STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT AND STEINER BUNDLES

CARLA DIONISI

Abstract. In this paper nondegenerate multidimensional matrices of boundary format in V_{0} ⊗ ⋯ ⊗ V_{p} are investigated by their link with Steiner vector bundles on product of projective spaces. In particular it is proved that for any nondegenerate matrix A there exists an explicit action of SL(2) on V_{0} ⊗ ⋯ ⊗ V_{p} and, describing the behaviour of its stabilizer in SL(V_{0}) × ⋯ × SL(V_{p}), we prove that Stab^{0}(A) ⊆ SL(2) and that the equality holds if and only if A belongs to a unique SL(V_{0}) × ⋯ × SL(V_{p})-orbit containing the identity matrices, according to [AO].

1. Introduction

Let V_{j} be a complex vector space of dimension k_{j} + 1 for j = 0, …, p with k_{0} = max_{i}{k_{i}}. Gelfand, Kapranov and Zelevinsky in [GKZ] proved that the dual variety of the Segre product ℙ(V_{0}) ×⋯× ℙ(V_{p}) is a hypersurface in (ℙ^{(k_{0}+1)...(k_{p}+1)−1})^{∨} if and only if k_{0} ≤ ∑_{i=1}^{p} k_{i}. The defining equation of this hypersurface is called the hyperdeterminant of format (k_{0} +1)×⋯×(k_{p} +1) and is denoted by Det. Moreover the hyperdeterminant is a homogeneous polynomial function on V_{0}^{∗} ⊗ ⋯ ⊗ V_{p}^{∗} so that the condition DetA ≠ 0 is meaningful for a (p + 1)-dimensional matrix A ∈ ℙ(V_{0} ⊗ ⋯ ⊗ V_{p}) of format (k_{0} + 1) × ⋯ × (k_{p} + 1). The hyperdeterminant is an invariant for the natural action of SL(V_{0}) × ⋯ × SL(V_{p})

1991 Mathematics Subject Classification. 14F05; 15A72.

Key words and phrases. Vector bundles, multidimensional matrices, theory of invariants.

2 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

on ℙ(V_{0} ⊗ ⋯ ⊗ V_{p}), and, in particular, if DetA ≠ 0 then A is semistable for this action.

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. The stabilizer are well known for p ≤ 1, so that in this paper we assume p ≥ 2. It easy to check (see [WZ], [DO]) that the degenerate matrices fill an irreducible variety of codimension k_{0} − ∑_{i=1}^{p} k_{i} + 1 and if k_{0} < ∑_{i=1}^{p} k_{i} then all matrices are degenerate. We will assume from now on that A is of boundary format i.e. that k_{0} = ∑_{i=1}^{p} k_{i}. (A self-contained approach to hyperdeterminant of boundary format matrices can be found in [DO]).

The main result of this paper is the following:

1.1. Theorem. Let A ∈ ℙ(V_{0} ⊗ ⋯ ⊗ V_{p}) be a 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). Moreover Stab(A)^{0} = SL(2) if and only if A is an identity (defined in 2.3).

1.2. Remark. 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 3dimensional 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 matrices. 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.

CARLA DIONISI 3

Ancona and Ottaviani in [AO] prove theorem 1.1 statement for p = 2 . We generalize their proof using the correspondence between nondegenerate boundary format matrices and vector bundles on a product of projective spaces. Indeed, for any fixed j ≠ 0, a (p+1)-dimensional matrix A ∈ V_{0}⊗⋯⊗V_{p} of format (k_{0} + 1) × ⋯ × (k_{p} + 1) defines a sheaf morphism f_{A}^{(j)} on the product X = ℙ^{k_{1}} × ⋯ × ℙ̂^{k_{j}} × ⋯ × ℙ^{k_{p}}

𝒪_{X} ⊗ V_{0}^{∨} ⟶^{f_{A}^{(j)}} 𝒪_{X}(1, …, 1) ⊗ V_{j}; (1)

it is easy to prove the following

1.3. Proposition. ([AO],[D]) If a matrix A is of boundary format, then DetA ≠ 0 if and only if for all j ≠ 0 the morphism f_{A}^{(j)} is surjective (so 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 it is a Steiner bundle as defined in [DK] (this case has been investigate in [AO]). We shall call to S_{A} with the name Steiner also for p ≥ 3. The main new technique introduced in this paper is the use of jumping hyperplanes for bundles on the product of p − 1 projective spaces. For p ≥ 2 there are two natural ways to introduce them that, by above correspondence, translate into two different conditions on the associated matrix and that we call weak and strong (see definition 3.1 and 3.7). They coincide when p = 2.

