# Schwarzenberger bundles on smooth projective varieties

Abstract: We define Schwarzenberger bundles on smooth projective varieties and we introduce the notions of jumping subspaces and jumping pairs of ( F 0 , O X ) -Steiner bundles. We determine a bound for the dimension of the set of jumping pairs. We classify those Steiner bundles whose set of jumping pairs has maximal dimension by proving that they are Schwarzenberger bundles.

## Summary (2 min read)

Jump to: [Introduction] – [1 Steiner bundles on smooth projective varieties] – [2 Generalized Schwarzenberger on smooth projective vari-] – [4 The tangent space of the jumping variety] – [5 The classification] and [In this case f0 = 1, J̃(E) is a rational normal curve and the natural projections are]

### Introduction

- Steiner vector bundles on projective spaces were first defined by Dolgachev and Kapranov in [DK93] as vector bundles E fitting in an exact sequence of the form 0→ OPn(−1)s → OtPn → E → 0.
- Next the authors outline the structure of the paper.
- In Section 4 the authors obtain an upper bound for the dimension of the jumping variety by studying its tangent space at a fixed jumping pair (see Theorem 4.1).
- Parts of this work were done in UCM-Madrid, IST-Lisbon and UNICAMPCampinas.

### 1 Steiner bundles on smooth projective varieties

- In this section the authors recall the definition of Steiner bundles on smooth projective varieties introduced in [MRS09] and they study some of their properties needed in the sequel.
- Observe that H0(F∨0 ) is globally generated and is the set of conics that passes through three points.
- This interpretation will play an essential role for studying Schwarzenberger bundles on X. Proposition 1.5.
- Then the following properties are equivalent: (i) ϕ is injective.
- This represents a different and challenging problem in the study of vector bundles and in this paper the authors will not deal with it.

### 2 Generalized Schwarzenberger on smooth projective vari-

- The authors goal in this section is to generalize Schwarzenberger bundles on the projective space and on the Grassmann variety G(k, n), as defined in [Arr10] and [AM14], respectively, to any smooth projective variety X. We first recall the definition of Schwarzenberger bundles on G(k, n), following [AM14].the authors.the authors.
- Given any x ∈ X, the composition (F0)x → H0(ψ∗U∨) of the first two maps (on the second factor) is obviously injective.
- This construction allows us to generalize the notion of a Schwarzenberger bundle to any smooth projective variety.

### 4 The tangent space of the jumping variety

- The authors result will allow us to classify in the next section all Steiner bundles such that J̃(E) has maximal dimension.
- Recall, furthermore, that Q denotes the vector bundle in (1).
- In [AM14] the authors proved that the tangent space of the jumping variety at a jumping pair can be also described as TΛJ̃(E) = { ψ ∈ (3) Using this description, the authors are able to obtain an upper bound for the dimension of TΛJ̃(E) and hence an upper bound for the dimension of J̃(E).
- The authors will prove the statement by defining independent elements in Hom ( Λ, T ∗ Λ ) which are also independent modulo TΛJ̃(E).
- Observe that the linear map ϕ1 also defines an (F0,OX)-Steiner bundle Ẽ. Consider the bundle morphism Oασ(X) g // ##.

### 5 The classification

- OX)-Steiner bundles whose jumping variety has maximal dimension.the authors.
- In particular, the authors prove that they are always Schwarzenberger bundles.
- The rest of the section will be devoted to its proof.
- Let E be a reduced (F0,OX)-Steiner bundle on a smooth projective variety X such that the jumping locus J̃(E) has maximal dimension.

### In this case f0 = 1, J̃(E) is a rational normal curve and the natural projections are

- Moreover, J̃(E) has maximal dimension if and only if J̃(Ē) has maximal dimension, according to the respective bounds.
- Therefore, when σ is surjective and J̃(E) has maximal dimension, all Steiner bundles on X are Schwarzenberger bundles given by the pullback of the corresponding Schwarzenberger bundle on G(f0−1,P(H0(F∨0 ))), classified in [AM14].
- Therefore, the authors can suppose, from now on, that σ is not surjective.
- The generic fibers of π1 and π′1, and the further first component projections given by the induction technique, have respectively dimension either 0 or at least f0.

