scispace - formally typeset

Journal ArticleDOI

Schwarzenberger bundles on smooth projective varieties

01 Sep 2016-Journal of Pure and Applied Algebra (North-Holland)-Vol. 220, Iss: 9, pp 3307-3326

AbstractWe 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)

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.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

Repositório ISCTE-IUL
Deposited in
Repositório ISCTE-IUL
:
2019-04-09
Deposited version:
Post-print
Peer-review status of attached file:
Peer-reviewed
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
https://dx.doi.org/10.1016/j.jpaa.2016.02.016. This article may be used for non-commercial
purposes in accordance with the Publisher's Terms and Conditions for self-archiving.
Use policy
Creative Commons CC BY 4.0
The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or
charge, for personal research or study, educational, or not-for-profit purposes provided that:
• a full bibliographic reference is made to the original source
• a link is made to the metadata record in the Repository
• the full-text is not changed in any way
The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
Serviços de Informação e Documentação, Instituto Universitário de Lisboa (ISCTE-IUL)
Av. das Forças Armadas, Edifício II, 1649-026 Lisboa Portugal
Phone: +(351) 217 903 024 | e-mail: administrador.repositorio@iscte-iul.pt
https://repositorio.iscte-iul.pt

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

Citations
More filters

Journal ArticleDOI
Abstract: We generalise Floystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.

2 citations


Journal ArticleDOI
Abstract: We generalise Floystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.

2 citations


References
More filters


Journal ArticleDOI
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


Posted Content
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


Journal ArticleDOI
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



Frequently Asked Questions (1)
Q1. What are the contributions in "Schwarzenberger bundles on smooth projective varieties" ?

The authors define Schwarzenberger bundles on smooth projective varieties and they introduce the notions of jumping subspaces and jumping pairs of ( F0, OX ) -Steiner bundles.