Moreover, weak and strong jumping hyperplanes are invariants of the action of SL(V_{0}) × … × SL(V_{p}) on matrices. By investigating these invariants we derive the proof of theorem 1.1 and also we obtain a characterization of a particular class of bundles called Schwarzenberger bundles (see [Sch] for the original definition in the case p = 2). Schwarzenberger bundles correspond exactly to such matrices A which verify the equality Stab^{0}(A) = SL(2) in theorem 1.1, called identity matrices.

4 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

I would like to thank G. Ottaviani for many useful discussions and for his invaluable guidance.

2. Multidimensional matrices of boundary format and geometric invariant theory

In this section we recall some results proved in [AO] about multidimensional matrices from the point of view of Mumford’s Geometric Invariant Theory.

2.1. Definition. [AO] 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, then we have A ∈ ⨁_{n≥0} W_{n}

2.2. Definition. [AO] 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^{∗}

2.3. Definition. [AO] 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

All identity matrices fill up a distinguished SL(V_{0})×⋯×SL(V_{p})-orbit in ℙ(V_{0} ⊗ … ⊗ V_{p}).

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

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

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

A is diagonalizable ⟺ ℂ^{∗} ⊂ Stab(A)

3. Jumping hyperplanes 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 unstable hyperplane of S if h^{0}(S_{|h}^{∗}) ≠ 0 (see [AO]).

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

Av_{0} = v_{1}h; (2)

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

6 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

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

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} ⊗ … (3)

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}).

3.2. Remark. The expression (3) 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]

3.3. Remark. A′_{j} is again of boundary format

3.4. Proposition. 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 (3.1) associated to ℋ, the map S_{A}^{(j)} → 𝒪_{h} induced by a non zero section of S_{A}^{(j)} is surjective (see [V2] prop.2.1). Since codim h = 1, then its kernel S′^{(j)} is locally free sheaf [Ser65] of rank k_{0} − k_{j} − 1 on X and it is the Steiner bundle associated to the matrix A^{(j)} as the snake-lemma applied to the following exact diagram show

0 ↓

S′^{(j)}

↓

0 ⟶ 𝒪_{X}(−1, …, −1) ⊗ V_{j}^{∨} ⟶^{f_{A}^{(j)}} 𝒪_{X} ⊗ V_{0}^{∨} ⟶ S_{A}^{(j)} ⟶ 0
↓ ↓ ↓
0 ⟶ 𝒪_{X}(−1, …, −1) ⟶ 𝒪_{X} ⟶ 𝒪_{h} ⟶ 0
↓  ↓  ↓
0  0  0

i.e. S′^{(j)} = S_{A′_{j}}^{(j)} and by proposition (1.3) the result follows.

3.5. Remark. If W (S_{A_{j}}) is the set of j-weak jumping hyperplanes of A , then the exact sequence (dual to the last column of the above diagram)

0 → S_{A}^{(j)}^{∨} → S_{A′_{j}}^{(j)}^{∨} → 𝒪_{X}(1, …, 1) → 0

show that W (S_{A_{j}}) ⊂ W (S_{A′_{j}}) ∪ {h}

3.6. Remark. An element g ∈ Stab(A) preserves h and it induces ḡ ∈ SL(V_{0}/_{<g(v_{0})>})×SL(V_{1})×⋯×SL(V_{j}/_{<g(v_{j})>})×⋯×SL(V_{p}) such that g · A projects to ḡ · A′_{j} and the elementary transformation behaves well with respect to the action of g.

3.7. 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} ⊗ …

8 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

3.8. 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.

3.9. Remark. For p = 2 the notations of strong jumping hyperplane and of weak jumping hyperplane coincide with each other (see [AO]).

3.10. Example. (the identity) Fixed a basis e_{0}^{(j)}, …, e_{k_{j}}^{(j)} in V_{j} for all j, 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_{1}}^{(1)} ⊗ ⋯ ⊗ 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 [Sch])(see also ([AO])

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

CARLA DIONISI 9

Proof. In the case p = 2 the statement is proved in [AO] (theorem 5.13) or in [V2] (theorem 3.1). 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 with 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. 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.

3.12. Proposition. Let A be a 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 proved in [AO] (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 with 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′

10 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

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 submatrices of A which are all uniquely determined, so also A is determined.

3.13. 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 [AO]) but in the case p ≥ 3 there is not an analog existence result.