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Citation for published item:

Arrondo, E., Marchesi, S. & Soares, H. (2016). Schwarzenberger bundles on smooth projective

varieties. Journal of Pure and Applied Algebra. 220 (9), 3307-3326

Further information on publisher's website:

10.1016/j.jpaa.2016.02.016

Publisher's copyright statement:

This is the peer reviewed version of the following article: Arrondo, E., Marchesi, S. & Soares, H.

(2016). Schwarzenberger bundles on smooth projective varieties. Journal of Pure and Applied

Algebra. 220 (9), 3307-3326, which has been published in final form at

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Schwarzenberger bundles on smooth projective varieties

Enrique Arrondo, Simone Marchesi and Helena Soares

Abstract

We deﬁne Schwarzenberger bundles on smooth projective varieties and we introduce the notions

of jumping subspaces and jumping pairs of (F

0

, O

X

)-Steiner bundles. We determine a bound

for the dimension of the set of jumping pairs. We classify those Steiner bundles whose set of

jumping pairs has maximal dimension by proving that they are Schwarzenberger bundles.

Keywords Projective varieties; Schwarzenberger bundles; Steiner bundles

2010 Mathematics Subject Classiﬁcation 14F05, 14N05

Introduction

Steiner vector bundles on projective spaces were ﬁrst deﬁned by Dolgachev and Kapranov

in [DK93] as vector bundles E ﬁtting in an exact sequence of the form

0 → O

P

n

(−1)

s

→ O

t

P

n

→ E → 0.

The authors use them to study logarithmic bundles E(H) = Ω(log H) associated to an arrange-

ment H of k hyperplanes with normal crossing. They show that logarithmic bundles are rank

n Steiner vector bundles on P

n

and get a Torelli type theorem. More precisely, they prove that

for k ≥ 2n + 3 the correspondence H → E(H) is bijective except when all hyperplanes osculate

the same rational normal curve. In this case, E(H) is the Schwarzenberger bundle associated

to this curve, as ﬁrst constructed in [Sch61].

In [Val00], Vallès generalized this result for k > n+1. Whereas the main tool in the ﬁrst paper

is the study of the jumping lines of E(H), Vallès focus on a special family of hyperplanes, called

unstable hyperplanes. He proves that if a Steiner bundle E has at least t+2 unstable hyperplanes

in general linear position then all hyperplanes osculate a rational normal curve and E is the

Schwarzenberger bundle associated to this curve. Sharing the same idea of unstable hyperplanes,

Ancona and Ottaviani show that a Steiner bundle is logarithmic if and only if it contains at

least t + 1 unstable hyperplanes (see [AO01]). Moreover, Vallès sees that this correspondence

between Schwarzenberger bundles and rational normal curves is also bijective in the following

sense: given a rational normal curve, one can construct the associated Schwarzenberger bundle

and reconstruct the curve from its set of unstable hyperplanes.

The generalization of the above correspondence was recently addressed by the ﬁrst author

in [Arr10]. Arrondo introduces the notion of Schwarzenberger bundles on P

n

of arbitrary rank

and the ensuing generalization of unstable hyperplanes, which he calls jumping subspaces. A

Schwarzenberger bundle will still be a Steiner bundle and in his paper Arrondo studies the

problems of when is the latter a Schwarzenberger bundle and the related Torelli-type theorem.

1

He gets a sharp bound for the dimension of the set of jumping hyperplanes and shows that in

the case of maximal dimension all Steiner bundles are Schwarzenberger bundles.

In the end of his work, Arrondo proposes to use the deﬁnition of Steiner bundles given in

[MRS09] to get a natural deﬁnition of Schwarzenberger bundles on other smooth projective

varieties. In [AM14], the results in [Arr10] are extended for the Grassmannian variety G(k, n)

and are the main motivation of the present paper. Our goal is to generalize the work in [Arr10]

and [AM14] to any smooth projective variety X.

An (F

0

, F

1

) - Steiner bundle E on X is a vector bundle on X deﬁned by an exact sequence

of the form 0 → F

s

0

→ F

t

1

→ E → 0, where (F

0

, F

1

) is a strongly exceptional ordered pair

of vector bundles on X such that F

∨

0

⊗ F

1

is generated by global sections. These bundles

were introduced in [Soa08] and a cohomological characterization can be found in [MRS09].

