundle on ℙ^{n}

h^{1}(S^{2}𝒮^{∗}) = n(n + 2k − 2) and h^{2}(S^{2}𝒮^{∗}) = \binom{k − 2}{2}\binom{n}{2}

In the second chapter of this thesis we restrict the base space to be ℙ^{2}, but we focus on a larger class of bundles ℰ, i.e. these bundles having free resolution

0 → ⨁_{i=1}^{k} 𝒪_{ℙ^{2}}(−a_{i}) → ⨁_{j=1}^{r+k} 𝒪_{ℙ^{2}}(−b_{j}) → ℰ → 0.  (2)

(By (1), Steiner bundles are a particular case).

Since Horrocks theorem [Hor64] asserts that any torsion-free sheaf ℱ on ℙ^{n} has homological dimension at most n−1, thus on ℙ^{2} any bundle has homological dimension 1, that is, has a free resolution of the type (2). Hence, the aim of the second chapter becomes to investigate stable vector bundles on the complex projective plane ℙ^{2} by means of their minimal free resolution.

Bohnhorst and Spindler, in the paper [BS92], develop interesting techniques for the study of minimal free resolution of rank-n stable vector bundles on ℙ^{n} having homological dimension 1. We partially adapt their results to the case of rank-r vector bundles on ℙ^{2} with r ≥ 2. In particular in proposition 2.2.4 we translate the condition of minimality of the resolution to a condition of admissibility of the pair (a, b), where a = (a_{1}, …, a_{k}) and b = (b_{1}, …, b_{k+r}).

As Bohnhorst and Spindler observe, admissible pairs (a, b) define a stratification of the moduli space 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) by constructible subsets. In the section 2.2.1 (lemma 2.2.19) we estimate the codimension of such strata and we characterize the pair of the general element of the moduli space. As a corollary, a quite easy proof of the irreducibility of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) follows (theorem 2.2.21) and some bounds about the regularity and the cohomology of its elements are found in 2.2.24 and 2.2.25.

All proofs work for rank-r vector bundles on ℙ^{2} having strongly admissible pairs (see definition 2.2.17). The moduli space 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) is irreducible for any r (the proofs with different techniques can be found in [Bar77a], [Ell83]), [HL93], [Pot79], [Mar78]) moreover we don’t have examples of rank r- stable vector bundles on ℙ^{2} which are not strongly admissible, so we give this conjecture: “the strong admissibility for the pairs of a holomorphic vector bundle ℰ on the projective space is a necessary condition for the stability of ℰ”.

A proof of this conjecture would give as corollary also the irreducibility of 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) for any rank r, by applying the same arguments of the proof of 2.2.20.

Finally, in the section 2.4, using the computer algebra system MACAULAY2 and an algorithm to compute admissible pair written in the Scheme dialect of Lisp, we give, in the cases r = 2, c_{1} = −1 and c_{2} ≤ 6, a positive answer to the open problem of the explicit description of the filtration in constructible open subsets of moduli spaces 𝔐_{ℙ^{2}}(r; c_{1}, c_{2}) and the results are summarized in a series of tables at the end of the thesis.

Many of the results from the second chapter are obtained in collaboration with Marco Maggesi.

Acknowledgements

I wish to thank Prof. Giorgio Ottaviani (Università di Firenze) for his invaluable scientific advice and his constant support and encouragement.

Many thanks to my colleague and friend Marco Maggesi who shared with me ideas and hours of work and discussions. I would like to thank him also for his computational support, since a large part of the results of the second part of this thesis were first conjectured and subsequently tested using computer algebra systems.

Chapter 1

Stabilizers for nondegenerate

matrices of boundary format and

Steiner bundles

1.1 Hyperdeterminants

1.1.1 Definition and basic properties

By an (p + 1)-dimensional matrix we shall means an array A = (a_{i_{0},...,i_{p}}) of complex numbers, where each index ranges over some finite set. It is possible to define an analog of determinant of square n × n matrices also for some multidimensional matrices, called hyperdeterminant. Its study was initiated by Cayley [Cay48] and Schläfli [Sch52] but was largely abandoned for 150 years. Recently it is undertaken by Gelfand, Kapranov, Zelevinsky and Weyman in [GKZ94], [GKZ92], [Wey94], [WZ96] and in this section we remind their results.

Let p ≥ 1 be an integer and 0 ≤ i_{j} ≤ k_{j} then A = (a_{i_{0},...,i_{p}}) is called (p + 1)- complex matrix of format (k_{0}+1)×⋯×(k_{p}+1) and let M = ℂ^{(k_{0}+1)×···×(k_{p}+1)} be the spaces of these matrices. With greater generality we can fix (p + 1) complex vector spaces V_{i} for i = 0, …, p of dimension k_{i} +1 and denote M = V_{0} ⊗⋯⊗V_{p} (the context allows to avoid any ambiguity with the previous definition of M). The definition of the hyperdeterminant of A can be stated in geometric, analytic and algebraic terms ([GKZ94]).

Geometrically, consider the product X = ℙ^{k_{0}} × ⋯ × ℙ^{k_{p}} of several projective spaces in Segre enbedding into the projective space ℙ^{(k_{0}+1)...(k_{p}+1)−1} (if ℙ^{k_{j}} is the projectivization of the complex vector space V_{j}^{∗} then the ambient projective space is ℙ^{(}V_{0}^{∗} ⊗⋯⊗V_{p}^{∗})). Let X^{∨} be the projective dual variety of X consisting of all hyperplanes in ℙ^{(k_{0}+1)...(k_{p}+1)−1} tangent to X at some point. If X^{∨} is a hypersurface in ℙ^{(k_{0}+1)...(k_{p}+1)−1}^{∗} then its defining equation, which is a homogeneous polynomial function on V_{0}^{∗} ⊗ ⋯ ⊗ V_{p}^{∗} is called the hyperdeterminant of format (k_{0} + 1) × ⋯ × (k_{p} + 1) and denoted by Det. As usual, if X^{∨} is not an hypersurface, we set Det equal to 1, and refer to this case as trivial. If V_{j} is equipped which a basis then the element f ∈ V_{0} ⊗ ⋯ ⊗ V_{p} is represented by a matrix A = (a_{i_{0},...i_{p}}) as above, and so DetA is a polynomial function of matrix entries. It is determined uniquely up to the sign by the requirement that the Det(A) has integer coefficients and is irreducible over ℤ.

Analytically, the hyperplane f = 0 belongs to X^{∨} if and only if f vanishes at some point of X with all first derivatives. If we consider the coordinate system x^{(j)} = (x_{0}^{(j)}, …, x_{k_{j}}^{(j)}) on each V_{j}^{∗} then f ∈ V_{0} ⊗ ⋯ ⊗ V_{p} is represented after restriction on X by a multilinear form

F (A, x) := f(x^{(0)}, …, x^{(p)}) = ∑_{(i_{0},...,i_{p})} a_{i_{0},...,i_{p}}x_{i_{0}}^{(0)} ⊗ ⋯ ⊗ x_{i_{p}}^{(p)}  (1.1)

In more invariant terms, we can think of A as an element of V_{0}^{∗} ⊗ ⋯ ⊗ V_{p}^{∗}, or more geometrically, as a section of the vector bundle 𝒪(1, …, 1) on the product of projective spaces X = ℙ^{k_{0}} × ⋯ × ℙ^{k_{p}}. There is a natural left action of the group G = GL(V_{0}) × ⋯ × GL(V_{p}) on V_{0} × ⋯ × V_{p} and a right action of G on M such that

F (Ag, x) = F (A, gx) (A ∈ M, x ∈ V_{0} × ⋯ × V_{p})

Therefore, the condition Det(A) = 0 means that the system of equations

f(x) = \frac{∂f(x)}{∂x_{i}^{(j)}} = 0  (1.2)

(for all i, j) has a solution x = (x^{(0)}, …, x^{(p)}) with all x^{(j)} ≠ 0. We say that a multilinear form f (or a matrix A) satisfying this condition is degenerate.

Algebraically, the degeneracy of a form f can be easily characterized as follows. We denote by 𝒦(f) (or 𝒦(A)) the set of points

x = (x^{(0)}, …, x^{(p)}) ∈ X = ℙ^{k_{0}} × ⋯ × ℙ^{k_{p}}

such that

f(x^{(0)}, …, x^{(j−1)}, y, x^{(j+1)}, …, x^{(p)} = 0

for every j = 1, …, p and y ∈ V_{j}^{∗}. We shall sometimes call 𝒦(A) the kernel. For a bilinear form f(x, y) there is a notion of left and right kernels

𝒦_{l}(f) = {x : f(x, y) = 0, ∀y}, 𝒦_{r}(f) = {y : f(x, y) = 0, ∀x}

and 𝒦(f) = 𝒦_{l}(f) × 𝒦_{r}(f).

1.1.1 Proposition. [GKZ94] A form f is degenerate if and only if 𝒦(f) is nonempty

Proof. Computing the differential of f we see that 𝒦(f) is exactly the set of solution of (1.2)

In particular, if p = 1 and so f is a bilinear form with a matrix A, the degeneracy of f just defined coincides with the usual notion of degeneracy and means that A is not of maximal rank. Obviously, this condition is of codimension 1 if and only if A is a square matrix, and in this case Det(A) coincides with the ordinary determinant det(A).

The fist natural question about hyperdeterminants is to describe all matrix for which Det(A) is non-trivial, i.e. X^{∨} is a hypersurface, or in other words, the degeneracy of A is a codimension one condition. The matrices of such formats can be viewed as multidimensional generalizations of ordinary square matrices.

1.1.2 Theorem. [GKZ94] The hyperdeterminant of format (k_{0}+1)×⋯×(k_{p}+1) exists if and only if

k_{j} ≤ ∑_{i≠j} k_{i}  (1.3)

for all j=0,…, p ((1.3) is called “polygon inequality”)

Assuming that (1.3) holds, i.e. the hyperdeterminant of a matrix A is nontrivial, then the next property of Det follows at once from any of the definitions.

1.1.3 Theorem. [GKZ94] The hyperdeterminant is relative invariant under the action of the group GL(V_{0}) × ⋯ × GL(V_{p}) (and so invariant under the action of SL(V_{0}) × ⋯ × SL(V_{p})).

We shall identify the set of matrix (multi-)indices I = {(i_{0}, …, i_{p}) : 0 ≤ i_{j} ≤ k_{j}} of a matrix A with the set of vertices of the product Δ^{k_{0}} ×⋯×Δ^{k_{p}} of (p + 1) standard simplices, thus the submatrix of A correspond to the faces of Δ^{k_{0}} × ⋯ × Δ^{k_{p}}. By a slice in the j-direction we mean the subset of all indices in I with the fixed j-th component, and also the corresponding submatrix of A. Two slice in the same direction are called parallel.

1.1.4 Corollary. [GKZ94]

(a) Interchanging two parallel slices leaves the hyperdeterminant invariant up to sign (which may equal 1);

(b) the hyperdeterminant is a homogeneous polynomial in the entries of each slice. The degree of homogeneity is the same for parallel slices;

(c) the hyperdeterminant does not change if we add to some slice a scalar multiple of a parallel slice;

(d) the gyperdeterminant of a matrix having two parallel slices proportional to each other is equal to 0. In particular, Det(A) = 0 if A has a zero slice.

1.1.5 Example. Now we will give an explicit form for the hyperdeterminant of the minimal 3-dimensional format 2 × 2 × 2 (in this case the hyperdetermiant was already known to A.Cayley [Cay45]).

Let A = (a_{ijk}) be a matrix with (i, j, k = 0, 1) then

\begin{aligned}Det(A) & = (a_{000}^{2}a_{111}^{2} + a_{001}^{2}a_{110}^{2} + a_{010}^{2}a_{101}^{2} + a_{011}^{2}a_{100}^{2}) \\ & − 2(a_{000}a_{001}a_{110}a_{111} + a_{000}a_{010}a_{101}a_{111} + a_{000}a_{011}a_{100}a_{111}+ \\ & + a_{001}a_{010}a_{101}a_{110} + a_{001}a_{011}a_{110}a_{100} + a_{010}a_{011}a_{101}a_{100})+ \\ & + 4(a_{000}a_{011}a_{101}a_{110} + a_{001}a_{010}a_{100}a_{111}). \end{aligned} (1.4)

Now we give a multidimensional generalization of the fact that the determinant is preserved by transposition of a matrix. For a matrix A = (a_{i_{0},...,i_{p}}) of format (k_{0} + 1) × ⋯ × (k_{p} + 1) and a permutation σ of indices 0, …, p we denotes by σ(A) the matrix of format (k_{σ^{−1}(0)} + 1) × ⋯ × (k_{σ^{−1}(p)} + 1), whose (j_{0}, j_{1}, …, j_{p})-th entry is equal to a_{j_{σ(0)},...,j_{σ(p)}}. The following result is an immediate consequence of definitions.

1.1.6 Proposition. [GKZ94] If A is degenerate then σ(A) is degenerate for every permutation σ. If Det(A) exists then Det(σ(A)) also exists and it is equal to Det(A).

Given A as above and B = (b_{j_{0},...,j_{q}}) of format (l_{0}+1)×⋯×(l_{q} +1), if k_{p} = l_{0} we may define the convolution (or product) A ∗ B of A and B as the (p + q − 1)- dimensional matrix C of format (k_{0} + 1) × ⋯ × (k_{p−1} + 1)(l_{1} + 1) × ⋯ × (l_{q} + 1) with entries

c_{i_{0},...,i_{p−1},j_{1},...,j_{q}} = ∑_{h=0}^{k_{p}} a_{i_{0},...,i_{p−1},h}b_{h,j_{1},...,j_{q}}.

Similarly, we can define the convolution A∗_{r,s}B with respect to a pair of indices r, s such that k_{r} = l_{s}.

1.1.7 Proposition. [GKZ94] If A, B are degenerate then A ∗ B is also degenerate. In particular there exist polynomials P (A, B) and Q(A, B) in entries of A and B such that

Det(A ∗ B) = P (A, B)Det(A) + Q(A, B)Det(B)

Fix p ≥ 2 and let N(k_{0}, …, k_{p}) be the degree of the hyperdeterminant of format k_{0} × ⋯ × k_{p}) (if this hyperdeterminant is not defined we make the convention N(k_{0}, …, k_{p}) = 0). 1.1.8 Theorem. [GKZ94] The generating function for the degree N(k_{0}, …, k_{p}) is given by

∑_{k_{0},...,k_{p}≥0} N(k_{0}, …, k_{p})z_{0}^{k_{0}} ⋯ z_{p}^{k_{p}} = (1 − ∑_{i=1}^{p} ie_{i}(z_{0}, …, z_{p}))^{−2}  (1.5)

where e_{i}(z_{0}, …, z_{p}) is the i-th elementary symmetric polynomial.

1.1.2 Hyperdeterminant of boundary format and its applications

1.1.9 Definition. A (p + 1)-dimensional matrix A = (a_{i_{0},...,i_{p}}) of format (k_{0} + 1) × ⋯ × (k_{p} + 1) is called boundary format if

k_{0} = ∑_{i=1}^{p} k_{i}

(interior format if k_{0} < ∑_{i=1}^{p} k_{i})

In this case, the hyperdeterminant exists (by theorem 1.1.2) and its degree can be explicitly evaluate by following formula ([GKZ94]):

N(k_{0}, …, k_{p}) = (k_{1} + ⋯ + k_{p} + 1)\binom{k_{1} + ⋯ + k_{p}}{k_{1}, …, k_{p}} = \frac{(k_{0} + 1)!}{k_{1}! ⋯ k_{p}!}  (1.6)

Note that for p = 1 the boundary is just that of ordinary square matrices and (1.6) expresses the fact that the (ordinary) determinant of a (k + 1) × (k + 1) matrix has degree k + 1.

In [GKZ94] Gelfand, Kapranov and Zelevinsky showed that if A is a boundary format matrix Det(A) can be interpreted as the resultant of the system of multilinear forms, and so they gave a number of explicit formulas for Det(A) similar to the classical Sylvester formula for the resultant of two binary forms. We consider p groups of variables x = (x_{0}^{(j)}, …, x_{k_{j}}^{(j)}) for 1 ≤ j ≤ p. Let S(m_{1}, …, m_{p}) := S^{m_{1}}V_{1} ⊗ ⋯ ⊗ S^{m_{p}}V_{p} denote the space of all polynomials in x = (x^{(1)}, …, x^{(p)}) which are homogeneous of degree m_{j} in the variables of each group x^{(j)}. We shall view a matrix A as a collection of (k_{0} + 1) multilinear forms f_{0}, f_{1}, …, f_{k_{0}} ∈ S(1, 1, …, 1) corresponding to the slices of A in the 0-th direction:

f_{i_{0}} = ∑_{i_{1},...,i_{p}} a_{i_{0},i_{1},...,i_{p}}x_{i_{1}}^{(1)} ⋯ x_{i_{p}}^{(p)}.  (1.7)

1.1.10 Theorem. [GKZ94] The hyperdeterminant Det(A) of the matrix of the boundary format is equal to the resultant of the system of multilinear forms f_{0}, f_{1}, …, f_{k_{0}}.In other words, A is degenerate if and only if the system of multilinear equations

f_{0}(x) = f_{1}(x) = ⋯ = f_{k_{0}}(x) = 0  (1.8)

has non trivial solution.

Analyzing the conditions of degeneracy, theorem 1.1.10 admits following easy generalization to the case when k_{0} > k_{1} + ⋯ + k_{p} 1.1.11 Theorem. [GKZ94] Suppose that k_{0} ≥ k_{1} + ⋯ + k_{p}.Then A is degenerate if and only if the system (1.8) has a non-trivial solution.The subvariety of degenerate matrices has codimension k_{0} − (k_{1} + ⋯ + k_{p}) + 1.