The following proposition is well known.

3.14. Proposition. All nondegenerate matrices of type 2 × k × (k + 1) are GL(2) × GL(k) × GL(k + 1) equivalent.

Proof. Let A, A′ two such matrices. They define two exact sequences on ℙ^{1}

0 → 𝒪(−k)⟶𝒪^{k+1}⟶^{A}𝒪(1)^{k} → 0

0 → 𝒪(−k)⟶𝒪^{k+1}⟶^{A′}𝒪(1)^{k} → 0

and easily we can find the following commutative diagram

0 → 𝒪(−k) ⟶ 𝒪^{k+1} ⟶^{A} 𝒪(1)^{k} → 0
↓^{1} ↘ ↓^{f}
0 → 𝒪(−k) ⟶ 𝒪^{k+1} ⟶^{A′} 𝒪(1)^{k} → 0

The above proposition can be reformulated as follows:

3.15. Proposition. Every surjective morphism of vector bundles on ℙ^{1}

𝒪_{ℙ^{1}}^{k+1} → 𝒪_{ℙ^{1}}(1)^{k}

is represented by an identity matrix.

Proof of theorem 1.1

CARLA DIONISI 11

Proof. We proceed by induction on k_{0}. If k_{0} = 2 the theorem is true by proposition 3.15. We may suppose that Stab(A)^{0} has dimension ≥ 1 then by theorem 2.4 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 3.11), 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}. Hence Stab(A)^{0} ⊆ SL(U) by the inductive hypothesis. Now, we want to show that the action is performed according to the isomorphisms V_{i} ≃ S^{k_{i}}U. 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})

can be identified with 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.

12 STABILIZERS FOR NONDEGENERATE MATRICES OF BOUNDARY FORMAT

In the case Stab(A)^{0} = SL(2), the action of SL(U) satisfies definition 2.3, proving that A is an identity.

3.16. Remark. Throughout this paper we work only on nondegenerate matrices. Indeed, in the proofs we apply the induction strategy (hence the results of [AO]) and the correspondence between matrices and vector bundles described in proposition 1.3. The characterization of the stabilizer of degenerate matrices is still an open problem. Another interesting problem is the study of the stabilizer of general multidimensional matrices (and not necessary of boundary format).

References

[AO] V. Ancona and G. Ottaviani, Unstable hyperplanes for Steiner bundles and multidimensional matrices, Advances in Geometry, 1 (2001), 165-192

[D] C. Dionisi, Multidimensional matrices and minimal resolutions of vector bundles Ph.D. thesis, Dip. Matematica ”R.Caccioppoli”, Università di Napoli, 2000.

[DO] C. Dionisi and G. Ottaviani, The Binet-Cauchy theorem for the hyperdeterminant of boundary format multdimensional matrices, pre-print 2001, math.AG/0104281

[DK] I. Dolgacev and M. M. Kapranov, Arrangement of hyperplanes and vector bundles on ℙ^{n}, Duke Math. J. 71 (1993), 633-664

[GKZ] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhaeuser, Boston 1994

[GKZ1] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinski, Hyperdeterminants, Adv. in Math. (1992), no. 96, 226–263

[Mar82] M. Maruyama, Elementary tranformations in the theory of algebraic vector bundles, Lect. Notes Math. (1982), no. 961, 241–266.

[Par] P. G. Parfenov, Orbits and their closures in the spaces ℂ^{k_{1}} ⊗ ⋯ ⊗ ℂ^{k_{r}}, Sbornik Math. (2001), 192, 89–112 (English transl.)

[Sch] R.L.E. Schwarzenberger, Vector bundles on the projective plane, Proc.London Math.Soc. 3 (1961), no. 11, 623–640.

[Ser65] J.P. Serre, Algèbre locale, multiplicités, Lecture Notes in Mathematics, no. 11, Springer-Verlag, 1965.

[WZ] J. Weyman and A. V. Zelevinsky, Singularities of hyperdeterminants, Ann. Inst. Fourier 46 (1996), 591-644.

[V1] J. Vallès, Fibrés de Schwarzenberger et coniques de droites sauteuses, Bull. Soc. Math. France, 128, (2000), 433-449.

[V2] J. Vallès, Nombre maximal d’hyperplans instables pour un fibré de Steiner, Math. Zeitschrift 233, 507-514 (2000).

Carla Dionisi, Dipartimento di Matematica Applicata “G. Sansone”, Via S. Marta, 3, I-50139, Firenze, Italy

   E-mail address: dionisi@dma.unifi.it