In the above cited papers, Arrondo and Marchesi deﬁne Schwarzenberger bundles on P

n

and

G(k, n). They are, respectively, (O

P

n

(−1), O

P

n

) and (U, O

G(k,n)

)-Steiner bundles obtained from

a triplet (Z, L, M), where Z is any projective variety, and L, M are globally generated vector

bundles on Z. In order to generalize these concepts in a natural way we will restrict our

study to (F

0

, O

X

)-Steiner bundles. Denoting f

0

= rk(F

0

), a Schwarzenberger on X will be

a Steiner bundle obtained from the data (Z, ψ, L), where Z is a projective variety provided

with a non-degenerate linearly normal morphism ψ : Z → G(f

0

, H

0

(F

∨

0

)), and L is a globally

generated vector bundle on Z. When X is the Grassmannian variety G(k, n) (which includes the

projective space case), we get a Schwarzenberger bundle according to Arrondo and Marchesi,

when ψ : Z → P

n

= P(H

0

(M)

∗

). Moreover, this deﬁnition will allow us to generalize the notion

of an (a, b)-jumping pair for an (F

0

, O

X

)-Steiner bundle E and prove a Torelli-type theorem

when a = 1 and b = f

0

. That is we will show that, in the case when the set

˜

J(E) of (1, f

0

)-

jumping pairs of E (endowed with a natural structure of a projective variety) has maximal

dimension, E is a (Z, ψ, L)-Schwarzenberger bundle and the Z =

˜

J(E).

Next we outline the structure of the paper. In Section 1 we recall the deﬁnition of Steiner

bundles on smooth projective varieties and their basic properties. In particular, we give an

equivalent deﬁnition in terms of linear algebra and get a low bound for the rank of an (F

0

, O

X

)-

Steiner bundle.

In Section 2 we recall the construction of Schwarzenberger bundles on the Grassmann variety

and deﬁne Schwarzenberger bundles on smooth projective varieties (Deﬁnition 2.1).

In Section 3 we introduce the notions of a jumping subspace and of a jumping pair of a

Steiner bundle on X (Deﬁnition 3.1) and endow the set of all jumping pairs with the structure

of a projective variety. We furthermore give a lower bound for its dimension.

In Section 4 we obtain an upper bound for the dimension of the jumping variety by studying

its tangent space at a ﬁxed jumping pair (see Theorem 4.1).

In Section 5 we provide a complete classiﬁcation of Steiner bundles whose jumping locus has

maximal dimension. In particular, we show that they all are Schwarzenberger bundles (Theorem

5.1).

Acknowledgements. The three authors were partially supported by Fundação para a Ciência e

Tecnologia, projects “Geometria Algébrica em Portugal”, PTDC/MAT/099275/2008 and “Comu-

nidade Portuguesa de Geometria Algébrica”, PTDC/MAT-GEO/0675/2012; and by Ministerio

de Educación y Ciencia de España, project "Variedades algebraicas y analíticas y aplicaciones",

MTM2009-06964. The second author was supported by the FAPESP postdoctoral grant num-

ber 2012/07481-1. The third author is also partially supported by BRU - Business Research

2

Unit, ISCTE-IUL. Parts of this work were done in UCM-Madrid, IST-Lisbon and UNICAMP-

Campinas. The authors would like to thank Margarida Mendes Lopes and Marcos Jardim for

the invitation and kind hospitality.

1 Steiner bundles on smooth projective varieties

In this section we recall the deﬁnition of Steiner bundles on smooth projective varieties

introduced in [MRS09] and we study some of their properties needed in the sequel.

Let us ﬁrst ﬁx some notation.

Notation 1.1. We will always work over a ﬁxed algebraically closed ﬁeld k of characteristic

zero and X will always denote a smooth projective variety over k.

The projective space P(V ) will be the set of hyperplanes of a vector space V over k or,

equivalently, the set of lines in the dual vector space of V , denoted by V

∗

.

We will write G(r − 1, P(V )) for the Grassmann variety of (r − 1)-linear subspaces of the

projective space P(V ). This is equivalent to consider the set G(r, V

∗

) of r-dimensional subspaces

of the vector space V

∗

.

The dual of a coherent sheaf E on X will be denoted by E

∨

. If E is a vector bundle on X

then, for each x ∈ X, E

x

is the ﬁbre over x and h

i

(E) denotes the dimension of H

i

(E).