Now, assuming that A is boundary format and

m_{j} = k_{1} + k_{2} + ⋯ + k_{j−1},  j = 1, …, p  (1.9)

(with the convection m_{1} = 0, we associate to our matrix A the linear operator)

∂_{A} : S(m_{1}, m_{2}, …, m_{p})^{k_{0}+1} → S(1 + m_{1}, 1 + m_{2}, …, 1 + m_{p})

given by ∂_{A}(g_{0}, …, g_{k_{0}}) = ∑_{i=0}^{k_{0}} f_{i}g_{i} 1.1.12 Example. In the case p = 2, A ∈ V_{0} ⊗ V_{1} ⊗ V_{2}

∂_{A} : V_{0}^{∨} ⊗ S^{k_{1}}V_{2} → V_{1} ⊗ S^{k_{1}+1}V_{2}

1.1.13 Remark. We remark that since A ∈ V_{0} ⊗ … ⊗ V_{p} can be regarded as a map V_{0}^{∨} → V_{1} ⊗ ⋯ ⊗ V_{p}, taken the dual map V_{1}^{∨} ⊗ … ⊗ V_{p}^{∨} → V_{0} (that we call also A), the theorem 1.1.10 asserts that

A is degenerate if and only if for all i ≠ 0 there exist non zero vectors v_{i} ∈ V_{i}^{∨} such that A(v_{1} ⊗ … ⊗ v_{p}) = 0.

Moreover, the map ∂_{A} is obtained by tensoring A by S^{m_{1}}V_{1} ⊗ … ⊗ S^{m_{p}}V_{p} (and by projecting on S^{m_{1}+1}V_{1} ⊗ … ⊗ S^{m_{p}+1}V_{p} in the natural way). 1.1.14 Proposition. [GKZ94] Each of the spaces S(m_{1}, m_{2}, …, m_{p})^{k_{0}+1} and S(1 + m_{1}, 1 + m_{2}, …, m_{p}) has the same dimension N = \frac{(k_{0}+1)!}{k_{1}!···k_{p}!} Proof. This follows at once from the standard fact that dim(S^{m}(ℂ^{k+1})) = \binom{k+m}{k}

Let us choose in each of the space S(m_{1}, m_{2}, …, m_{p})^{k_{0}+1} and S(1 + m_{1}, 1 + m_{2}, …, m_{p} + 1) the basis consisting of monomials. We will denote by the same symbol ∂_{A} the matrix of the operator ∂_{A} in these bases, that by above proposition is square.

1.1.15 Lemma. [GKZ94] The polynomial det(∂_{A}) is non-zero, and it is irreducible over ℤ

1.1.16 Theorem. [GKZ94] We have Det(A) = det(∂_{A})

1.1.3 Singularities of hyperdeterminants

Let Y = (V_{0} − {0}) × ⋯ × (V_{p} − {0}). We say that x ∈ W is a critical point of a matrix A ∈ M if (1.2) is verified for all i, j (by definition A ∈ M is degenerate if it has at least one critical point in W). In more geometric way there is a natural projection pr : Y → X (Y = ℙ^{k_{0}} × ⋯ × ℙ^{k_{p}}), so the coordinates x_{i}^{(j)} of a point y ∈ Y are the homogeneous coordinates of pr(y) ∈ X. This projection makes Y a principal fiber bundle over X with the structure group (ℂ^{∗})^{p}. It is clear that for every A ∈ M the set of critical points of A in Y is a union of fibers of the projection pr : Y → X. We shall say that a point x ∈ X is a critical point of A if the fiber pr^{−1}(x) ⊂ Y consists of critical points of A. Now we consider the incidence variety

Z = {(A, x) ∈ M × X : x is a critical point ofA}

Then the variety Δ := X^{∨} ⊂ M of degenerate matrices is the image pr_{1}(Z), where pr_{1} is the projection (A, x) → A. This description implies at once that Δ is irreducible(the irreducibility of Δ follows from that of Z, and Z is irreducible, since it is a vector bundle over an irreducible variety X). We know that Δ is a hypersurface in M if and only if the matrix format satisfies the polygon inequality (1.3) and the equation of Δ is the hyperdeterminant. Assuming that this holds Weyman and Zelevinsky in [WZ96] describe the irreducible components of the singular locus Δ_{sing} of the hypersurface Δ. They prove that for matrices of dimension ≥ 3 and format different from 2 × 2 × 2, Δ_{sing} has codimension 1 in Δ and as shown in [GKZ92], this gives a complete description of the matrix formats for which the hyperdeterminant can be computed by Schläfli’s methods of iterating discriminants [Sch52]. Moreover they classify the irreducible components of Δ_{sing} for all matrix format. 1.1.17 Remark. The ordinary square matrices have a natural stratification according to their rank, the set Δ is the largest closed stratum of this stratification, and the next closed stratum (that is, the set of matrices of corank ≥ 2)is exactly the singular locus Δ_{sing}.

In particular, in [WZ96] is proved that for boundary format matrices the singular locus Δ_{sing} is always an irreducible hypersurface in Δ. This is a sharp contrast with the case of interior format, in fact in this case Δ_{sing} has two irreducible components, both of codimension 1 in Δ. The origin of this difference between the interior and boundary formats lies in the fact that for boundary format the hyperdeterminant can be interpreted as the resultant of a system of multilinear forms (1.1.10).

1.1.4 Multidimensional diagonal matrices and Vandermonde matrix

1.1.18 Definition. [AO99] A (p + 1)-dimensional matrix of boundary format A ∈ V_{0} ⊗ ⋯ ⊗ V_{p} is called triangulable if one of the following equivalent conditions holds:

i) there exist bases in V_{j} such that a_{i_{o},...,i_{p}} = 0 for i_{0} > ∑_{t=1}^{p} i_{t} ii) there exist a vector space U of dimension 2, a subgroup ℂ^{∗} ⊂ SL(U) and isomorphisms V_{j} ≃ S^{k_{j}}U such that if V_{0}⊗⋯⊗V_{p} = ⨁_{n∈ℤ} W_{n} is the decomposition into direct sum of eigenspaces of induced representation, we have A ∈ ⨁_{n≥0} W_{n}

proof of the equivalence between i) and ii) Let x, y be a basis of U such that t ∈ ℂ^{∗} acts on x and y as tx and t^{−1}y. Set e_{k}^{(j)} := x^{k}y^{k_{j}−k}\binom{k_{j}}{k} ∈ S^{k_{j}}U for j > 0 and e_{k}^{(0)} := x^{k_{0}−k}y^{k}\binom{k_{0}}{k} ∈ S^{k_{0}}U so that e_{i_{0}}^{(0)} ⊗ … ⊗ e_{i_{p}}^{(p)} is a basis of S^{k_{0}}U ⊗ … ⊗ S^{k_{p}}U which diagonalizes the action of ℂ^{∗}. The weight of e_{i_{0}}^{(0)} ⊗ … ⊗ e_{i_{p}}^{(p)} is 2 (∑_{t=1}^{p} i_{t} − i_{0}), hence ii) implies i). The converse is trivial.

1.1.19 Definition. [AO99] A (p + 1)-dimensional matrix of boundary format A ∈ V_{0} ⊗ ⋯ ⊗ V_{p} is called diagonalizable if one of the following equivalent conditions holds:

i) there exist bases in V_{j} such that a_{i_{o},...,i_{p}} = 0 for i_{0} ≠ ∑_{t=1}^{p} i_{t} ii) there exist a vector space U of dimension 2, a subgroup ℂ^{∗} ⊂ SL(U) and isomorphisms V_{j} ≃ S^{k_{j}}U such that A is a fixed point of the induced action of ℂ^{∗} 1.1.20 Definition. [AO99] A (p + 1)-dimensional matrix of boundary format A ∈ V_{0} ⊗ ⋯ ⊗ V_{p} is an identity if one of the following equivalent conditions holds:

i) there exist bases in V_{j} such that

a_{i_{o},...,i_{p}} = \begin{cases}0 & for i_{0} ≠ ∑_{t=1}^{p} i_{t} \\ 1 & for i_{0} = ∑_{t=1}^{p} i_{t} \end{cases}

ii) there exist a vector space U of dimension 2 and isomorphisms V_{j} ≃ S^{k_{j}}U such that A belongs to the unique one dimensional SL(U)-invariant subspace of S^{k_{0}}U ⊗ ⋯ ⊗ S^{k_{p}}U

The equivalence between i) and ii) follows easily from the following remark: the matrix A satisfies the condition ii) if and only if it corresponds to the natural multiplication map S^{k_{1}}U ⊗ … ⊗ S^{k_{p}}U → S^{k_{0}}U (after a suitable isomorphism U ≃ U^{∗} has been fixed).

The above definitions agrees with some in [WZ96] where a (p + 1)-complex matrix A = (a_{i_{0},...,i_{p}}) of format (k_{0} + 1) × ⋯ × (k_{p} + 1) is called diagonal if a_{i_{0},...,i_{p}} = 0 unless i = (i_{0}, …, i_{p}) is a diagonal multi-index for k = (k_{0}, …, k_{p}) i.e. if i and (k − i) satisfy the “polygon” inequality (1.3). We shall see that a generic diagonal matrix A is nondegenerate as proved in [WZ96]. This is a consequence of the following statement: there exists an extreme monomial

∏_{i=(i_{0},...,i_{p})} a_{i}^{d(i,k)}  (1.10)

appearing in Det(A) such that d(i, k) > 0 if and only if i is diagonal for k. (Recall that a monomial is extreme if it corresponds to a vertex of the Newton polytope of Det(A)). To construct such a monomial we recall that in [GKZ94] and [GKZ92] is shown that any coefficient of the degree N(k_{0}, …, k_{p}) in the formula (1.6) has a combinatorial expression as a sum of positive summands; furthermore, a combinatorial argument shows that N(k_{0}, …, k_{p}) > 0 exactly when k = (k_{0}, …, k_{p}) is diagonal. Now consider the square root of the generating function (1.6), i.e., the series

∑_{k_{0},...,k_{p}≥0} M(k_{0}, …, k_{p})z_{0}^{k_{0}} ⋯ z_{p}^{k_{p}} = (1 − ∑_{i=1}^{p} ie_{i}(z_{0}, …, z_{p}))^{−1}  (1.11)

For any two non-negative integer vectors i = (i_{0}, …, i_{p}), k = (k_{0}, …, k_{p}) we set

d(i, k) = M(i_{0}, …, i_{p})M(k_{0} − i_{0}, …, k_{p} − i_{p})  (1.12)

1.1.21 Theorem. [WZ96] For every interior or boundary matrix format, the monomial (1.10) with the exponent given by (1.12) is an extreme monomial in Det(A) appearing with the coefficient ±1. The exponent d(i, k) is positive if and only if i is diagonal for k.

This theorem proves that the diagonal matrices are nondegenerate and since the hyperdeterminant was defined only up to the sign, then it gives a natural choice of the sign, by requiring that the monomial given by (1.10),(1.12) occurs in Det(A) with coefficient 1. 1.1.22 Remark. If the matrix format is boundary, then the determinantal formula Det(A) given by (1.1.16), implies that the hyperdeterminant of diagonal matrix is just the monomial (1.10). This is no longer true for interior format. Using computer algebra system MACAULAY, Weyman and Zelevinsky found that the hyperdeterminant of a diagonal 3 × 3 × 3 matrix is given by

\begin{matrix}Det(A) = & (a_{000}a_{222})^{8}(a_{110}a_{101}a_{011}a_{112}a_{121}a_{211})^{2}(a_{000}^{2}a_{111}^{4}a_{222}^{2}+ \\ + 8a_{000}a_{111}^{2}a_{222}(a_{000}a_{112}a_{121}a_{211} + +a_{222}a_{110}a_{101}a_{011}))+ \\ + 16(a_{000}a_{112}a_{121}a_{211})^{2} + 16(a_{222}a_{110}a_{101}a_{011})^{2}+ \\ − 32a_{000}a_{110}a_{101}a_{011}a_{112}a_{121}a_{211}a_{222}) \end{matrix} (1.13)

Here the diagonal monomial is the only one occurring with coefficient 1. There are two other extreme monomials occurring in (1.13): both of them have coefficient 16 and not contain the variable a_{111}. It follows that there exists a nondegenerate diagonal matrix having a_{111} = 0.

Now we assume that the matrix format is boundary.We shall construct another special class of matrices analogs of the classical Vandermonde matrix. Let Λ = (λ_{i,j})_{0≤i≤k_{0},1≤j≤p} be a (k_{0} + 1) × (p) complex matrix. We define Vandermonde-type matrix A = A(Λ) of format (k_{0} + 1) × ⋯ × (k_{p} + 1) by the formula

a_{i_{0},i_{1},...,i_{p}} = λ_{i,1}^{i_{1}}λ_{i,2}^{i_{2}} ⋯ λ_{i,p}^{i_{p}}  (1.14)

If p = 1, k_{0} = k_{1} = k then A = A(Λ) is the usual (k + 1) × (k + 1) Vandermonde matrix

(\begin{matrix}1 & λ_{0} & λ_{0}^{2} & … & λ_{0}^{k} \\ 1 & λ_{1} & λ_{1}^{2} & … & λ_{1}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & λ_{k} & λ_{k}^{2} & … & λ_{k}^{k} \end{matrix})

1.1.23 Proposition. The matrix A(Λ) is nondegenerate if and only if for each j = 2, …, p the numbers λ_{0,j}, λ_{1,j}, …, λ_{k_{0},j} are mutually distinct.

1.2 Geometric invariant theory and nondegen-

erate matrices of boundary format

We consider the natural action of SL(V_{0}) × … × SL(V_{p}) on ℙ(V_{0} ⊗ … ⊗ V_{p}). We may suppose p ≥ 2. The definitions of triangulable, diagonalizable and identity boundary format matrix, given in the last section, apply to elements of ℙ(V_{0} ⊗ … ⊗ V_{p}) as well. In particular all identity matrices fill a distinguished orbit in ℙ(V_{0}⊗…⊗V_{p}). Moreover, we know that the hyperdeterminant of multidimensional matrices is a homogeneous polynomial function over V_{0} ⊗ … ⊗ V_{p} then the condition Det A ≠ 0 is meaningful for A ∈ ℙ(V_{0}⊗…⊗V_{p}). The function Det is SL(V_{0}) × … × SL(V_{p})-invariant (see theorem (1.1.3)), in particular if Det A ≠ 0 then A is semistable for the action of SL(V_{0}) × … × SL(V_{p}). We denote by Stab (A) ⊂ SL(V_{0}) × … × SL(V_{p}) the stabilizer subgroup of A and by Stab (A)^{0} its connected component containing the identity.

We remember that if e_{0}^{(j)}, …, e_{k_{j}}^{(j)} is a basis in V_{j} so that every A ∈ V_{0}⊗⋯⊗V_{p} has coordinate form

A = ∑ a_{i_{0},...,i_{p}}e_{i_{0}}^{(0)} ⊗ ⋯ ⊗ e_{i_{p}}^{(p)}  (1.15)

and if x_{0}^{(j)}, …, x_{k_{j}}^{(j)} are the coordinates in V_{j}, then A has the following different descriptions:

1. A multilinear form

A = ∑_{(i_{0},...,i_{p})} a_{i_{0},...,i_{p}}x_{i_{0}}^{(0)} ⋯ x_{i_{p}}^{(p)}

2. For any fixed j ≠ 0 an ordinary matrix M_{A}^{(j)} = (m_{i_{j}i_{0}}) of size (k_{j} + 1) × (k_{0} + 1) whose entries are multilinear forms

m_{i_{j}i_{0}} = ∑_{(i_{1},...î_{j}...,i_{p})} a_{i_{0},...,i_{p}}x_{i_{1}}^{(1)} ⊗ ⋯ ⊗ x̂_{i_{j}}^{(j)} ⊗ ⋯ ⊗ x_{i_{p}}^{(p)}  (1.16)

3. For any fixed j ≠ 0 a sheaf morphism f_{A}^{(j)} on the product X = ℙ^{k_{1}} × ⋯ × ℙ̂^{k_{j}} × ⋯ × ℙ^{k_{p}}

𝒪_{X} ⊗ V_{0}^{∨} → 𝒪_{X}(1, …, 1) ⊗ V_{j}  (1.17)

1.2.1 Theorem. Let A be a matrix of boundary format, let f_{A}^{(1)} be a surjective map and S^{∗} := kerf_{A}^{(1)} a vector bundle then

h^{0}(S^{∗}(k_{1}, k_{1} + k_{2}, …, k_{1} + … + k_{p−1}) = 0  (1.18)

Proof. We have that S^{∗} is a vector bundle on X = ℙ^{k_{2}} × ⋯ × ℙ^{k_{p}} of rank equal to dimX = k_{0} − k_{1} and detS^{∗} = 𝒪_{X}(−k_{1} − 1, …, −k_{1} − 1), hence

S^{∗}(k_{1}, k_{1}+k_{2}, …, k_{1}+…+k_{p−1}) ≃ ∧^{k_{0}−k_{1}−1}S(−1, k_{2}−1, …, k_{2}+k_{3}+⋯+k_{p−1}−1)

The result follows from the (k_{0} −k_{1} −1)-power of the sequence dual to (1.17):

\begin{aligned}0 & → S^{k_{0}−k_{1}−1}V_{1}^{∨} ⊗ 𝒪(−k_{2} − ⋯ − k_{p}, , −k_{3} − ⋯ − k_{p}, …, −k_{p}) → … \\ … & → V_{0} ⊗ S^{(k_{0}−k_{1}−2)}V_{1}^{∨} ⊗ 𝒪(−k_{2} − ⋯ − k_{p} − 1, −k_{3} − ⋯ − k_{p} − 1, …, −k_{p} − 1) → … \\ … & → ∧^{k_{0}−k_{1}−1}V_{0}^{∨} ⊗ 𝒪(−1, k_{2} − 1, …, k_{2} + k_{3} + ⋯ + k_{p−1} − 1) → … \\ … & → ∧^{k_{0}−k_{1}−1}S(−1, k_{2} − 1, …, k_{2} + ⋯ + k_{p−1} − 1) → 0 \end{aligned} ; (1.19)

1.2.2 Theorem. The following properties are equivalent

i) The system f_{0}(x) = ⋯ = f_{k_{0}}(x) has only trivial solution on ii) f_{A} is surjective and S^{∗} = Kerf_{A}^{(1)} is a vector bundle on X = ℙ^{k_{2}} × ⋯ × ℙ^{k_{p}} of rank k_{0} − k_{1} iii) det(∂_{A}) ≠ 0. Proof.

i) ⟹ ii) It follows obviously by definition of f_{i}. ii) ⟹ iii) By proposition 1.1.14 ∂_{A} is a map between finite vector spaces having same dimension \frac{(k_{0}+1)!}{k_{1}!k_{2}!...k_{p}!}. If f_{A} is surjective the kernel of ∂_{A} is equal to H^{0}(S^{∗}(k_{1}, k_{1} + k_{2}, …, k_{1} + ⋯ + k_{p−1})) which vanishes by the above lemma, then ∂_{A} is an injective map and det(∂_{A}) ≠ 0. iii) ⟹ i) If the system has non trivial solution then there exist nonzero vectors v_{i} ∈ V_{i}^{∨} , i = 1, … p such that A(v_{1} ⊗ ⋯ ⊗ v_{p}) = 0 and the nonzero tensor v_{1}^{⊗(m_{1}+1)} ⊗ v_{2}^{⊗(m_{2}+1)} ⊗ ⋯ ⊗ v_{p}^{⊗(m_{p}+1)} maps to zero, hence det(∂_{A}) = 0. 1.2.3 Remark. Above theorem suggests to give as definition of hyperdeterminant of a matrix A the usual determinant of the square matrix ∂_{A}.

