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Journal ArticleDOI

Schwarzenberger bundles on smooth projective varieties

TL;DR: Schwarzberger bundles on smooth projective varieties were introduced in this article, and the notions of jumping subspaces and jumping pairs of (F 0, O X ) -Steiner bundles were defined.
About: This article is published in Journal of Pure and Applied Algebra.The article was published on 2016-09-01 and is currently open access. It has received 2 citations till now.

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.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, the existence of monads on the projective space was shown to be irreducible under certain conditions on the cohomology sheaf of a monad.
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.

5 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of monads on the projective space was shown to be irreducible under certain conditions on the cohomology sheaf of a monad.
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
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Journal ArticleDOI
TL;DR: In this paper, the authors study the bundles of logarithmic 1-forms corresponding to such divisors from the point of view of classification of vector bundles on $P^n.
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.

110 citations

Posted Content
TL;DR: In this article, it was shown that the symmetry group of a Steiner bundle is contained in SLO2U and that the Steiner bundles are exactly the bundles introduced by Schwarzen- berger (Schw), which correspond to ''identity'' matrices.
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.

39 citations

Journal ArticleDOI
TL;DR: In this article, Dolgachev et Kapranov et al. present a suite exacte result on the fibres logarithmiques of Steiner's fibres, showing that (n+k+1) hyperplans distincts are (n +k+2) infeasible.
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.

38 citations

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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.