In order to deﬁne Steiner bundles on a smooth projective variety X we will need the following

deﬁnition.

Deﬁnition 1.2. Let X be a smooth projective variety. A coherent sheaf E on X is exceptional

if

Hom(E, E) ' k,

Ext

i

(E, E) = 0, for all i ≥ 1.

An ordered pair (E, F) of coherent sheaves on X is called an exceptional pair if both E and F

are exceptional and

Ext

p

(F, E) = 0, for all p ≥ 0.

If, in addition,

Ext

p

(E, F ) = 0 for all p 6= 0,

we say that (E, F ) is a strongly exceptional pair.

Deﬁnition 1.3. Let X be a smooth projective variety. An (F

0

, F

1

)-Steiner bundle E on X is

a vector bundle on X deﬁned by an exact sequence of the form

0 → S ⊗ F

0

→ T ⊗ F

1

→ E → 0,

where S and T are vector spaces over k of dimensions s and t, respectively, and (F

0

, F

1

) is an

ordered pair of vector bundles on X satisfying the two following conditions:

(i) (F

0

, F

1

) is strongly exceptional;

(ii) F

∨

0

⊗ F

1

is generated by global sections.

Examples 1.4.

3

(a) A Steiner bundle, as deﬁned by Dolgachev and Kapranov in [DK93], is an (O

P

n

(−1), O

P

n

)-

Steiner bundle in the sense of Deﬁnition 1.3. More generally, vector bundles E with a

resolution of type

0 → O

P

n

(a)

s

→ O

P

n

(b)

t

→ E → 0,

where 1 ≤ b − a ≤ n, are (O

P

n

(a), O

P

n

(b))-Steiner bundles on P

n

(see [MRS09]).

(b) Consider the smooth hyperquadric Q

n

⊂ P

n+1

, n ≥ 2, and let Σ

∗

denote the Spinor bundle

Σ on Q

n

if n is odd, and one of the Spinor bundles Σ

+

or Σ

−

on Q

n

if n is even. The

vector bundle E on Q

n

deﬁned by an exact sequence of the form

0 → O

Q

n

(a)

s

→ Σ

∗

(n − 1)

t

→ E → 0,

for any 0 ≤ a ≤ n − 1, is an (O

Q

n

(a), Σ

∗

(n − 1))-Steiner bundle (see [MRS09]).

(c) Any exact sequence of vector bundles on the Grassmann variety G := G(r − 1, P(V )) of

the form

0 → U

s

→ O

t

G

→ E → 0,

where U denotes the rank r universal subbundle of G, deﬁnes a (U, O

G

)-Steiner bundle E

on G. These bundles were studied by Arrondo and Marchesi in [AM14].

(d) Let X =

f

P

2

be the blow-up of P

2

at three points p

1

, p

2

and p

3

. Let K

X

= −3L + E

1

+

E

2

+ E

3

denote the canonical divisor, where L is the divisor corresponding to a line not

passing through any of the three points, and E

i

is the exceptional divisor of the blow-up

at the point p

i

, i = 1, 2, 3. Take F

0

= −2L + E

1

+ E

2

+ E

3

and F

1

= O

X

.

Observe that H

0

(F

∨

0

) is globally generated and is the set of conics that passes through

three points. In particular, h

0

(F

∨

0

) = 3 = dim X + 1.

Let us now prove that the pair of vector bundles (F

0

, F

1

) is strongly exceptional. Since

both F

0

and F

1

are line bundles on a projective variety, the fact that Hom(F

0

, F

0

) =

Hom(O

X

, O

X

) = C and Ext

i

(F

0

, F

0

) = Ext

i

(O

X

, O

X

) = 0, i = 1, 2, is straightforward.

Using Riemann-Roch formula we obtain χ(F

∨

0

) = 3 and thus h

1

(F

∨

0

) = h

2

(F

∨

0

). Since

H

2

(F

∨

0

) = H

0

(K

X

+F

0

)

∗

= H

0

(−5L+2E

1

+2E

2

+2E

3

)

∗

= 0 we get that Ext

i

(F

0

, O

X

) =

H

i

(F

∨

0

) = 0, for i = 1, 2.