More in general, given an opportune order on the integers k_{i} in the similar way of (1.9), theorem 1.1.10 easily translates into: 1.2.4 Theorem. The following properties are equivalent

i) DetA ≠ 0; ii) for all j ≠ 0 the matrix M_{A}^{(j)} has constant rank k_{j} + 1 on X = ℙ^{k_{1}} × … ℙ̂^{k_{j}} × ⋯ × ℙ^{k_{p}}; iii) for all j ≠ 0 the morphism f_{A}^{(j)} is surjective so that S^{∗}_{A}^{(j)} = Kerf_{A}^{(j)} is a vector bundle of rank k_{0} − k_{j}

In the particular case p = 2 the (dual) vector bundle S_{A}^{(1)} lives on the projective space ℙ^{n}, n = k_{2} and is a Steiner bundle as defined in [DK93] (this case has been investigate in [AO99]). We can keep to S_{A} the name Steiner also for p ≥ 3.

Moreover, the action of SL(V_{0}) × ⋯ × SL(V_{p}) on A translates for all j to an action on the corresponding bundle S^{∗}_{A}^{(j)} in two steps: first the action of SL(V_{0})×SL(V_{j}) leaves the bundle in the same isomorphism class, then SL(V_{1})× ⋯ × { \atop SL(V_{j})} × ⋯ × SL(V_{p}) acts on the classes, i.e. on the moduli space of Steiner bundles. It follows that the invariants of the matrices for the action of SL(V_{0}) × ⋯ × SL(V_{p}) coincide with the invariants of the action of SL(V_{1}) × ⋯ × { \atop SL(V_{j})} × ⋯ × SL(V_{p}) on the moduli space of the corresponding bundles and the stable points of both actions correspond to each other. In particular, investigating the properties and the invariant of both the above action, Ancona and Ottaviani in [AO99], proved the following main results.

1.2.5 Theorem. [AO99] Let A ∈ ℙ(V_{0} ⊗ … ⊗ V_{p}) of boundary format such that Det A ≠ 0. Then

A is triangulable ⟺ A is not stable for the action of SL(V_{0}) × … × SL(V_{p})

1.2.6 Theorem. [AO99] Let A ∈ ℙ(V_{0} ⊗ … ⊗ V_{p}) of boundary format such that Det A ≠ 0. Then

A is diagonalizable ⟺ ℂ^{∗} ⊂ Stab(A)

In the case p = 2 in the following proposition we give an alternative simple proof of this result.

1.2.7 Proposition. Let A ∈ ℙ(V_{0} ⊗ V_{1} ⊗ V_{2}) of boundary format such that Det A ≠ 0. If ℂ^{∗} ⊆ GL(V_{0}) × GL(V_{1}) × GL(V_{2}) stabilizes A then A is diagonalizable.

Proof. For all i=0,1,2 we choose a basis of V_{i} such that

a_{i0j} = 0 if i ≠ j and ∀i ≥ k_{2} + 1

and we put

\begin{aligned}α_{0} & ≤ ⋯ ≤ & α_{k_{0}} \\ β_{0} & ≥ ⋯ ≥ & β_{k_{1}} \\ γ_{0} & ≥ ⋯ ≥ & γ_{k_{2}} \end{aligned}

the weights of the action of ℂ^{∗} respectively on V_{0},V_{1} and V_{2}, hence we have a region on left-up of the matrix M_{A} corresponding to maximal weight. Claim: α_{0} < α_{1} and β_{0} > β_{1}.

Assume, as a contradiction, that there exist two integers s ≥ 1 and t ≥ 1 such that

\begin{aligned}α_{0} & = ⋯ = & α_{s} \\ β_{0} & = ⋯ = & β_{t}. \end{aligned}  (1.20)

Since ℂ^{∗} ⊂ Stab(A), then (1.20) implies also that γ_{0} = ⋯ = γ_{s} and in the leftup minor of M_{A} of order (t + 1) × (s + 1) we have only the variables x_{0}, …, x_{s} where x_{0}, …, x_{k_{2}} is the coordinate system of ℙ(V_{2}). Moreover, in the right-up minor of order (t + 1) × (k_{0} − s) we have only the variables x_{s+1}, …, x_{k_{2}}, the left down minor of order (k_{1} − t) × (s + 1) is the zero matrix and in the last minor, that we call A′, we have all the variables x_{i}.

If s < t the left-up minor has rank less than (t+1), hence, the first (t+1)-rows of M_{A} are linearly dependent by putting x_{i} = 0 for i ≥ s + 1. This contradicts, by theorem 1.2.4, the hypothesis DetA ≠ 0.

If s = t, since s, t ≤ 1, by calculating the zero of the determinant of the left-up minor, we get a point of ℙ(V_{2}) where the first s rows are dependent as in the above case.

If s > t the transposed matrix of A′ gives on ℙ^{k_{2}} the sheaf morphism

𝒪_{ℙ^{k_{2}}}^{k_{1}−t} → 𝒪_{ℙ^{k_{2}}}^{k_{0}−s}(1)

whose rank drops on a subvariety of codimension ≤ k_{0} − s − k_{1} + t + 1 ≤ k_{2} = dim ℙ(V_{2}), then we get, as contradiction, that A is degenerate. So, (1.20) cannot occur.

Now, we suppose that k_{1} = 1 then for all i = 0, .., k_{2} + 1 m_{1,i} = a_{i,1,i−1}x_{i−1} i.e. A is diagonal. In fact, by comparing: the weights of m_{1,i} with the weight of m_{0,k} we get a_{i1i} = 0; the weights of m_{1,i} we get a_{ikj} = 0 ∀j ≥ i + 2; the weights of m_{1,i} with the weights of m_{0,t} , we get a_{i1(i+1)} = 0 if t > i and a_{i1j} = 0 ∀j ≤ i − 2 if t < i.

Also in the case k_{1} > 1, the same comparisons gives that m_{j0} = 0 for all j > 0. The minor obtained by cutting-out the first rows and the first column has maximal rank (since A is nondegenerate), is boundary format and it is stabilized by ℂ^{∗}, then, by induction on k_{1}, it is diagonal. Hence, also A is diagonal.

1.2.8 Theorem. [AO99] Let A ∈ ℙ(V_{0} ⊗ V_{1} ⊗ V_{2}) of boundary format such that Det A ≠ 0. Then there exists a 2-dimensional vector space U such that SL(U) acts over V_{i} ≃ S^{k_{i}}U and according to this action on V_{0} ⊗ V_{1} ⊗ V_{2} we have Stab (A)^{0} ⊂ SL(U). Moreover the following cases are possible

Stab (A)^{0} ≃ \begin{cases}0 \\ ℂ \\ ℂ^{∗} \\ SL(2) (this case occurs if and only if A is an identity) \end{cases}

1.2.9 Remark. When A is an identity then Stab (A) ≃ SL(2).

In the next section we extend the first part of this result to the case p ≥ 2 and using jumping hyperplane we characterize Schwarzenberger bundles on product of projective spaces.

1.3 Jumping hyperplanes, Steiner bundles and

stabilizers

Let p = 2 and S := S^{1} be the Steiner bundle on ℙ(V_{2}) defined by a matrix A ∈ V_{0} ⊗ V_{1} ⊗ V_{2} boundary format, an hyperplane h ∈ ℙ(V_{2}^{∗}) is an instable hyperplane of S if h^{0}(S_{|h}^{∗}) ≠ 0 (see [AO99]).

In particular, H^{0}(S^{∗}(t)) identifies the spaces of (k_{0} + 1) × 1-column vectors v with entries in S^{t}V_{2} such that Av = 0, and an hyperplane h is instable for S if and only if there are nonzero vectors v_{0} of size (k_{0} + 1) × 1 and (k_{1} + 1) × 1 both with constant coefficients such that

Av_{0} = v_{1}h.  (1.21)

(the tensor ℋ = v_{0} ⊗v_{1} is called instable (or jumping) hyperplane for the matrix A)

For p ≥ 3 there are at least two ways to define a jumping hyperplane. We will them weak and strongly jumping hyperplanes.

1.3.1 Definition. ℋ = v_{0} ⊗ v_{j} ∈ V_{0} ⊗ V_{j} is a (j)-weak jumping hyperplane for A if ∃ v_{0}, w_{1}, …, w_{k_{0}} basis of V_{0} such that

A = v_{0} ⊗ v_{j} ⊗ h + ∑_{i=1}^{k_{0}} w_{i} ⊗ …  (1.22)

where h ∈ V_{1} ⊗⋯⊗V̂_{j} ⊗⋯⊗V_{p} is an hyperplane for ℙ^{k_{1}} ×⋯×ℙ̂^{k_{j}} ×⋯×ℙ^{k_{p}} ⊂ ℙ(V_{1} ⊗ ⋯ ⊗ V̂_{j} ⊗ ⋯ ⊗ V_{p}).

1.3.2 Remark. The expression (1.22) means, as in the case p = 2, that H^{0}(Kerf_{A}_{|h}^{(j)}) ≠ 0.

If ℋ = v_{0} ⊗ v_{j} is a (j)-weak jumping hyperplane for A then the map:

\begin{matrix}V_{0} ⊗ ⋯ ⊗ V_{p} & → (V_{0}/_{<v_{0}>}) ⊗ ⋯ ⊗ (V_{j}/_{<v_{j}>}) ⊗ ⋯ ⊗ V_{p} \\ A & ↦ A′_{j} \end{matrix}

gives an elementary transformation [Mar82] 1.3.3 Remark. A′_{j} is again of boundary format

1.3.4 Theorem. If A′_{j} is defined as above

DetA ≠ 0 ⇒ DetA′_{j} ≠ 0

Proof. If X := ℙ^{k_{1}} ×⋯×ℙ̂^{k_{j}} ×⋯×ℙ^{k_{p}} and h is the hyperplane defined in (1.3.1) associated to ℋ, the map 𝒪_{h} → S_{A}^{∗}_{|h}^{(j)} induces a surjective map S_{A}^{∗}^{(j)} → 𝒪_{h} and an exact sequence:

0 → S′^{(j)} → S_{A}^{(j)} → 𝒪_{h} → 0

where by definitions S′^{(j)} = S_{A′_{j}}^{(j)}. Since codim h = 1 then it is locally free [Ser65] and by theorem (1.2.4) the result follows.

1.3.5 Remark. An element g ∈ SL(V_{0}) × SL(V_{j}) preserves h and it induces ḡ ∈ SL(V_{0}/_{<g(v_{0})>}) × SL(V_{j}/_{<g(v_{j})>}) such that g · A projects to ḡA′_{j} and the elementary transformation is well behaved with respect the action of g.

1.3.6 Definition. ℋ = v_{0} ⊗ v_{1} ⊗ ⋯ ⊗ v_{p} is a strong jumping hyperplane for A if ∃v_{0}, w_{1}, …, w_{k_{0}} basis of V_{0} such that

A = v_{0} ⊗ v_{1} ⊗ ⋯ ⊗ v_{p} + ∑_{i=1}^{k_{0}} w_{i} ⊗ …

1.3.7 Remark. If ℋ is a strong jumping hyperplane then ℋ is a (j)-weak jumping hyperplane for all j = 1, …, p. In particular for a strong jumping hyperplane there are many elementary transformations. 1.3.8 Remark. For p = 2 the notation of strong jumping hyperplane coincides to weak jumping hyperplane (see [AO99])

1.3.9 Example. (the identity) The identity matrix is represented by

I := ∑_{\begin{matrix}i_{0}=i_{1}+···+i_{p} \\ 0≤i_{j}≤k_{j} \end{matrix}} e_{i_{0}}^{(0)} ⊗ ⋯ ⊗ e_{i_{p}}^{(p)}

Let t_{0}, …, t_{k_{0}} be any distinct complex numbers. Let w be the (k_{0} + 1) × (k_{0} + 1) Vandermonde matrix whose (i, j) entry is t_{j}^{(i−1)}, so acting with w over V_{0} we have:

e_{j}^{(0)} = ∑_{s=0}^{k_{0}} ē_{s}^{(0)}t_{s}^{j}

Then substituting

\begin{aligned}I & = ∑_{\begin{matrix}i_{0}=i_{1}+···+i_{p} \\ s=0,...k_{0} \end{matrix}} ē_{i_{0}}^{(0)}t_{s}^{i_{0}} ⊗ ⋯ ⊗ e_{i_{p}}^{(p)} \\ & = ∑_{s=0}^{k_{0}} ē_{s}^{0} ⊗ (∑_{i_{1}=0}^{k_{1}} e_{i_{1}}^{(1)}t_{s}^{i_{1}}) ⊗ ⋯ ⊗ (∑_{i_{p}=0}^{k_{p}} e_{i_{p}}^{p}t_{s}^{i_{p}}) \end{aligned}

Thus, since t_{i} have no restrictions, I has infinitely many strong jumping hyperplane. We call Schwarzenberger bundle the vector bundle associated to I (in fact in the case p = 2 it is exactly the same introduced by Schwarzenberger in [Sch61])(see also ([AO99])

1.3.10 Proposition. Let A be boundary format matrix with DetA ≠ 0. If A has N ≥ k_{0} + 3 strong jumping hyperplanes then it is an identity.

Proof. In the case p = 2 the statement is prove in [AO99] (theorem 5.13). Chosen V_{0} and other two vector spaces among V_{1}, …, V_{p} (say V_{1} and V_{2}), one may perform several elementary transformations whit V_{0} and all the others so that we get A′ ∈ V′_{0} ⊗ V_{1} ⊗ V_{2} boundary format matrix with DetA′ ≠ 0 and N′ ≥ k′_{0} + 3 strong jumping hyperplanes, then A′ is an identity. (Hence in ℙ(V_{1}) and ℙ(V_{2}) there are rational normal curves because S_{A′} is a Schwarzenberger bundle). As in the above example, one can change the hyperplane giving the elementary transformation, so that for all N strong jumping hyperplanes we get t_{1}, …, t_{N} distinct complex numbers and corresponding suitable basis of V_{1} and V_{2} :

\begin{matrix}ē_{0}^{(1)} & … ē_{k_{1}}^{(1)} \\ ē_{0}^{(2)} & … ē_{k_{2}}^{(2)} \end{matrix}

such that the hyperplanes are given by

∑_{i=0}^{k_{1}} ē_{i}^{(1)}t_{j}^{i} and ∑_{i=0}^{k_{2}} ē_{i}^{(2)}t_{j}^{i} forj = 1, … N

Now, changing V_{1} and V_{2} with the pairs V_{1}, V_{j} (j = 1, … p ) we get

A := ∑_{s=0}^{k_{0}} ē_{s}^{0} ⊗ (∑_{i_{1}=0}^{k_{1}} e_{i_{1}}^{(1)}t_{s}^{i_{1}}) ⊗ ⋯ ⊗ (∑_{i_{p}=0}^{k_{p}} e_{i_{p}}^{p}t_{s}^{i_{p}})

showing that A is an identity.

1.3.11 Proposition. Let A be boundary format matrix with DetA ≠ 0. If A has k_{0} + 2 strong jumping hyperplanes then it is uniquely determined by these hyperplanes.

Proof. In the case p = 2 the statement is prove in [AO99] (theorem 5.3). Chosen V_{0} and other two vector spaces among V_{1}, …, V_{p} (say V_{1} and V_{2}), one may perform several elementary transformations whit V_{0} and all the others so that we get A′ ∈ V′_{0} ⊗V_{1} ⊗V_{2} boundary format matrix with DetA′ ≠ 0 and N′ = k′_{0} +2 strong jumping hyperplanes, then A′ is uniquely determined. Now, changing V_{1} and V_{2} with the pairs V_{1} and V_{j} (j = 2, …, p) we detect all the 3-dimensional submatrix of A which are all uniquely determined, then also A is.

1.3.12 Remark. In the case p = 2 we know that k_{0} + 2 jumping hyperplanes give an existence condition for the bundle S_{A} (it is a logarithmic bundle, see [AO99]) but in the case p ≥ 3 there is not analog existence result. Moreover, by definitions, there exists a unique matrix A with k_{0} + 1 strong jumping hyperplanes given.

1.3.13 Theorem. Let A be boundary format matrix with DetA ≠ 0. Then there exists a 2-dimensional vector space U such that SL(U) acts over V_{i} ≃ S^{k_{i}}U and according to this action on V_{0} ⊗ ⋯ ⊗ V_{p} we have Stab(A)^{0} ⊆ SL(U).

Proof. We proceed by induction on k_{0}. If k_{0} = 2 then p = 2 and V_{0} ⊗ V_{1} ⊗ V_{2} ≃ ℂ^{3} ⊗ ℂ^{2} ⊗ ℂ^{2} and a Steiner bundle on the line ℙ^{1} is Schwarzenberger (A = I), then the result follows. We may suppose that Stab(A)^{0} has dimension ≥ 1 then by theorem 1.2.5 the matrix A is triangulable and this implies that there exists a strong jumping hyperplane ℋ. We may, also, suppose that the number of jumping hyperplanes is finite otherwise A is an identity (proposition 1.3.10), hence ℋ is Stab(A)^{0}- invariant. Let A′_{1} the image of A by the elementary transformation associated to the weak jumping hyperplane v_{0} ⊗v_{1} defined by ℋ (we choose j = 1 to have simpler notations). The matrix A_{1} belongs to V′_{0} ⊗V′_{1} ⊗V_{2} ⊗⋯⊗V_{p} where V′_{0} = V_{0}/_{<v_{0}>} and V′_{1} = V_{1}/< v_{1} >, it is nondegenerate and of boundary format then, by induction, there exists a 2-dimensional vector space U such that

V′_{0} ≃ S^{k_{0}−1}(U), V′_{1} ≃ S^{k_{1}−1}(U) and V_{i} = S^{k_{i}}(U) for all i ≥ 2

and Stab(A′_{1})^{0} ⊆ SL(U). (By using essentially the same argument we could work in GL(V_{0} × ⋯ × GL(V_{p}))

Since A′_{1} is obtained from the matrix A after the choice of two directions, any element which stabilizes A also stabilizes A′_{1}, so Stab(A)^{0} ⊆ Stab(A′)^{0}.

Moreover, we remark that the above considered elementary transformation gives the decomposition V_{0} = V′_{0} ⊕ ℂ and V_{1} = V′_{1} ⊕ ℂ.

Since no other morphism of SL(U) in SL(V_{0}) × SL(V_{1}) × SL(S^{k_{2}}U) × ⋯ × SL(S^{k_{p}}U) can give S^{k_{0}−1}U⊗S^{k_{1}−1}U⊗S^{k_{2}}U⊗…⊗S^{k_{p}}U as an invariant summand of V_{0} ⊗ V_{1} ⊗ S^{k_{2}}U ⊗ S^{k_{p}}U, then the inclusion

Hom(V′_{0}, V′_{1} ⊗ V_{2} ⊗ ⋯ ⊗ V_{p}) ⊂ Hom(V_{0}, V_{1} ⊗ V_{2} ⊗ … V_{p})

identifies to the SL(U)-invariant inclusion

S^{k_{0}−1}U ⊗ S^{k_{1}−1}U ⊗ S^{k_{2}}U ⊗ … ⊗ S^{k_{p}}U ⊂ S^{k_{0}}U ⊗ S^{k_{1}}U ⊗ S^{k_{2}}U ⊗ … ⊗ S^{k_{p}}U

according to the natural actions, hence also V_{0} = S^{k_{0}}U and V_{1} = S^{k_{1}}U.

1.4 Cohomology of Schwarzenberger bundles

In [AO94] it was proved that a Steiner bundles S on ℙ^{n} defined by the sequences

0 → I ⊗ 𝒪(−1) → W ⊗ 𝒪 → S → 0

where I and W are vector spaces of dimension k and k + n respectively, are stable, give smooth points in the moduli space of all stable bundles with the same rank and Chern classes and they belong to an irreducible component of dimension (k − 1)(n − 1)(n + k + 1). Moreover, in [AO99], it was proved that for a Steiner bundle S as above, h^{0}(S^{∗}(t)) = 0 if and only if t ≤ k − 1.

In the class of Steiner bundle we want to give a cohomological characterization of Schwarzenberger bundles, hence, we calculate the dimension of the cohomology of their second symmetric power.

Throughout this section 𝕂 denotes an algebraically closed field of characteristic zero. U denotes a 2-dimensional 𝕂 vector space (U =< s, t >), S_{n} = S^{n}U its n-th symmetric power (S_{n} =< s^{n}, s^{n−1}t, …, t^{n} >) and ℙ^{n} = ℙ(S_{n}). First, we prove the existance of a special Steiner bundles (or Schwarzenberger bundle) 𝒮 on ℙ^{n} defined by the following exact sequence

0 → 𝒮^{∗} → S_{n+k−1}^{∨} ⊗ 𝒪 ⟶^{B} S_{k−1}^{∨} ⊗ 𝒪(1) → 0  (1.23)

where

B = ( \begin{matrix}s^{n} & s^{n−1}t & ⋯ & st^{n−1} & t^{n} \\ & ... & ... & & ... & ... \\ & & s^{n} & s^{n−1}t & ⋯ & st^{n−1} & t^{n} \end{matrix} )

and second we calculate its cohomology.

There is a natural exact sequence of GL(U)-equivariant maps for any k ≥ 2, n ≥ 1 (Clebsch-Gordan sequence):

0 →∧^{2} U ⊗ S_{k−2} ⊗ S_{n−1} ⟶^{β} S_{k−1} ⊗ S_{n} ⟶^{μ} S_{k+n−1} → 0  (1.24)

where μ is the multiplication map and β is defined by (s ∧ t) ⊗ f ⊗ g → (sf ⊗ tg − tf ⊗ sg).

1.4.1 Lemma. Let B, C be vector spaces of dimension k and 2n(k − 1). A linear map B ⊗ S_{n}^{∨} ⟶^{b} C induces a sheaf homomorphism

B ⊗ Ω^{1}(1) ⟶^{b̃} C ⊗ 𝒪

Proof. By the dual Euler sequence:

0 → Ω^{1}(1) → 𝒪 ⊗ S_{n}^{∨} → 𝒪(1) → 0

we obtain

\begin{aligned}0 → Ω^{2}(2) → 𝒪 & ⊗ & ∧^{2} S_{n}^{∨} & ⟶ & 𝒪(1) & ⊗ S_{n}^{∨} → 𝒪(2) → 0 \\ & ↘ & ↗ \\ & & Ω^{1}(2) \\ & ↗ & ↘ \\ 0 & & & 0 \end{aligned}  (1.25)

i.e the following two exact sequences:

0 → Ω^{2}(2) → 𝒪⊗ ∧^{2} S_{n}^{∨} ⟶^{f} Ω^{1}(2) → 0  (1.26)
0 → Ω^{1}(2) ⟶^{g} 𝒪(1) ⊗ S_{n}^{∨} → 𝒪(2) → 0  (1.27)

Then b̃ is defined by the composition:

0 → B ⊗ Ω^{1}(1) ⟶^{g⊗id_{𝒪(−1)}} B ⊗ S_{n}^{∨} ⊗ 𝒪 ⟶^{b⊗id_{𝒪}} C ⊗ 𝒪  (1.28)

i.e b̃ = (b ⊗ id_{𝒪}) ∘ (g ⊗ id_{𝒪(−1)} ) where g ⊗ id_{𝒪(−1)} is an injective map.

1.4.2 Remark. It is true also the conversely of the previous lemma. In fact, given the homomorphism b̃ the linear map b can be defined as b = H^{0}(b̃^{∨})^{∨} where b̃^{∨} : C^{∨} ⊗ 𝒪 → B^{∨} ⊗ T_{ℙ^{n}}(−1). By Euler sequence we get:

0 → H^{0}(𝒪 ⊗ S_{n}) ⟶^{∼} H^{0}(T_{ℙ^{n}}(−1)) → 0

⇒ H^{0}(b̃^{∨}) : H^{0}(C^{∨} ⊗ 𝒪) → H^{0}(B^{∨} ⊗ T_{ℙ^{n}}(−1)) with : H^{0}(C^{∨} ⊗ 𝒪) ≃ C^{∨} ⊗ ℂ

H^{0}(B^{∨} ⊗ T (−1)) ≃ B^{∨} ⊗ ℂ ⊗ V_{n} Then dualizing it is induced the map: b : B ⊗ S_{n} → C

Now, if we denote

B := S_{k−1}^{∨} and C :=∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ S_{n−1}^{∨}  (1.29)

then, by previous lemma we can define the morphism

b̃: S_{k−1}^{∨} ⊗ Ω^{1}(1) →∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ S_{n−1}^{∨} ⊗ 𝒪

where b = β^{∨} and β is defined in (1.24).

1.4.3 Lemma. b̃ is a surjective map.

Proof. We consider the following exact commutative diagram:

0  0
↓  ↓
0 ⟶  N  ⟶ S_{k−1}^{∨} ⊗ Ω^{1}(1) ⟶^{b̃} ∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ S_{n−1}^{∨} ⊗ 𝒪
↓  f↓  ∥
0 ⟶ S_{k+n−1}^{∨} ⊗ 𝒪 ⟶^{μ^{∨}} S_{k−1}^{∨} ⊗ S_{n}^{∨} ⊗ 𝒪 ⟶^{β^{∨}} ∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ S_{n−1}^{∨} ⊗ 𝒪 ⟶ 0
φ↓  g↓
S_{k−1}^{∨} ⊗ 𝒪(1)  S_{k−1}^{∨} ⊗ 𝒪(1)

↓ 0

(1.30)

where φ := g ∘ μ^{∨} and N = Kerφ ≅ Kerb̃ In fact, if x ∈ Kerb̃ ⇒ b̃(x) = β^{∨}(f(x)) = 0 ⇒ f(x) ∈ Kerβ^{∨} = Imμ^{∨} ⇒ ∃a ∈ S_{k+n−1}^{∨} ⊗ 𝒪 such that f(x) = μ^{∨}(a) ∈ Imf = Kerg g(μ^{∨}(a)) = 0 ⇒ a ∈ Kerφ. then we can define the morphism Φ :Kerb̃ → Kerφ defined by

x ↦ a Claim: Φ is an isomorphism. Obviously Φ is surjective by definition. Moreover, since f and μ^{∨} are injective maps, if x ≠ x′ and Φ(x) ≠ Φ(x′) then f(x) = μ^{∨}(Φ(x)) ≠ μ^{∨}(Φ(x′)) = f(x′) (absurde) then Φ is also injective. Now, by Snake lemma applied to exact diagram (1.30) it sufficient to prove that φ is surjective, but this follows immediately observing that the map

φ^{∨} : S_{k−1} ⊗ 𝒪(−1) ⟶^{g^{∨}} S_{k−1} ⊗ S_{n} ⊗ 𝒪 ⟶^{μ} S_{k+n−1} ⊗ 𝒪  (1.31)

is represented by the k × (k + n) matrix:

(\begin{matrix}s^{n} & s^{n−1}t & ⋯ & st^{n−1} & t^{n} \\ & ... & ... & & ... & ... \\ & & s^{n} & s^{n−1}t & ⋯ & st^{n−1} & t^{n} \end{matrix} )

which has maximal rank.

1.4.4 Remark. The bundle N in the previous lemma is exactly the Schwarzenberger bundle 𝒮^{∗} that we wanted.

Now, by diagram (1.30) and identifications (1.29) the Schwarzenberger bundle 𝒮^{∗} verifies:

0 → 𝒮^{∗} → B ⊗ Ω^{1}(1) → C ⊗ 𝒪 → 0  (1.32)

By using the same technique of [Dio98], we perform its second symmetric and its alternating power and we get

0 → S^{2}𝒮^{∗} →Ã ⟶ B ⊗ C ⊗ Ω^{1}(1) →∧^{2} C ⊗ 𝒪 → 0

↘ ↗

M′

↗ ↘ 0 0

(1.33) where Ã := S^{2}(B ⊗ Ω^{1}(1)) = (S^{2}B ⊗ S^{2}(Ω^{1}(1))) ⊕ (∧^{2} B ⊗ Ω^{2}(2)) and

0 →∧^{2} 𝒮^{∗} →\overline{A} ⟶ B ⊗ C ⊗ Ω^{1}(1) → 𝒪 ⊗ S^{2}C → 0

↘ ↗

M

↗ ↘ 0 0

(1.34)

where \overline{A} :=∧^{2} (B ⊗ Ω^{1}(1)) = (∧^{2} B ⊗ S^{2}(Ω^{1}(1))) ⊕ (S^{2}B ⊗ Ω^{2}(2))

Diagram (1.33) gives the following two exact sequences:

O → H^{0}(M′) → H^{1}(S^{2}𝒮^{∗}) → H^{1}(Ã) → H^{1}(M′) → H^{2}(S^{2}(𝒮^{∗})) → H^{2}(Ã) → ⋯ (1.35)

O → H^{0}(M′) → B ⊗ C ⊗ H^{0}(Ω^{1}(1)) →∧^{2} C → H^{1}(M′) → B ⊗ C ⊗ H^{1}(Ω^{1}(1)) → ⋯

∥ ∥ 0 0

(1.36)

Sequence (1.36) implies: H^{0}(M′) = 0 and H^{1}(M′) ≅∧^{2} C Then, by using the two formulas:

H^{1}(Ã) = (S^{2}B ⊗ H^{1}(S^{2}Ω^{1}(1))) ⊕ (∧^{2} B ⊗ H^{1}(Ω^{2}(2)) = S^{2}B⊗ ∧^{2} S_{n}^{∨}

and: H^{2}(Ã) = (S^{2}B ⊗ H^{2}(S^{2}Ω^{1}(1))) ⊕ (∧^{2} B ⊗ H^{2}(Ω^{2}(2)) = 0 sequence (1.35) becomes:

0 → H^{1}(S^{2}𝒮^{∗}) → H^{1}(Ã) → H^{1}(M′) → H^{2}(S^{2}(𝒮^{∗})) → 0

i.e.

0 → H^{1}(S^{2}𝒮^{∗}) → S^{2}B⊗ ∧^{2} S_{n}^{∨} ⟶^{Φ̃}∧^{2} C → H^{2}(S^{2}𝒮^{∗}) → 0

⟹ H^{2}(S^{2}𝒮^{∗}) ≅ Coker(Φ̃) = (Ker(Φ̃^{∨}))^{∨}

Then:

H^{2}(S^{2}𝒮^{∗})^{∨} = Ker [∧^{2} (S_{k−2} ⊗ S_{n−1}) ⟶^{Φ̃^{∨}} S^{2}(S_{k−1})⊗ ∧^{2} S_{n}]

Moreover, diagram (1.34) gives the following two exact sequences:

O → H^{0}(M) → H^{1}(∧^{2} 𝒮^{∗}) → H^{1}(\overline{A}) → H^{1}(M) → H^{2}(∧^{2} 𝒮^{∗}) → H^{2}(\overline{A}) → ⋯ (1.37) O → H^{0}(M) → B ⊗ C ⊗ H^{0}(Ω^{1}(1)) → S^{2}C⊗ H^{0}(𝒪) → H^{1}(M) → 0 → ⋯ ∥  ∥ 0  S^{2}C (1.38) and,

from sequence (1.38), we get H^{0}(M) = 0 and H^{1}(M) ≃ S^{2}C Then, since :

H^{1}(\overline{A}) = (H^{1}(S^{2}(Ω^{1}(1))⊗ ∧^{2} B) ⊕ (S^{2}B ⊗ H^{1}(Ω^{2}(2))) =∧^{2} B⊗ ∧^{2} S_{n}^{∨}

and H^{2}(\overline{A}) = 0 sequence (1.37) becomes:

\begin{matrix}O → H^{0}(M) → H^{1}(∧^{2} 𝒮^{∗})→ H^{1}(\overline{A}) → H^{1}(M) → H^{2}(∧^{2} 𝒮^{∗}) → 0 \\ ∥ \\ 0 \end{matrix} i.e. 0 → H^{1}(∧^{2}

𝒮^{∗}) →∧^{2} B⊗ ∧^{2} S_{n}^{∨} ⟶^{\overline{Φ}} S^{2}C → H^{2}(∧^{2} 𝒮^{∗}) → 0

⟹ H^{2}(∧^{2} 𝒮^{∗}) ≅ Coker(\overline{Φ}) = (Ker(\overline{Φ}^{∨}))^{∨}

Then we obtain :

(H^{2}(∧^{2} 𝒮^{∗}))^{∨} = Ker [S^{2}(S_{k−2} ⊗ S_{n−1}) ⟶^{\overline{Φ}^{∨}}∧^{2} S_{k−1}⊗ ∧^{2} S_{n}]

Same reasonings of [Dio98] can be repeated in this case and we get that there are two injective SL(2)-equivariant morphisms

ε̃ : ∧^{2} S_{k−3} ⊗ S^{2}S_{n−2} →∧^{2} (S_{k−2} ⊗ S_{n−1})

ε̄ : S^{2}S_{k−3} ⊗ S^{2}S_{n−2} → S^{2}(S_{k−2} ⊗ S_{n−1})

such that: Imε̃ ⊂ Ker Φ̃^{∨} and Imε̄ ⊂ Ker Φ̄^{∨} Moreover, analogously to [OT94](page 202) we get

H^{2}(𝒮^{∗} ⊗ 𝒮^{∗}) ≅ Ker(Φ^{∨})^{∨}

where

Φ^{∨} : S_{k−2}^{⊗2} ⊗ S_{n−1}^{⊗2} → S_{k−1}^{⊗2}⊗ ∧^{2} S_{n}

and there is an isomorphism of SL(2)-representations

ε : S_{k−3}^{∨} ⊗ S_{k−3}^{∨} ⊗ S^{2}S_{n−2}^{∨} → Ker(Φ^{∨})

1.4.5 Theorem. For any special Steiner bundle 𝒮^{∗}

\begin{aligned}& H^{2}(S^{2}𝒮^{∗}) ≃ ∧^{2} (S_{k−3})^{∨} ⊗ S^{2}(S_{n−2})^{∨} \\ and & H^{2}(∧^{2} 𝒮^{∗}) ≃ S^{2}(S_{k−3})^{∨}⊗ ∧^{2} (S_{n−2})^{∨} \end{aligned}

Proof. We can consider the following diagram with exact rows and columns:

0  0  0  0
↓  ↓  ↓  ↓

0 → H^{2}(∧^{2} 𝒮^{∗})^{∨} → S^{2}(S_{k−2} ⊗ S_{n−1}) ⟶^{Φ̄^{∨}}∧^{2} S_{k−1}⊗ ∧^{2} S_{n} → H^{1}(∧^{2} 𝒮^{∗})^{∨} → 0

↓ ↓ ↓ ↓

0 → H^{2}(𝒮^{∗} ⊗ 𝒮^{∗})^{∨} → S_{k−2}^{⊗2} ⊗ S_{n−1}^{⊗2} ⟶^{Φ^{∨}} S_{k−1}^{⊗2}⊗ ∧^{2} S_{n} → H^{1}(𝒮^{∗} ⊗ 𝒮^{∗})^{∨}

↓ ↓ ↓ ↓

0 → H^{2}(S^{2}𝒮^{∗})^{∨} →∧^{2} (S_{k−2} ⊗ S_{n−1}) ⟶^{Φ̃^{∨}} S^{2}S_{k−1}⊗ ∧^{2} S_{n} → H^{1}(S^{2}𝒮^{∗})^{∨} → 0

↓  ↓  ↓  ↓
0  0  0  0

Then, we can consider the following diagram:

0  0  0
↓  ↓  ↓

0 → S^{2}S_{k−3} ⊗ S^{2}S_{n−2} → S_{k−3}^{⊗2} ⊗ S^{2}S_{n−2} →∧^{2} S_{k−3} ⊗ S^{2}S_{n−2} → 0

↓ ε̄ ↓ ε ↓ ε̃

0 → H^{2}(∧^{2} 𝒮^{∗})^{∨} → H^{2}(𝒮^{∗} ⊗ 𝒮^{∗})^{∨} → H^{2}(S^{2}𝒮^{∗})^{∨} → 0

↓ 0 and by the Snake-Lemma there is the exact sequence : 0 → Ker ε̄ → Ker ε → Ker ε̃ → Coker ε̄ → Coker ε → Coker ε̃ → 0

∥  ∥  ∥  ∥
0  0  0  0

⇒ Coker ε̄ = 0 ⇒ ε̄ is an isomorphism ⇒ ε̃ is an isomorphism. Thus:

\begin{aligned}& H^{2}(S^{2}𝒮^{∗})^{∨} ≅ ∧^{2} (S_{k−3}) ⊗ S^{2}(S_{n−2}) \\ and & H^{2}(∧^{2} 𝒮^{∗})^{∨} ≅ S^{2}(S_{k−3})⊗ ∧^{2} (S_{n−2}) \end{aligned}

as we wanted.