From the fact that H

0

(F

∨

0

) 6= 0 and Hom(F

0

, F

0

) = H

0

(F

0

⊗ F

∨

0

) 6= 0 it follows that

Hom(O

X

, F

0

) must be trivial. Furthermore, Ext

2

(O

X

, F

0

) = H

2

(F

0

) = H

0

(−F

0

+K

X

)

∗

=

H

0

(−L)

∗

= 0. Then, it also holds Ext

1

(O

X

, F

0

) = H

1

(F

0

) = 0, for we have χ(F

0

) = 0.

So, we have just proved that any vector bundle E ﬁtting in a sequence of type

0 → F

s

0

→ O

t

X

→ E → 0

is an (F

0

, O

X

)-Steiner bundle on the blow-up X.

The following proposition gives a characterization of (F

0

, F

1

)-Steiner bundles on a smooth

projective variety X by means of linear algebra (recall also Lemma 1.2 in [Arr10] or Lemma 1.7

in [AM14]). This interpretation will play an essential role for studying Schwarzenberger bundles

on X.

Proposition 1.5. To give an (F

0

, F

1

)-Steiner bundle on a smooth projective variety X

0 → S ⊗ F

0

→ T ⊗ F

1

→ E → 0,

4

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##### References

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Abstract: Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector bundles on $P^n$. It turns out that all such bundles are stable. The study of jumping lines of these bundles gives a unified treatment of several classical constructions associating a curve to a collection of points in $P^n$. The main result of the paper is \"Torelli theorem\" which says that the collection of hyperplanes can be recovered from the isomorphism class of the corresponding logarithmic bundle unless the hyperplanes ocsulate a rational normal curve. In this latter case our construction reduces to that of secant bundles of Schwarzenberger.

103 citations

•

Abstract: We study some properties of the natural action of SLOV0U SLOVpU on non- degenerate multidimensional complex matrices A A POV0 n n VpU of boundary format (in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non-stable ones as the matrices which are in the orbit of a ''triangular'' matrix, and the matrices with a stabilizer containing C as those which are in the orbit of a ''diagonal'' matrix. For pa 2i t turns out that a non-degenerate matrix A A POV0 n V1 n V2U detects a Steiner bundle SA (in the sense of Dolgachev and Kapranov) on the projective space P n , na dimOV2Uˇ1. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SLO2U and that the SLO2U-invariant Steiner bundles are exactly the bundles introduced by Schwarzen- berger (Schw), which correspond to ''identity'' matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of AutOP n U, answering in the first nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of SA (counting multiplicities) produces an interesting discrete invariant of A, which can take the values 0; 1; 2; .. .; dim V0a 1o ry; the y case occurs if and only if SA is Schwarzenberger (and A is an identity). Finally, the Gale transform for Steiner bun- dles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.

38 citations

••

Abstract: Soit
${\cal S}_{n,k}$
la famille des fibres de Steiner S sur
${\bf P}_n$
definis par une suite exacte (
$k>0$
)
\[ 0\rightarrow kO_{{\bf P}_n}(-1) \longrightarrow (n+k)O_{{\bf P}_n}\longrightarrow S \rightarrow 0 \] Nous montrons le resultat suivant : Soient
$S\in{\cal S}_{n,k}$
et
$H_1,\cdots,H_{n+k+2}$
des hyperplans distincts tels que
$h^0(S^{\vee}_{H_i})
eq 0$
. Alors il existe une courbe rationnelle normale
$C_n\subset{\bf P}_{n}^{\vee}$
telle que
$H_{i}\in C_n$
pour
$i=1, ..., n+k+2$
et
$S\simeq E_{n+k-1}(C_n)$
, ou
$E_{n+k-1}(C_n)$
est le fibre de Schwarzenberger sur
${\bf P}_n$
appartenant a
${\cal S}_{n,k}$
associea la courbe
$C_n\subset{\bf P}_{n}^{\vee}$
. On en deduit qu'un fibre de Steiner
$S\in{\cal S}_{n,k}$
, s'il n'est pas un fibre de Schwarzenberger, possede au plus (n+k+1) hyperplans instables; ceci prouve dans tous les cas un resultat de Dolgachev et Kapranov ([DK], thm. 7.2) concernant les fibres logarithmiques.

32 citations