By above results we get:

h^{2}(S^{2}𝒮^{∗}) = \binom{k − 2}{2}\binom{n}{2} and h^{2}(∧^{2} 𝒮^{∗}) = \binom{k − 1}{2}\binom{n − 1}{2} (1.39)

Now, by performing second symmetric power of first column of diagram (1.30) we obtain:

0 → S^{2}𝒮^{∗} → S^{2}S_{n+k−1}^{∨} ⊗ 𝒪 → S_{n+k−1}^{∨} ⊗ S_{k−1}^{∨} ⊗ 𝒪(1) →∧^{2} S_{k−1}^{∨} ⊗ 𝒪(2) → 0

then the Euler-Poincaré characteristic of S^{2}𝒮^{∗} is

χ(S^{2}𝒮^{∗}) = \binom{n + k + 1}{2} + \binom{k}{2}\binom{n + 2}{2} − k(n + k)(n + 1)

Moreover, since for Steiner bundles h^{0}(𝒮^{∗}) = 0 (see [AO99] proposition 3.4), then H^{0}(S^{2}𝒮^{∗}) = 0 and after calculation we get 1.4.6 Theorem. Any Schwarzenberger bundle 𝒮^{∗} verifies:

h^{1}(S^{2}𝒮^{∗}) = h^{2}(S^{2}𝒮^{∗}) − χ(S^{2}𝒮^{∗}) = n(n + 2k − 2)

Chapter 2

Minimal resolution of general

stable vector bundles on ℙ^{2}

2.1 Generalities and notations

Let us write a free resolution F_{•} → ℰ → 0 of a vector bundle ℰ in the form

⋯ → ⨁_{q∈ℤ}B_{1,q} ⊗_{ℂ} 𝒪(−q) → ⨁_{q∈ℤ}B_{0,q} ⊗_{ℂ} 𝒪(−q) → ℰ → 0,  (2.1)

where the complex vector spaces B_{p,q} are zero but a finite number. Then, we denote by b_{p,q}(ℰ) := dim Tor_{p}(ℰ, ℂ)_{q} the Betti numbers of ℰ and with b_{p,q}(F_{•}) := dim B_{p,q} the Betti numbers of the resolution (2.1). We recall that, if F_{•} is minimal, then b_{p,q}(F_{•}) = b_{p,q}(ℰ) for all p, q.

A coherent sheaf ℱ on ℙ^{n} is said m-regular if H^{q}(ℙ^{n}, ℱ(m − q)) = 0 for all q > 0.

2.1.1 Theorem (Castelnuovo-Mumford). [Mum66] An m-regular sheaf is (m + 1)-regular.

The regularity of a sheaf is defined as the minimum integer r for which ℱ is r-regular. By the previous theorem and the semicontinuity of cohomology groups, the regularity is a upper semicontinuous function on the related moduli space.

2.1.2 Theorem ([Gre89]). For all coherent sheaves ℱ on ℙ^{n},

reg(ℱ) = max{q − p | b_{p,q}(ℱ) ≠ 0}.  (2.2)

Another parameter that measures the “complexity” of a resolution is its length: we say that 0 → 𝒜_{n} → ⋯ → 𝒜_{0} → ℱ → 0 is a resolution of ℱ of length n. The homological dimension hd(ℱ) of a sheaf ℱ is the length of its minimal free resolution. One fundamental tool to control the homological dimension is the following result that can be regarded as a refined version of the Hilbert syzygy theorem:

2.1.3 Theorem (Horrocks [Hor64]). A torsion-free sheaf ℱ on ℙ^{n} has homological dimension at most n − 1.

2.2 Admissible pairs and resolutions

Let ℰ be a rank r vector bundle on ℙ^{2}. By Horrocks theorem [Hor64], ℰ has homological dimension at most 1, that is, it has a free resolution of the form

0 → ⨁_{i=1}^{k} 𝒪_{ℙ^{2}}(−a_{i}) ⟶^{Φ} ⨁_{j=1}^{r+k} 𝒪_{ℙ^{2}}(−b_{j}) → ℰ → 0.  (2.3)

We do not assume that the resolution (2.3) is minimal. Vector bundles of homological dimension 0, i.e., splitting vector bundles, are a special case in which either k = 0 or the resolution (2.3) is not minimal; however, they are of marginal interest to us since they are not stable.

We suppose that the two sequences a_{i} and b_{i} are indexed in nondecreasing order

\begin{aligned}a_{1} & ≤ a_{2} ≤ ⋯ ≤ a_{k}, \\ b_{1} & ≤ b_{2} ≤ ⋯ ≤ b_{k} ≤ ⋯ ≤ b_{r+k}. \end{aligned}  (2.4)

We call (a, b) = ((a_{1}, …, a_{k}), (b_{1}, …, b_{r+k})) the associated pair to the resolution (2.3). If the resolution (2.3) is minimal, we call (a, b) the pair associated to the bundle ℰ. Notice that the associated pair and the Betti numbers of a resolution encode exactly the same information; in particular max(a_{k} − 1, b_{r+k}) is the Castelnuovo-Mumford regularity of ℰ.

The Chern classes c_{1}, c_{2} of ℰ are determined by a_{i} and b_{j} with the formulas

c_{1} = ∑_{i=1}^{k} a_{i} − ∑_{i=1}^{k+r} b_{i},  (2.5)
2c_{2} − c_{1}^{2} = ∑_{i=1}^{k} a_{i}^{2} − ∑_{i=1}^{k+r} b_{i}^{2}.  (2.6)

2.2.1 Definition. The pair (a, b) is said to be admissible if

a_{i} > b_{2+i} for all i = 1, …, k  (2.7)

For brevity we say that the resolution (2.3) is admissible if the associated pair (a, b) is. 2.2.2 Example. By Euler sequence

0 → 𝒪 → 𝒪(1)^{⊕3} → T_{ℙ^{2}} → 0, the tangent bundle of ℙ^{2} has associated pair ((0), (−1, −1, −1)) which is admissible. 2.2.3 Example. The resolution

0 → 𝒪 → 𝒪(−1) ⊕ 𝒪(1)^{⊕3} → T_{ℙ^{2}} ⊕ 𝒪(−1) → 0,

is not admissible. Remark that T_{ℙ^{2}} ⊕ 𝒪(−1) is not a stable bundle.

More generally, we can consider the associated pair (a, b) to any vector bundle of homological dimension 1 on ℙ^{n} with n ≥ 2. In that case, we say that (a, b) is admissible if a_{i} > b_{n+i} for i = 1, …, k, as in [BS92]. 2.2.4 Proposition. If r ≥ 2, resolution (2.3) is minimal if and only if it is admissible and every constant entry of the matrix (ϕ_{i,j}) is zero. Proof. Obviously, if the pair is admissible and every constant entry of the matrix (ϕ_{i,j}) is zero then (2.3) is minimal. Conversely, for r = 2 the statement was proved by Bohnhorst and Spindler ([BS92] proposition 2.3). Now suppose that r > 2 and (2.3) is minimal. Since ℰ(b_{r+k}) is globally generated, Bertini’s theorem ensures that a generic map f: 𝒪(−b_{r+k}) → ℰ is injective. Then in the following commutative diagram columns and rows are exact and ℰ′′ is locally free:

\begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & 0 & ⟶ & 𝒪(−b_{r+k}) & ⟶^{Id} & 𝒪(−b_{r+k}) & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & ⊕_{i=1}^{k}𝒪(−a_{i}) & ⟶ & ⊕_{i=1}^{r+k}𝒪(−b_{i}) & ⟶ & ℰ & ⟶ & 0 \\ & & ↓_{Id} & & ↓ & & ↓ \\ 0 & ⟶ & ⊕_{i=1}^{k}𝒪(−a_{i}) & ⟶ & ⊕_{i=1}^{r+k−1}𝒪(−b_{i}) & ⟶ & ℰ′′ & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned} (2.8)

The minimality of the middle row yields the minimality of the last row. Using induction on r, we may assume that the last row is admissible. Then the middle row is also admissible.

2.2.5 Theorem. Let r = 2 and suppose that the resolution (2.3) is admissible. Let c_{1} = ∑ a_{i} − ∑ b_{j} be the first Chern class and μ be the slope of ℰ. Then ℰ is semistable (respectively stable) if and only if

b_{1} ≥ −μ  (resp. b_{1} > −μ).  (2.9)

Proof. This is theorem 2.7 of [BS92] in the case of rank-2 vector bundles on ℙ^{2}.

For the case of higher rank, we have no chances to extend the above arithmetical characterization since stable and unstable vector bundles may have the same associated pair.

2.2.6 Proposition. If resolution (2.3) is admissible (in particular if it is minimal) and ℰ is semistable (resp. stable) then the associated pair (a, b) verifies b_{1} ≥ −μ (resp. b_{1} > −μ).

Proof. If ℰ is semistable (resp. stable), then

H^{0}(ℰ(m)) = 0 ∀m < −μ(ℰ) (resp. ∀m ≤ −μ(ℰ))  (2.10)

but, from the exact sequence

0 → ⨁_{i=1}^{k} 𝒪(−a_{i} + b_{1}) → ⨁_{j=1}^{r+k} 𝒪(−b_{j} + b_{1}) → ℰ(b_{1}) → 0,  (2.11)

we have H^{0}(ℰ(b_{1})) ≠ 0 then b_{1} > −μ(ℰ) (resp. b_{1} ≥ −μ(ℰ)).

2.2.7 Lemma. If resolution (2.3) is admissible and ℰ is semistable, then

b_{r+k} < a_{k}.

Proof. If b_{r+k} ≥ a_{k} then ℰ split as ℰ = 𝒪(−b_{k+r}) ⊕ ℰ′′ and by admissibility

\begin{aligned}∑_{i=1}^{k} a_{i} − ∑_{j=1}^{r+k−1} b_{j} & = − ∑_{i=1}^{2} b_{i} + ∑_{i=1}^{k}(a_{i} − b_{n+i}) −∑_{i=k+3}^{r+k−1} b_{i} \\ & ≥ −2b_{r+k} + k − (r − 3)b_{r+k} \\ & > (1 − r)b_{r+k} \end{aligned}  (2.12)

then we have μ(ℰ′′) > −b_{r+k} = μ(𝒪(−b_{r+k})) which contradicts the semistability of ℰ.

We denote by ℑ the set of all admissible pairs (a, b) associated to rank rvector bundles on ℙ^{2} with Chern classes c_{1}, c_{2} satisfying the conditions b_{1} > −μ = (∑ a_{i} − ∑ b_{j})/r and b_{r+k} < a_{k}. Proposition 2.2.6 shows that the set ℑ contains the set of all possible associated pair to a stable vector bundle in 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) and coincides exactly with it for r = 2. Then

𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) = ∐_{(a,b)∈ℑ}𝔐(a, b)  (2.13)

where 𝔐(a, b) will be the subset (possibly empty) of 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) of vector bundles with associated pair (a, b). 2.2.8 Remark. Let us fix a pair (a, b) ∈ ℑ and define F_{0} = ⊕_{j=1}^{k+r}𝒪(−b_{j}), F_{1} = ⊕_{i=1}^{k}𝒪(−a_{i}). We consider the linear subspace V of Hom(F_{1}, F_{0}) consisting of those homomorphisms ϕ : F_{1} → F_{0} such that every constant entry of the matrix (ϕ_{i,j}) is zero and The set X = {ϕ ∈ V |ϕ is injective, cokerϕ is locally free} is a Zariski open subspace of V. The cokernel of the universal homomorphism

Φ : pr_{2}^{∗}F_{1} → pr_{2}^{∗}F_{0} on X × ℙ^{2}

induces a morphism

τ : X → 𝔐

By definition 𝔐(a, b) = τ(X). Especially 𝔐(a, b) is a constructible subset of 𝔐.

The following result was stated and proved by Bohnhorst and Spindler [BS92] for rank-n vector bundles on ℙ^{n} with homological dimension 1, but their proof works on ℙ^{2} for vector bundles of any rank without modifications. 2.2.9 Theorem. For all (a, b) ∈ ℑ, the closed set \overline{𝔐(a, b)} is an irreducible algebraic subset of 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) of dimension:

dim \overline{𝔐(a, b)} = dim Hom(F_{1}, F_{0}) + dim Hom(F_{0}, F_{1})
− dim End(F_{1}) − dim End(F_{0}) + 1 − #{(i, j) : a_{i} = b_{j}},

(2.14)

where F_{0} = ⊕_{j=1}^{k+r}𝒪(−b_{j}), F_{1} = ⊕_{i=1}^{k}𝒪(−a_{i}).

We introduce a partial order on ℑ, namely

(ã, b̃) ⪯ (a, b)  ⟺  ∃ α_{1}, ⋯, α_{t} ∈ ℤ such that up to order
ã = (a_{1}, ⋯, a_{k}, α_{1}, ⋯, α_{t})

and b̃ = (b_{1}, ⋯, b_{k+n}, α_{1}, ⋯, α_{t})

2.2.10 Lemma. If (a, b), (ã, b̃) ∈ ℑ then

(ã, b̃) ⪯ (a, b) ⟹ 𝔐^{1}(ã, b̃) ⊂ \overline{𝔐^{1}(a, b)}

In the following lemma we find an upper bound on the regularity of semistable vector bundles on ℙ^{2} of rank 2. A lower bound is given in corollary 2.2.24.

2.2.11 Theorem. A normalized semistable rank 2 bundle ℰ on ℙ^{2} is c_{2}-regular.

Proof. For brevity’s sake, we set ξ_{i} := a_{i} − b_{i+2} and t_{i} := b_{i+2} − b_{2}. Obviously, ξ_{i} ≥ 1 and t_{i} ≥ 0. We rewrite (2.5) as

∑_{i=1}^{k} ξ_{i} = ∑_{i=1}^{2} b_{i} + c_{1}  (2.15)

and by (2.6), using inequalities (2.4), (2.7), (2.9) we get

\begin{aligned}∑_{i=1}^{2} b_{i}^{2} + 2c_{2} − c_{1}^{2} & = ∑_{i=1}^{k}(a_{i}^{2} − b_{i+2}^{2}) = ∑_{i=1}^{k} ξ_{i}(2b_{2} + 2t_{i} + ξ_{i}) \\ & ≥ 2b_{2} ∑_{i=1}^{k} ξ_{i} + ∑_{i=1}^{k}(2t_{i} + ξ_{i}) \\ & = (2b_{2} + 1)(∑_{i=1}^{2} b_{i} + c_{1}) + 2 ∑_{i=1}^{k} t_{i}. \end{aligned}  (2.16)

If we suppose that b_{2} + ∑_{i=1}^{k} t_{i} ≥ c_{2} + 1, then

∑_{i=1}^{2} b_{i}^{2} + 2b_{2} − c_{1}^{2} − (2b_{2} + 1)(∑_{i=1}^{2} b_{i} + c_{1}) ≥ 2.

Since the left side is non increasing with respect to b_{1}, we may restrict ourselves to the case b_{1} = −c_{1}. But ∑_{i=2}^{2} b_{i}^{2} + 2b_{2} − (2b_{2} + 1)(∑_{i=2}^{2} b_{i}) ≥ 2 it easily seen to be impossible. Then ∑_{i=1}^{k} t_{i} must be at most c_{2} − b_{2} and in particular

b_{k+2} = b_{2} + t_{k} ≤ b_{2} + ∑ t_{i} ≤ c_{2}.  (2.17)

Now, we must show that a_{k} ≤ c_{2} + 1. We rewrite (2.5) as ∑_{i=1}^{k−1} ξ_{i} = ∑_{i=1}^{2} b_{i} + b_{k+2} − a_{k} + c_{1} and by (2.6)

∑_{i=1}^{2} +b_{i}^{2} + b_{k+2}^{2} − a_{k}^{2} + 2c_{2} − c_{1}^{2} = ∑_{i=1}^{k−1}(a_{i}^{2} − b_{i+2}^{2}) = ∑_{i=1}^{k−1} ξ_{i}(2b_{2} + 2t_{i} + ξ_{i}) ≥

≥ 2b_{2} ∑_{i=1}^{k−1} ξ_{i} + ∑_{i=1}^{k−1} ξ_{i} ≥ (2b_{2} + 1)(∑_{i=1}^{2} b_{i} + b_{k+2} − a_{k} + c_{1}) (2.18)

that can be put in the form

\begin{matrix}∑_{i=1}^{2} b_{i}^{2} − (2b_{2} + 1)(∑_{i=1}^{2} b_{i} + c_{1}) + 2c_{2} − c_{1}^{2} ≥ \\ ≥ (a_{k} − b_{k+2})(a_{k} + b_{k+2} − 2b_{2} − 1). \end{matrix}  (2.19)

Suppose that a_{k} ≥ c_{2} + 2. By (2.17) we have a_{k} − b_{k+2} ≥ c_{2} + 2 − c_{2} = 2 and we observe also that a_{k} + b_{k+2} − 2b_{2} − 1 ≥ c_{2} − b_{2} + 1. Substituting and simplifying, (2.19) becomes

∑_{i=1}^{2} b_{i}^{2} − (2b_{2} + 1)(∑_{i=1}^{2} b_{i} + c_{1}) − c_{1}^{2} ≥ 2 + c_{1}^{2}  (2.20)

As before, we can restrict ourselves to the case b_{1} = −c_{1} obtaining

∑_{i=2}^{2} b_{i}^{2} − (2b_{2} + 1) ∑_{i=2}^{2} b_{i} ≥ 2

that do not have solution for b_{i} positive. Then a_{k} ≤ c_{2} + 1.

2.2.12 Remark. The above theorem is sharp in the sense that the admissible pairs ((c_{2} + 1), (0, 1, c_{2})) and ((c_{2} + 1), (1, 1, c_{2})) are associated to rank 2 semistable bundles of Chern classes c_{1}, c_{2} that have regularity c_{2}. 2.2.13 Remark. It is also possible to prove that a semistable rank 2 bundle on ℙ^{2} is c_{2} regular if c_{1} = 0 and (c_{2} + 1)-regular if c_{1} = −1 using the bounds on dimension of cohomology groups proved by Elencwajg and Forster (proposition 2.18 [EF80]) and Grauert-Mülich theorem.

2.2.14 Remark. From (2.15) and the thesis of previous theorem, the value k in (2.3) is bounded by:

k ≤ ∑_{i=1}^{k} ξ_{i} = ∑_{i=1}^{2} b_{i} + c_{1} ≤ 2c_{2} + c_{1}  (2.21)

Hence, for fixed Chern classes c_{1}, c_{2}, there are only a finite number of admissible pair of rank 2 vector bundles and we can write an algorithm to enumerate such pairs restricting the search to a finite domain and to list them with the help of a computer. 2.2.15 Example. Associated admissible pairs for some Chern classes

1. Each bundle ℰ ∈ 𝔐_{ℙ^{2}}(2; −1, 2) has a resolution of the form:

0 → 𝒪(−3) → 𝒪(−1) ⊕ 𝒪(−2)^{⊕2} → ℰ → 0

and ((3), (1, 1, 2)) is the associated pair. This is the only admissible pair in this case. In the section 2.3 we will describe explicitly this moduli space.

2. Admissible pairs associated to ℰ ∈ 𝔐_{ℙ^{2}}(2; −1, 5):

           a_i           b_j            Codim.
       ========================================
         (3 4)         (2 2 2 2)       |  0
         (3 3 4)       (2 2 2 2 3)     |  2
         (4 4)         (1 2 3 3)       |  2
         (3 5)         (1 2 2 4)       |  4
         (6)           (1 1 5)         |  6

3. Admissible pairs associated to 𝔐_{ℙ^{2}}(2; −1, 9)

  a_i           b_j            Codim.
========================================
(4 4 4 4 4)   (3 3 3 3 3 3 3) |  0
(4 4 5)       (2 3 3 3 3)     |  1
(4 4 4 5)     (2 3 3 3 3 4)   |  4
(5 5)         (2 2 3 4)       |  4
(5 5 5)       (1 3 4 4 4)     |  6
(4 5 5)       (2 2 3 4 4)     |  6
(4 4 6)       (2 2 3 3 5)     |  6
(4 7)         (2 2 2 6)       |  8
(3 6)         (2 2 2 4)       |  8
(6)           (1 3 3)         |  8
(3 5 6)       (2 2 2 4 5)     |  9
(3 4 7)       (2 2 2 3 6)     |  9
(5 6)         (1 3 3 5)       |  9
(4 6)         (1 3 3 4)       |  9
(3 3 8)       (2 2 2 2 7)     |  10
(4 4 7)       (1 3 3 3 6)     |  10
(4 5 6)       (1 3 3 4 5)     |  10
(6 6)         (1 2 5 5)       |  10
(4 8)         (1 2 3 7)       |  11
(5 7)         (1 2 4 6)       |  11
(3 9)         (1 2 2 8)       |  12
(10)          (1 1 9)         |  14

2.2.1 Natural pairs and general vector bundles

We say that (a, b) = ((a_{1}, …, a_{k}), (b_{1}, …, b_{r+k})) is a natural pair if it is admissible and

b_{r+k} < a_{1}, a_{k} ≤ b_{1} + 2. (2.22)

The above inequalities imply a_{i} ≠ b_{j} for all i and j.

We observe that natural pairs are parametrized by three integers s, k, α such that

k ≥ 1 and − k + 1 ≤ α ≤ k + r (2.23)

as follows: the pair (a, b)_{s,k,α} corresponding to the triple (s, k, α) is the pair associated to a resolution of the form

0 → 𝒪(−s − 1)^{k} → 𝒪(−s)^{α} ⊕ 𝒪(−s + 1)^{r+k−α} → ℰ → 0  (2.24)

if α ≥ 0, or of the form

0 → 𝒪(−s − 1)^{k+α} ⊕ 𝒪(−s)^{−α} → 𝒪(−s + 1)^{r+k} → ℰ → 0  (2.25)

if α < 0. We have excluded the case α = −k so that s is the regularity of the pair, i.e. s = max(a_{k} − 1, b_{r+k}).

Through this section we are going to show that resolutions of general vector bundles have natural pairs. 2.2.16 Theorem. If r = 2, one has codim \overline{𝔐(a, b)} = 0 if and only if (a, b) is a natural pair.

As a remarkable consequence we will derive a quite simple proof of the irreducibility of moduli spaces of stable vector bundles on ℙ^{2} and we will compute the regularity and the cohomology of their general elements.

We recall that, since dim Ext^{2}(ℱ, ℱ) = 0 for any stable vector bundle ℱ on ℙ^{2}, the relative moduli 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) space is smooth of dimension

dim Ext^{1}(ℱ, ℱ) = 2rc_{2} − (r − 1)c_{1}^{2} − r^{2} + 1.  (2.26)

Let us consider the function A(t) := h^{2}(𝒪(t)) and the finite differences of first and second order (Δ_{u}A)(t) := A(t + u) − A(t) and (Δ_{v}Δ_{u}A)(t) := (Δ_{u}A)(t + v) − (Δ_{u}A)(t).

2.2.17 Definition. The pair (a, b) is said to be strongly admissible if

a_{i} > b_{r+i} for all i = 1, …, k  (2.27)

For r = 2 strongly admissibility and admissibilility coincide. 2.2.18 Example. The map Φ in (2.3) can be expressed by a (r + k) × k matrix of forms (ϕ_{i,j}) of degree deg(ϕ_{i,j}) = (b_{i} − a_{j}). If (a, b) is a strongly admissible pair and ω_{0} … ω_{r} are linear forms in general position on ℙ^{n}, then the (r + k) × k matrix

(ϕ_{ij}) := [\begin{matrix}ω_{0}^{a_{1}−b_{1}} & … & 0 \\ ⋮ & ... & ⋮ \\ ⋮ & & ω_{0}^{a_{k}−b_{k}} \\ ω_{r}^{a_{1}−b_{r+1}} & & ⋮ \\ ⋮ & ... & ⋮ \\ 0 & … & ω_{r}^{a_{k}−b_{r+k}} \end{matrix} ]  (2.28)

defines a minimal free resolution and ℰ := coker Φ is a vector bundle with associated pair (a, b).

2.2.19 Lemma. If a strongly admissible pair (a, b) is associated to a stable vector bundle ℰ on ℙ^{2}, Then

\begin{aligned}codim \overline{𝔐(a, b)} & = ∑_{i=1}^{r} h^{1}(ℰ(b_{i})) + #\{(i, j) : a_{i} = b_{j}\} \\ & + ∑_{i,j=1}^{r}(Δ_{b_{i+r}−a_{i}}Δ_{b_{j+r}−a_{j}}A)(a_{i} − b_{j+r}) \end{aligned}  (2.29)

Proof. Let

0 → F_{1} → F_{0} → ℰ → 0  (2.30)

be the minimal resolution of ℰ where

F_{0} = ⨁_{j=1}^{k+r} 𝒪(−b_{j}),  F_{1} = ⨁_{i=1}^{k} 𝒪(−a_{i}).  (2.31)

The stability of ℰ ensures the vanishing dim(Ext^{2}(ℰ, ℰ)) = h^{2}(ℰ^{∗} ⊗ ℰ) = 0 so that h^{2}(F_{0}^{∗} ⊗ ℰ) = h^{2}(F_{1}^{∗} ⊗ ℰ). Then from (2.30) we easily find the following data:

\begin{aligned}h^{0}(F_{0}^{∗} ⊗ ℰ) & = h^{0}(F_{0}^{∗} ⊗ F_{0}) − h^{0}(F_{0}^{∗} ⊗ F_{1}), \\ h^{0}(F_{1}^{∗} ⊗ ℰ) & = h^{0}(F_{1}^{∗} ⊗ F_{0}) − h^{0}(F_{1}^{∗} ⊗ F_{1}), \\ dim(Ext^{1}(ℰ, ℰ)) & = h^{1}(ℰ^{∗} ⊗ ℰ) = \\ & = h^{1}(F_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ)+ \\ & + h^{0}(F_{1}^{∗} ⊗ ℰ) − h^{0}(F_{0}^{∗} ⊗ ℰ) + 1 \end{aligned}  (2.32)

and from (2.14) we have

codim \overline{𝔐(a, b)} = dim(Ext^{1}(ℰ, ℰ)) − dim \overline{𝔐(a, b)} =
= h^{1}(F_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ) + #{(i, j) : a_{i} = b_{j}}

(2.33)

Now, by splitting F_{0} as 𝒪(−b_{1}) ⊕ ⋯ ⊕ 𝒪(−b_{r−1}) ⊕ F̃_{0} with F̃_{0} := ⊕_{i=r+1}^{k+r}𝒪(−b_{i}), the above formula becomes

\begin{aligned}codim \overline{𝔐(a, b)} & = ∑_{i=1}^{r} h^{1}(ℰ(b_{i})) + #\{(i, j) : a_{i} = b_{j}\} \\ & + h^{1}(F̃_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ). \end{aligned}  (2.34)

Since h^{2}(F̃_{0}^{∗} ⊗ F_{0}) = h^{2}(F̃_{0}^{∗} ⊗ F̃_{0}) and h^{2}(F_{1}^{∗} ⊗ F_{0}) = h^{2}(F_{1}^{∗} ⊗ F̃_{0}) so

\begin{aligned}h^{1}(F̃_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ) & = h^{2}(F̃_{0}^{∗} ⊗ F_{1}) − h^{2}(F̃_{0}^{∗} ⊗ F_{0}) \\ & − h^{2}(F_{1}^{∗} ⊗ F_{1}) + h^{2}(F_{1}^{∗} ⊗ F_{0}) = \\ & = h^{2}(F̃_{0}^{∗} ⊗ F_{1}) − h^{2}(F̃_{0}^{∗} ⊗ F̃_{0}) \\ & − h^{2}(F_{1}^{∗} ⊗ F_{1}) + h^{2}(F_{1}^{∗} ⊗ F̃_{0}) = \\ & = ∑_{i,j=1}^{r} [h^{2}(𝒪(b_{i+r} − a_{j})) − h^{2}(𝒪(b_{i+r} − b_{j+r})) \\ & − h^{2}(𝒪(a_{i} − a_{j})) + h^{2}(𝒪(a_{i} − b_{j+r}))] \end{aligned} ; (2.35)

Finally the equation (2.34) becomes

\begin{aligned}codim \overline{𝔐(a, b)} & = ∑_{i=1}^{r} h^{1}(ℰ(b_{i})) + #\{(i, j) : a_{i} = b_{j}\} \\ & + ∑_{i,j=1}^{r}(Δ_{b_{i+r}−a_{i}}Δ_{b_{j+r}−a_{j}}A)(a_{i} − b_{j+r}) \end{aligned}  (2.36)

Proof of theorem 2.2.16. It can be verified by direct computation from theorem 2.2.9 that, if ℰ has natural pair, then the codimension of \overline{𝔐(a, b)} is zero.

Conversely, let u, v be two non-negative integers. Since all finite difference (Δ_{u}A)(t) := A(t + u) − A(t) are non decreasing functions of t, then

(Δ_{v}Δ_{u}A)(t) ≥ 0  (2.37)

and by the previous lemma

codim \overline{𝔐(a, b)} ≥ ∑_{i=1}^{2} h^{1}(ℰ(b_{i})) + #\{(i, j) : a_{i} = b_{j}\}.  (2.38)

If codim \overline{𝔐(a, b)} = 0, we have a_{k} ≤ b_{1} + 2 and #\{(i, j) : a_{i} = b_{j}\} = 0, since h^{1}(ℰ(b_{1})) = 0 implies h^{2}(F_{1}(b_{1})) = 0. This forces (a, b) to be a natural pair.

2.2.20 Proposition. Let 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) be nonempty and

s := max{ρ ∈ ℤ : 2ρ^{2} + 2c_{1}ρ − 2ρ ≤ 2c_{2} − c_{1}^{2} + c_{1} − 1},  (2.39)

or, equivalently,

s := min{ρ ∈ ℤ : 2ρ^{2} + 2c_{1}ρ + 2ρ ≥ 2c_{2} − c_{1}^{2} − c_{1}}.  (2.39bis)

If α and k are defined by

\begin{aligned}α & := 2c_{2} − c_{1}^{2} + 2 − 2s^{2} − 2c_{1}s, \\ k & := (2s + c_{1} − 2 + |α|)/2, \end{aligned}  (2.40)

then (a, b)_{s,k,α} is the only natural pair of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}).

Proof. This is a verification; we outline the main steps of the computation. In the first place, one must ensure that the natural pair (a, b)_{s,k,α} is actually associated to vector bundles in 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}). This amount to show that, according with equations (2.5) and (2.6), the pair (a, b)_{s,k,α} has the appropriate Chern classes and that conditions (2.23) hold.

From theorem 2.2.16, any pair (a, b) such that dim \overline{𝔐(a, b)} = 0 is a natural pair of the form (a, b)_{s,k,α}. From resolutions (2.24) and (2.25) with r = 2 we find that α, k must satisfy (2.40). Then it remains to verify that s is uniquely determined from c_{1}, c_{2} and satisfy (2.39). By substitution, the inequalities −k < α ≤ k + 2 turn into

2s^{2} + 2c_{1}s − c_{1} − 2s + 1 ≤ 2c_{2} − c_{1}^{2} ≤ 2s^{2} + 2c_{1}s + c_{1} + 2s.  (2.41)

Since the intervals [2s^{2} +2c_{1}s−c_{1} −2s+1, 2s^{2} +2c_{1}s+c_{1} +2s] are disjoint for s varying in ℤ, then equations (2.39) and (2.39bis) give the only suitable value for s.

2.2.21 Theorem. Moduli spaces of stable rank 2 vector bundles on ℙ^{2} are irreducible.

Proof. Moduli spaces of stable rank 2 vector bundles on ℙ^{2} are smooth. By the previous proposition they can have only one connected component.

2.2.22 Corollary. The general element of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) has natural cohomology.

The above corollary justify the terminology “natural pair”. A different proof for it, working also for higher rank, can be found in [HL93], by using sophisticated techniques of stacks theory. 2.2.23 Example. If r = 2 and c_{1} = −1, in the following table we put the natural pairs for c_{2} ≤ 15.

         c2    a_i              b_j
  ============================================
         1  |  (2)             (1 1 1)
         2  |  (3)             (1 1 2)
         3  |  (3 3)           (1 2 2 2)
         4  |  (3 3 3)         (2 2 2 2 2)
         5  |  (3 4)           (2 2 2 2)
         6  |  (4 4)           (2 2 2 3)
         7  |  (4 4 4)         (2 2 3 3 3)
         8  |  (4 4 4 4)       (2 3 3 3 3 3)
         9  |  (4 4 4 4 4)     (3 3 3 3 3 3 3)
        10  |  (4 4 4 5)       (3 3 3 3 3 3)
        11  |  (4 5 5)         (3 3 3 3 3)
        12  |  (5 5 5)         (3 3 3 3 4)
        13  |  (5 5 5 5)       (3 3 3 4 4 4)
        14  |  (5 5 5 5 5)     (3 3 4 4 4 4 4)
        15  |  (5 5 5 5 5 5)   (3 4 4 4 4 4 4 4)

Now, we are going to give some inequalities on the regularity and the cohomology of stable vector bundles using proposition 2.2.20. In particular, for rank 2 vector bundles, the next two corollaries give respectively a refined version of corollary 5.4 in [Bru80] and proposition 7.1 in [Har78].

2.2.24 Corollary. A general vector bundle ℰ in 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) has regularity reg(ℰ) = s, where s is given by (2.39).

2.2.25 Corollary. Let [ℰ] be a vector bundle in 𝔐 = 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) and s defined by (2.39). Then H^{0}(ℰ(t)) ≠ 0 if

t ≥ s  when 2s^{2} + 2c_{1}s + 2s = 2c_{2} − c_{1}^{2} − c_{1},
t ≥ s − 1 otherwise.

The above inequality is sharp, in the sense that, if ℰ is general, it gives a necessary and sufficient condition.

Proof. Let ((a_{1}, …, a_{k}), (b_{1}, …, b_{k+2})) be the admissible pair associated to a vector bundle ℰ in 𝔐. Then one has H^{0}(ℰ(t)) ≠ 0 if and only if t − b_{1} ≥ 0. By semicontinuity of cohomology groups and theorem 2.2.21, it is enough to restrict ourselves to the case where ℰ is general. So, by (2.24) and (2.25) one has H^{0}(ℰ(t)) ≠ 0 if and only if

t ≥ s  when α = k + 2
t ≥ s − 1 otherwise

and the condition α = k + 2 is equivalent to 2s^{2} + 2c_{1}s + 2s = 2c_{2} − c_{1}^{2} − c_{1} by (2.40).

2.2.26 Remark. It is known that the moduli space 𝔐 = 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) is irreducible for any rank r (various proofs of this result can be found in [Bar77a], [Ell83]), [HL93], [Pot79], [Mar78]). If (a, b) is a strongly admissible natural pair and there exists a rank r -vector bundle ℰ ∈ 𝔐(a, b) then \overline{𝔐}(a, b) = 0 hence, by irreducibility of 𝔐, as in the proof of proposition 2.2.20, we get that (a, b) is uniquely determinated, i.e. (a, b) = (a, b)_{s,k,α} where

s := max{ρ ∈ ℤ : rρ^{2} + 2c_{1}ρ − rρ ≤ 2c_{2} − c_{1}^{2} + c_{1} − 1},  (2.42)

and

\begin{aligned}α & := 2c_{2} − c_{1}^{2} + r − rs^{2} − 2c_{1}s, \\ k & := (rs + c_{1} − r + |α|)/2. \end{aligned}  (2.43)

Some remarks about strong admissibility

Let r = 3, a = (2, 2, 4, 4, 4) and b = (1, 1, 1, 1, 3, 3, 3, 3). Coherent sheaves on ℙ^{2} of type (a, b) form an irreducible family and the general sheaf in the family is a vector bundle because (a, b) is admissible. Moreover, the family contains sheaves of the form I_{P} ⊕ I_{Q} ⊕ I_{Z} where P, Q are point in ℙ^{2} and Z is a general set of six point of ℙ^{2}. These sheaves are semistable because they are ideal sheaves and because direct sums of semistable sheaves of same slope are semistable [OSS80]. By openness of semistability, the general sheaf of type (a, b) is a semistable vector bundle, with (a,b) admissible but not strongly admissible (by Hoppe criterion it is not stable).

So far we did not succeed in finding stable vector bundles with admissible pairs not strongly admissible. This fact and the above remark suggest to formulate following conjecture: ”If ℰ is a stable rank r-vector bundle on the projective space then its minimal resolution is strongly admissible”.

A proof of this conjecture would give as corollary also the irreducibility of 𝔐_{ℙ^{2}}(r, c_{1}, c_{2}) by applyng the same arguments of the proof of 2.2.20.

2.3 The sheaves ℱ(r, f_{1}, f_{2})

Now we turn our attention on a specific example. We want to give an explicit description of the moduli space of rank-2 stable vector bundles on ℙ^{2} with Chern classes c_{1} = −1, c_{2} = 2 in the spirit of the previous sections. With this aim we are going to introduce the class of sheaves ℱ(r, f_{1}, f_{2}). 2.3.1 Lemma. Any stable rank-2 vector bundle ℰ on ℙ^{2} with Chern classes c_{1} = −1, c_{2} = 2 is presented by a minimal resolution of the form

0 → 𝒪(−3) → 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} → ℰ → 0.  (2.44)

Proof. After few tries, with the aid of theorem 2.2.11, one sees that ((3), (1, 1, 2)) is the only admissible pair with the prescribed rank and Chern classes.

Let U be a complex vector space of dimension 3 and let ℙ^{2} = ℙ(U) be the projective space equipped with the SL_{ℂ}(3) = SL_{ℂ}(U) action. An element (r, f_{1}, f_{2}) ∈ H^{0}(𝒪(1) ⊕ 𝒪(2)^{⊕2}) defines a morphism 𝒪(−3) → 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} of sheaves on ℙ^{2}. We suppose r ≠ 0 to avoid uninteresting pathologies and we denote by ℱ(r, f_{1}, f_{2}) the sheaf defined by the following exact sequence:

0 → 𝒪(−3) →^{(r,f_{1},f_{2})} 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} → ℱ(r, f_{1}, f_{2}) → 0.  (2.45)

The rank 2 sheaf ℱ(r, f_{1}, f_{2}) has Chern classes c_{1}(ℱ) = −1 and c_{2}(ℱ) = 2. In this section we are going to study some properties of the sheaves ℱ(r, f_{1}, f_{2}).

The matrix

B = [\begin{matrix}λ & 0 & 0 \\ ω_{1} & α_{1,1} & α_{1,2} \\ ω_{2} & α_{2,1} & α_{2,2} \end{matrix}] ,  (2.46)

where λ, α_{i,j} are constants and ω_{1}, ω_{2} linear forms on ℙ^{2}, defines an isomorphism of exact sequences (of sheaves)

0 ⟶ 𝒪(−3) ⟶_{(r,f_{1},f_{2})} 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} ⟶ ℱ(r, f_{1}, f_{2}) ⟶ 0
↓^{Id} ↓^{B} ↓^{ϕ_{B}}
0 ⟶ 𝒪(−3) ⟶_{(s,g_{1},g_{2})} 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} ⟶ ℱ(s, g_{1}, g_{2}) ⟶ 0

(2.47)

and, in particular, a morphism ϕ_{B} between ℱ(r, f_{1}, f_{2}) and ℱ(s, g_{1}, g_{2}) where s, g_{1}, g_{2} satisfy:

t(s, g_{1}, g_{2}) = B.^{t}(r, f_{1}, f_{2}).  (2.48)

Conversely, every morphism between sheaves ϕ: ℱ(r, f_{1}, f_{2}) → ℱ(s, g_{1}, g_{2}) is induced by a morphism of corresponding exact sequences i.e. ϕ = ϕ_{B} for a suitable matrix B.

2.3.2 Lemma. Two sheaves ℱ(r, f_{1}, f_{2}), ℱ(s, g_{1}, g_{2}) are isomorphic if and only if (r, f_{1}, f_{2}) and (s, g_{1}, g_{2}) satisfy

\begin{aligned}s & = λr \\ g_{1} & = ω_{1}r + α_{1,1}f_{1} + α_{1,2}f_{2} \\ g_{2} & = ω_{2}r + α_{2,1}f_{1} + α_{2,2}f_{2} \end{aligned}  (2.49)

where λ, α_{i,j} are constants and ω_{i} linear forms on ℙ^{2} such that

λ det(α_{i,j}) ≠ 0.  (2.50)

Proof. The map ϕ_{B} is an isomorphism if and only if B has maximal rank.

Let ℱ be the sheaf ℱ = ℱ(r, f_{1}, f_{2}). We denote by V (f) the zeros set of a form f on ℙ^{2}, by Σ(r) the net of degenerated conics containing V (r) and by Σ(r, f_{1}, f_{2}) the linear system of conics of the form V (ωr + α_{1}f_{1} + α_{2}f_{2}) where ω is a linear form and α_{1}, α_{2} are scalars. By the previous lemma, V (r) and Σ(r, f_{1}, f_{2}) are intrinsically determined by ℱ. Note that the pencil generated by V (f_{1}) and V (f_{2}) is not uniquely individuated, but its dimension is an invariant canonically associated to ℱ. 2.3.3 Lemma. Let us fix a system of homogeneous coordinates x_{0}, x_{1}, x_{2} on ℙ^{2}. Up to the action of SL_{ℂ}(3) every sheaf ℱ(r, f_{1}, f_{2}) (with r not identically zero) is isomorphic to one of the following sheaves:

(i) ℱ(x_{0}, x_{1}^{2}, x_{2}^{2}),

(ii) ℱ(x_{0}, x_{1}x_{2}, x_{2}^{2}),

(iii) ℱ(x_{0}, x_{1}x_{2}, x_{1}x_{2}),

(iv) ℱ(x_{0}, x_{1}^{2}, x_{1}^{2}),

(v) ℱ(x_{0}, x_{1}x_{2}, 0),

(vi) ℱ(x_{0}, x_{1}^{2}, 0),

(vii) ℱ(x_{0}, 0, 0). Proof. We can choose an homogeneous coordinate system such that r = x_{0}. The linear system Σ(r, f_{1}, f_{2}) is given by the conics V (β_{0}x_{0}^{2}+β_{1}x_{0}x_{1}+β_{2}x_{0}x_{2}+α_{1}f_{1}+ α_{2}f_{2}) depending on β and α. Without loss of generality, we may assume that the monomials x_{0}^{2}, x_{0}x_{1}, x_{0}x_{2} do not appear in f_{1} e f_{2} i.e. V (f_{i}) are conics of rank 2 with a singularity in (1, 0, 0). We are going to study the sheaf ℱ(r, f_{1}, f_{2}) in dependence of the dimension d(r, f_{1}, f_{2}) of Σ(r, f_{1}, f_{2}).

If d(r, f_{1}, f_{2}) = 5, the dimension of the linear system is maximal and the conics x_{0}^{2}, x_{0}x_{1}, x_{0}x_{2}, f_{1}, f_{2} are in general position. In this case the pencil of conics generated by V (f_{1}) and V (f_{2}) contains two (distinct or not) rank 1 conics, i.e. double line through (1, 0, 0). If the two conics are distinct, we can assume, up to change of coordinates, that they are defined by the equations x_{1}^{2} = 0, x_{2}^{2} = 0. This corresponds to the case (i) of the lemma. When the two conics coincide, Σ(r, f_{1}, f_{2}) has a base point on the line V (r) (case (ii)). If d(r, f_{1}, f_{2}) = 4, either f_{1} and f_{2} are coincident (cases (iii) and (iv)) or distinct (cases (v) and (vi)). By looking at the base locus of Σ(r, f_{1}, f_{2}) we can distinguish case (iii) from case (iv) and case (v) from (vi). Finally, if d(r, f_{1}, f_{2}) = 3 both conics V (f_{1}), V (f_{2}) are in Σ(r) and we get case (vii).

2.3.4 Remark. In the classification of the last lemma we observe that

• case (i) is the unique locally free sheaf and, in particular, it is stable (Hoppe’s criterion),

• case (ii) is stable because one can show that its dual is the bundle T ℙ^{2}(−1),

• cases (iii) and (iv) are not stable, in fact, if V (f_{1}), V (f_{2}) coincide, we may assume f_{1} = f_{2} and give explicitly a destabilizing subsheaf J:

0 ⟶ 𝒪(−3) ⟶_{(r,f_{1},f_{1})} 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} ⟶ ℱ(r, f_{1}, f_{2}) ⟶ 0
↑^{Id}  ↑[\begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix}]  ↑
0 ⟶ 𝒪(−3) ⟶_{(r,f_{1})} 𝒪(−2) ⊕ 𝒪(−1) ⟶  J  ⟶ 0

(2.51)

• case (v) and (vi) are not stable and now a destabilizer subsheaf is the following J:

0 ⟶ 𝒪(−3) ⟶_{(r,f_{1},0)} 𝒪(−2) ⊕ 𝒪(−1)^{⊕2} ⟶ ℱ(r, f_{1}, f_{2}) ⟶ 0
↑^{Id}  ↑[\begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}]  ↑
0 ⟶ 𝒪(−3) ⟶_{(r,f_{1})} 𝒪(−2) ⊕ 𝒪(−1) ⟶  J  ⟶ 0

(2.52)

• case (vii) is not torsion free sheaf.

As a consequence, we get the following

2.3.5 Lemma. ℱ(r, f_{1}, f_{2}) is a stable sheaf if and only if Σ(r, f_{1}, f_{2}) has dimension 5.

2.3.1 Jumping lines of ℱ(x_{0}, x_{1}^{2}, x_{2}^{2})

In this section we are going to study the jumping lines of first and second kind of a locally free sheaf ℱ= ℱ(r, f_{1}, f_{2}). After lemma 2.3.3, it is enough for us to consider the case of ℱ= ℱ(x_{0}, x_{1}^{2}, x_{2}^{2}).

Jumping lines of second kind were introduced by Hulek in [Hul79] where it is also proved that they are a powerful tool for the classification of rank-2 vector bundles on ℙ^{2} with odd first Chern class.

2.3.6 Proposition. Let ℱ be the sheaf ℱ(x_{0}, x_{1}^{2}, x_{2}^{2}).

1. The line {x_{0} = 0} is the unique jumping line of ℱ.

2. The jumping lines of second kind of ℱ are only the lines through the points (0, 1, 0), (0, 0, 1).

Proof. The canonical isomorphism of 2-dimensional vector spaces V ≃ V^{∗} ⊗ det V give the isomorphism of vector bundles ℱ ≃ ℱ^{∗}(−1). Then ℱ has the following injective free resolution:

0 → ℱ → 𝒪 ⊕ 𝒪(1)^{2} →^{ϕ} 𝒪(2) → 0  (2.53)

where ϕ is given by the matrix (x_{0}, x_{1}^{2}, x_{2}^{2}). For a jumping line ℓ, the map:

H^{0}(ϕ_{ℓ}(−1)) : H^{0}(𝒪_{ℓ} ⊕ 𝒪(−1)^{2}) → 𝒪(1)

is not injective and this is true if and only if ℓ is the line {x_{0} = 0}. Moreover, ℓ is a jumping line of second kind if and only if the map:

H^{0}(ϕ_{ℓ^{2}}) : H^{0}(𝒪_{ℓ^{2}}(1) ⊕ 𝒪_{ℓ^{2}}^{2}) → H^{0}(𝒪_{ℓ^{2}}(2))

has not maximal rank ad by calculating this is verified is and only if ℓ is a line through (0, 1, 0) or (0, 0, 1).

2.3.2 The moduli space 𝔐_{Sheaf,ℙ^{2}}(−1, 2)

Let 𝔐_{Sheaf,ℙ^{2}}(−1, 2) be the moduli space of stable rank 2 sheaves on ℙ^{2} with Chern classes c_{1} = −1, c_{2} = 2 and let 𝔐^{∘} ⊂ 𝔐_{Sheaf,ℙ^{2}}(−1, 2) be the open set which appears in a minimal resolution same as (2.45). Since the first Chern class is odd, semistable sheaves are stable.

Let M be the incidence variety:

M := {((P_{1}, P_{2}), ℓ) ∈ S^{2}ℙ^{2} × ℙ^{2}^{∗} | P_{1}, P_{2} ∈ ℓ; ℓ ⊂ ℙ^{2}} .  (2.54)

where S^{2}ℙ^{2} is the space parameterizing all pairs of points of ℙ^{2}. If ℙ^{5} denotes the space of non zero symmetric matrices 3 × 3, since S^{2}ℙ^{2}^{∗} is isomorphic to the space of degenerated (non zero) conics on ℙ^{2},

M ≃ {(x, A) ∈ ℙ^{2} × ℙ^{5} : det A = 0, Ax = 0} .  (2.55)

2.3.7 Theorem. The varieties 𝔐^{∘} and M are isomorphic.

Proof. By lemma 2.3.5 every point of 𝔐^{∘} is a class of sheaves of type ℱ(r, f_{1}, f_{2}) where the linear system Σ(r, f_{1}, f_{2}) has dimension 5. The pencil of conics generated by V (f_{1}) and V (f_{2}) contains two conics (possibly coincident) tangent to V (r) in the points P_{1}, P_{2}. Then we get the map

\begin{aligned}ϕ: & 𝔐^{∘} & → & M \\ & [ℱ(r, f_{1}, f_{2})] & ↦ & (P_{1}, P_{2}, V (r)) \end{aligned}

The map ϕ is bijective since, given a conic g_{1} (not divisible by r) tangent at V (r) in P_{1} (or P_{2}), every other conic tangent at V (r) in P_{1} (or P_{2}) is V (ωr + g_{1}) with an appropriate ω.

2.3.8 Corollary. The moduli space of vector bundles 𝔐_{ℙ^{2}}(−1, 2) is isomorphic to S^{2}ℙ^{2} \ Δ where Δ is the diagonal of S^{2}ℙ^{2}.

Proof. By lemma 2.3.1, the space 𝔐_{ℙ^{2}}(−1, 2) consists precisely of locally free sheaf of type ℱ(r, f_{1}, f_{2}) that is elements of 𝔐^{∘} corresponding to points (P_{1}, P_{2}, r) of M where P_{1} and P_{2} are distinct.

2.3.9 Remark. The same description of 𝔐_{ℙ^{2}}(−1, 2) obtained with a different approach can be found in [OSS80] page 344.

2.3.10 Remark. With the the isomorphism of the previous theorem, toric sheaves in 𝔐^{∘} admit the following characterization. The action of the torus T^{2} := ℂ^{∗} × ℂ^{∗} on ℙ^{2}, which in coordinates is given by:

\begin{aligned}α: & T^{2} × ℙ^{2} & → & ℙ^{2} \\ & ((t_{1}, t_{2}), (x_{0}, x_{1}, x_{2})) & ↦ & (x_{0}, t_{1}x_{1}, t_{2}x_{2}) \end{aligned}

has fixed points (1, 0, 0), (0, 1, 0), (0, 0, 1) and fixed lines x_{0} = 0, x_{1} = 0, x_{2} = 0. The induced action on M is given by

α_{t_{1},t_{2}}(P_{1}, P_{2}, r)) = (α_{t_{1},t_{2}}(P_{1}), α_{t_{1},t_{2}}(P_{2}), α_{t_{1},t_{2}}(r)).  (2.56)

A point (P_{1}, P_{2}, r) ∈ M corresponds at a toric sheaf [ℱ] ∈ 𝔐^{∘} if and only if P_{1} and P_{2} are fixed points and r is a fixed line. In conclusion we have 3 toric bundles presented by

ℱ(x_{i}, x_{j}^{2}, x_{k}^{2}) i, j, k distinct

and 6 toric (not locally free) sheaves presented by

ℱ(x_{i}, x_{j}x_{k}, x_{k}^{2}) i, j, k distinct.

2.4 Monads of stable rank-2 bundles on ℙ^{2}

For stable rank-2 vector bundles on ℙ^{2} with fixed Chern classes c_{1}, c_{2}, theorem 2.2.5 and equations (2.5), (2.6) give sufficient conditions to find relative admissible pairs. Remark 2.2.14 ensures the finiteness of ℑ and permit to write an algorithm to enumerate such pairs by restricting the search to a finite domain. The sets 𝔐(a, b) for all pairs (a, b) varying in ℑ give a partition of the moduli space 𝔐 := 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) in constructible subsets. The structure of this partition is in general not known. In fact we ignore which numerical conditions must be fulfilled by two admissible pairs (a, b) and (a′, b′) for 𝔐(a, b) to be in the closure of 𝔐(a′, b′). Theorem 2.2.10 provides some information in this regard and in the previous section we found the admissible pair of general bundle (i.e., the pair of the open dense subset of the moduli space), but, even in the simpler cases, these results are too weak to complete the analysis of 𝔐 (see examples 2.2.15).

Even if the general problem still remains open, some more help in this investigation comes from the following application of Beilinson theorem (see [OSS80]).

2.4.1 Proposition. A normalized r-bundle ℰ over ℙ^{2} is the cohomology of a monad

0 → H ⊗ 𝒪(−1) ⟶^{A} K ⊗ 𝒪 ⟶^{B} L ⊗ 𝒪(1) → 0  (2.57)

where H = H^{1}(ℙ^{2}, ℰ(−2)), K = H^{1}(ℙ^{2}, ℰ ⊗ Ω^{1}) and L = H^{1}(ℙ^{2}, ℰ(−1)).

Hence, for fixed Chern classes, different admissible pairs correspond to different maps (or matrices) A and B of the “same” monad. This allows us to construct vector bundles on ℙ^{2} starting from a suitable surjective maps B: K ⊗ 𝒪 → L ⊗ 𝒪(1) with the following steps:

(i) find a generic element A of the algebraic set of maps {T: H ⊗ 𝒪(−1) → L ⊗ 𝒪 | BT = 0} ;

(ii) test if A is injective (otherwise choose a new B); (iii) take the cohomology of the monad (2.57). We denote by ℰ(B) a vector bundle obtained with this method.

For c_{1} = −1 and some values of c_{2} we can describe the structure of the moduli space completely by writing explicitly the matrices A and B of some deformation of vector bundles.

For now, we don’t have a unique method to write all the possible deformations. In the next section we give an explicit description of the moduli space in the case resumed in the following figure. (We restrict our attention at the case c_{1} = −1 since for odd first Chern class the notions of semistability and stability coincide).

All matrices are found using computer algebra system MACAULAY2 and the algorithm to compute admissible pairs and their ”codimension” is written in the Scheme dialect of lisp and available in [Mag99].

c_{2}  a  b  codim
2  (3) (1 1 2)  0
3  (3 3) (1 2 2 2)  0
(4) (1 1 3)  2
4 (3 3 3) (2 2 2 2 2)  0
(4) (1 2 2)  1
(3 4) (1 2 2 3)  2
(5) (1 1 4)  4
5 (3 4) (2 2 2 2) 0
(3 3 4) (2 2 2 2 3)  2
(4 4) (1 2 3 3)  2
(3 5) (1 2 2 4)  4
(6) (1 1 5)  6
6 (4 4) (2 2 2 3) 0
(3 4 4) (2 2 2 3 3)  2
(4 4 4) (1 3 3 3 3)  3
(3 3 5) (2 2 2 2 4)  4
(5) (1 2 3)  4
(4 5) (1 2 3 4)  5
(3 6) (1 2 2 5)  6
(7) (1 1 6)  8

Admissible pair of 𝔐_{ℙ^{2}}(2; −1, c_{2})

The cases c_{2} = 2, 3 are obvious.

2.4.1 𝔐_{ℙ^{2}}(2; −1, 4)

A vector bundle ℰ in 𝔐_{ℙ^{2}}(2; −1, 4) is the cohomology of a monad

0 → 𝒪(−1)^{3} ⟶^{A} 𝒪^{9} ⟶^{B} 𝒪(1)^{4} → 0

Let B^{1} be the matrix

B^{1} = (\begin{matrix}x_{0} & x_{1} & x_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} \end{matrix}) .

With the aid of a computer algebra system one easily find that the vector bundle ℰ^{1} = ℰ(B) belongs to to the stratum 𝔐_{1} := 𝔐((5), (1, 1, 4)).

Analogously, the matrix

B_{ξ}^{2} = (\begin{matrix}x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} \end{matrix})

defines an element ℰ(B_{ξ}^{2}) in 𝔐_{2} := 𝔐((3, 4), (1, 2, 2, 3)) for any sufficiently generic linear form ξ. By choosing ξ = ε(x_{0} + x_{1} + x_{2}) we find a family of vector bundles ℰ_{ε}^{2} := ℰ(B_{ε(x_{0}+x_{1}+x_{2})}^{2}) for a generic ε ∈ ℂ^{∗}. Since for ε = 0 we have B_{0}^{2} = B^{1}, then ℰ_{ε}^{2} is a family of vector bundles having limit in 𝔐^{1}, that is, 𝔐_{1} ⊂ \overline{𝔐}_{2}.

One can show that 𝔐_{1} ⊂ \overline{𝔐}_{3} similarly by taking the limit of the family ℰ_{ε}^{3} := ℰ(B_{ε(x_{0}+x_{1}+x_{2})}^{3}) of vector bundles in 𝔐_{3} := 𝔐((4), (1, 2, 2)) defined by

B_{ξ}^{3} = (\begin{matrix}x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} \end{matrix}) .

Moreover, by lemma 2.2.10, we have 𝔐_{2} ⊂ \overline{𝔐}_{3} and all the previous subset are on the boundary of 𝔐((3, 3, 3), (2, 2, 2, 2, 2)) which is the stratum of the general element.

2.4.2 𝔐_{ℙ^{2}}(2; −1, 5)

For the moduli space 𝔐_{ℙ^{2}}(2; −1, 5) the monad (2.57) take the form

0 → 𝒪(−1)^{4} ⟶^{A} 𝒪^{11} ⟶^{B} 𝒪(1)^{5} → 0

We are going to consider the following family of matrices depending on two linear forms ξ, η on ℙ^{2}:

B_{ξ,η}^{1} = ( \begin{matrix}x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & x_{1} & x_{2} & η & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} \end{matrix} )

B_{ξ,η}^{2} = ( \begin{matrix}x_{0} & ξ & x_{0} & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ η & x_{0} & 0 & x_{0} & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 \\ 0 & 0 & x_{0} & 0 & x_{0} & 0 & 0 & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & x_{0} & 0 & x_{0} & 0 & 0 & x_{1} & x_{2} & 0 \\ 0 & 0 & 0 & 0 & x_{0} & 0 & x_{0} & 0 & 0 & x_{1} & x_{2} \end{matrix} )

Then one may easly verify with a computer algebra system that when ξ and η are null ℰ(B_{0,0}^{1}) and ℰ(B_{0,0}^{1}) belong to 𝔐((6), (1, 1, 5)). For generic ξ one has

ℰ(B_{ξ,0}^{1}) belongs to 𝔐((3, 5), (1, 2, 2, 4)),
ℰ(B_{ξ,ξ}^{1}) belongs to 𝔐((4, 4), (1, 2, 3, 3)),
ℰ(B_{ξ,ξ}^{2}) belongs to 𝔐((3, 3, 4), (2, 2, 2, 2, 3)).

It follows, by taking the limit for ε → 0 in the matrices B_{εξ,0}^{1}, B_{ξ,εξ}^{1} and B_{εξ,εξ}^{2} that

𝔐((6), (1, 1, 5)) ⊂ \overline{𝔐}((3, 5), (1, 2, 2, 4)),
𝔐((3, 5), (1, 2, 2, 4)) ⊂ \overline{𝔐}((4, 4), (1, 2, 3, 3)),
𝔐((3, 5), (1, 2, 2, 4)) ⊂ \overline{𝔐}((3, 3, 4), (2, 2, 2, 2, 3)).

Moreover, all the strata are in the closure of the 0-codimensional stratum 𝔐((3, 4), (2, 2, 2, 2)) while the closures of 𝔐((4, 4), (1, 2, 3, 3)) and 𝔐((3, 3, 4), (2, 2, 2, 2, 3)) are disjoint since they have the same dimension. This complete the description of the strata in 𝔐_{ℙ^{2}}(2; −1, 5).

2.4.3 𝔐_{ℙ^{2}}(2; −1, 6)

Let us suppose that ℰ be a vector bundle in 𝔐_{ℙ^{2}}(2; −1, 6) and apply prop 2.4.1 to both ℰ and ℰ(1). Then one obtain ℰ and ℰ(1) respectively as cohomology of monads

0 → 𝒪(−1)^{5} ⟶^{A}𝒪^{13} ⟶^{B} 𝒪(1)^{6} → 0,  (2.58)
0 → 𝒪(−1)^{6} ⟶^{A}𝒪^{13} ⟶^{B} 𝒪(1)^{5} → 0.  (2.59)

This give us more freedom to describe ℰ since upon twist of ℰ we can start the construction of the previous sections from a matrix B that fits in (2.58) or in (2.59). For brevity we denote the eight strata according to the following list

\begin{aligned}𝔐_{1} & = 𝔐((7), (1, 1, 6)), & 𝔐_{5} & = 𝔐((3, 3, 5), (2, 2, 2, 2, 4)), \\ 𝔐_{2} & = 𝔐((3, 6), (1, 2, 2, 5)), & 𝔐_{6} & = 𝔐((4, 4, 4), (1, 3, 3, 3, 3)), \\ 𝔐_{3} & = 𝔐((4, 5), (1, 2, 3, 4)), & 𝔐_{7} & = 𝔐((3, 4, 4), (2, 2, 2, 3, 3)), \\ 𝔐_{4} & = 𝔐((5), (1, 2, 3)), & 𝔐_{8} & = 𝔐((4, 4), (2, 2, 2, 3)). \end{aligned}

We begin by using (2.59). Let ξ = (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}) and η = (η_{1}, η_{2}, η_{3}, η_{4}, η_{5}) be vectors of linear form on ℙ^{2} and consider the matrix

B_{ξ,η}^{1} = ( \begin{matrix}x_{0} & 0 & ξ_{1} & 0 & 0 & 0 & η_{1} & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & x_{0} & 0 & ξ_{2} & 0 & 0 & 0 & η_{2} & x_{1} & x_{2} & 0 & 0 & 0 \\ 0 & 0 & x_{0} & 0 & ξ_{3} & 0 & 0 & 0 & η_{3} & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & x_{0} & 0 & ξ_{4} & 0 & 0 & 0 & η_{4} & x_{1} & x_{2} & 0 \\ 0 & 0 & 0 & 0 & x_{0} & 0 & ξ_{5} & 0 & 0 & 0 & η_{5} & x_{1} & x_{2} \end{matrix} )

As in the previous sections, one may play with various possibilities for ξ and η with a computer algebra program. We call ℰ(B_{ξ,η}) the resulting vector bundles in 𝔐_{ℙ^{2}}(2; −1, 6). It turns out that for generic forms α_{i} and β_{j} with i, j = 1 … 5 one has

• for ξ = 0 and η = 0 the vector bundle ℰ(B_{0,0}) belongs to 𝔐_{1};

• ℰ(B_{(0...0,α_{5}),0}) and ℰ(B_{0,(β_{1},0...0)}) belong to 𝔐_{2}; • ℰ(B_{(α_{1},α_{2},0,0,0),(β_{1},β_{2},0,0,0)}) belongs to 𝔐_{3};

• ℰ(B_{(α_{1},α_{2},α_{3},0,α_{5}),0}) belongs to 𝔐_{4};

• ℰ(B_{(0,0,0,α_{1},α_{2}),(0,0,0,β_{1},β_{2})}) belongs to 𝔐_{5}.

• ℰ(B_{(α_{1},α_{2},α_{3},α_{4},0),0}) and ℰ(B_{(α_{1},α_{2},α_{3},0,α_{5}),(β_{1},β_{2},0,0,0)}) belong to 𝔐_{6};

• ℰ(B_{(α_{1},α_{2},α_{3},α_{4},0),(β_{1},β_{2},0,0,0)}) and ℰ(B_{(0,α_{2},0,α_{4},α_{5}),(0,0,0,β_{4},β_{5})}) belong to 𝔐_{7}. Moreover for general linear forms ω the matrix

B^{2} = ( \begin{matrix}x_{0} & 0 & 0 & 0 & 0 & ω & 0 & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & x_{0} & 0 & 0 & 0 & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 \\ 0 & 0 & x_{0} & 0 & 0 & 0 & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & x_{0} & 0 & 0 & 0 & 0 & 0 & 0 & x_{1} & x_{2} & 0 \\ 0 & 0 & 0 & 0 & x_{0} & 0 & α_{5} & 0 & 0 & 0 & 0 & x_{1} & x_{2} \end{matrix} )

gives the vector bundle ℰ(B^{2}) belongs to 𝔐_{5}. Then

• 𝔐_{1} is contained in the closure of any other constructible subsets;

• 𝔐_{2} is contained in \overline{𝔐}_{3}, \overline{𝔐}_{5}, \overline{𝔐}_{6}, \overline{𝔐}_{7} and \overline{𝔐}_{8};

• 𝔐_{3} ⊂ \overline{𝔐}_{6}, 𝔐_{4} ⊂ \overline{𝔐}_{6}, 𝔐_{6} ⊂ \overline{𝔐}_{7} and 𝔐_{5} ⊂ \overline{𝔐}_{7}

• 𝔐_{4} and 𝔐_{5} so they are unrelated each other.

Now, let us look a the monad (2.58). If

B_{ξ,η}^{2} = ( \begin{matrix}x_{0} & x_{1} & x_{2} & η & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & ξ & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} & ξ & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{0} & x_{1} & x_{2} \end{matrix} ) .

and

B_{ξ,η}^{3} = ( \begin{matrix}x_{0} & 0 & η & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & x_{0} & 0 & η & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & x_{0} & 0 & ξ & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & x_{0} & 0 & ξ & 0 & 0 & 0 & x_{1} & x_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{0} & 0 & ξ & 0 & 0 & 0 & x_{1} & x_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & x_{0} & 0 & ξ & 0 & 0 & 0 & x_{1} & x_{2} \end{matrix} )

then, for a generic linear form ω on ℙ^{2} and ε ∈ ℂ^{∗} one has

[ℰ(B_{ω,0}^{2})] ∈ 𝔐_{2},  [ℰ(B_{ω,εω}^{2})] ∈ 𝔐_{4},
[ℰ(B_{ω,0}^{3})] ∈ 𝔐_{3},  [ℰ(B_{ω,εω}^{3})] ∈ 𝔐_{7}.

It follows 𝔐_{2} ⊂ \overline{𝔐}_{4} and 𝔐_{3} ⊂ \overline{𝔐}_{7}. Also remember that 𝔐_{3} ⊂ \overline{𝔐}_{4} by 2.2.10.

In general, by semicontinuity of the the dimension of the cohomology groups h^{q}, if X and Y are constructible subsets of 𝔐_{ℙ^{2}}(c_{1}, c_{2}) and Y ∈ \overline{X}, then h^{q}(ℰ(t)) ≥ h^{q}(ℱ(t)) for all t ∈ ℤ, ℰ ∈ Y, ℱ ∈ X.

If ℰ ∈ 𝔐_{6} then h^{1}(ℰ(4)) = 3 and h^{1}(ℰ(5)) = 0, while if ℱ ∈ 𝔐_{5} then h^{1}(ℱ(4)) = 2 and h^{1}(ℰ(5)) = 1 hence 𝔐_{6} and 𝔐_{5} are not related.

So far, it remains unclear if 𝔐_{3} is in the closure of 𝔐^{5} or not.

2.4.4 Tables

The results of the above section can be resumed in the following pictures

Filtration of 𝔐_{ℙ^{2}}(2; −1, 4)

𝔐((3, 4), (1, 2, 2, 3)) 𝔐((4), (1, 2, 2)) 𝔐((3, 3, 3), (2, 2, 2, 2, 2)) 𝔐((5), (1, 1, 4))

Filtration of 𝔐_{ℙ^{2}}(2; −1, 5)

[figure: 𝔐((6), (1, 1, 5)); 𝔐((3, 5), (1, 2, 2, 4)); 𝔐((3, 3, 4), ((2, 2, 2, 2, 3)); 𝔐((4, 4), (1, 2, 3, 3)); 𝔐((3, 4), (2, 2, 2, 2))]

Filtration of 𝔐_{ℙ^{2}}(2; −1, 6)

[figure: 𝔐((7), (1, 1, 6)); 𝔐((3, 6), (1, 2, 2, 5)); 𝔐((4, 5), (1, 2, 3, 4)); 𝔐((3, 3, 5), (2, 2, 2, 2, 4)); 𝔐((5), (1, 2, 3)); 𝔐(4, 4, 4), (1, 3, 3, 3, 3)); 𝔐((3, 4, 4), (2, 2, 2, 3, 3)); 𝔐((4, 4), (2, 2, 2, 3)); ?]

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