On vector bundles whose general sections have all projectively equivalent zero-loci
type: Item
instance of: scholarly article
title: On vector bundles whose general sections have all projectively equivalent zero-loci
zbMATH Open document ID: 0795.14012
author: E. Ballico
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1994-09-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108290
review text: By a well-known lemma of Serre, if a vector bundle \(E\) of rank \(r\) on a variety \(X\) is spanned by global sections, then there is a subspace \(W\) of \(H^ 0(X,E)\) with \(\dim W\leq r+\dim X\) such that \(W\) spans \(E\). The author studies here the case \(\dim W=4\), \(r=2\), \(X\) an integral projective surface. Let \(i\) be the morphism from \(X\) to the Grassmannian \(G(2,4)\subset \mathbb{P}^ 5\) associating to \(x\) in \(X\) the quotient \(E_ x\) of \(W\), \(E_ x=\) fibre of \(E\) at \(x\). Assume that (1) \(i\) is an embedding, \(i(X)\) is ordinary (i.e. reflexive with dual a hypersurface); (2) there is \(s\in W\) with zero scheme \(s_ 0\) union of \(C_ 2(E)-2\) smooth points of \(X\) (with reduced structure) and a length 2 scheme supported by another smooth point of \(X\); (3) all general \(s_ 0\) are reduced and projectively equivalent. The main result in the first part of the paper says that under these assumptions \(C_ 2(E)\leq 28\). The second part of the paper is devoted to plane curves \(C\) with all general tangent lines to \(C\) intersecting \(C\) in projectively equivalent subsets (under \(\Aut \mathbb{P}^ 1\)). It is shown that if \(\text{Card}(C\cap\ell)_{\text{red}})\geq 3\) for every line \(\ell\), then varying line \(\ell\) in \((\mathbb{P}^ 2)^*\) the projective equivalence classes of sets \(C\cap\ell\) vary in a two dimensional family.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 492563
zbMATH Keywords: vector bundles
zbMATH Keywords: global sections
zbMATH Keywords: zero locus
zbMATH Keywords: Chern class
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
cites work: Vector Bundles Over an Elliptic Curve
cites work: Vector bundles, reflexive sheaves and algebraic surfaces
cites work: On pairs of plane curvers with projectively equivalent linear sections
cites work: On Projective Varieties with Projectively Equivalent Zero-Dimensional Linear Sections
cites work: On the galois group associated to a generically etale morphism
cites work: Q3032389
cites work: Galois groups of enumerative problems
cites work: Q4749943
cites work: Q4775956
cites work: Q3735865
cites work: Wronskians and Plücker formulas for linear systems on curves
cites work: The uniform position principle for curves in characteristic p
cites work: Quotient spaces modulo reductive algebraic groups
Recommended article: Q4855283
Recommended article: Q4799922
Recommended article: Q3504487
Recommended article: On Projective Varieties with Projectively Equivalent Zero-Dimensional Linear Sections
Recommended article: Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles
Recommended article: Q4202172
Recommended article: Q4391147
Recommended article: Q4463401
Recommended article: Q4884054
Recommended article: Ample vector bundles with zero loci of sectional genus two
description: scientific article; zbMATH DE number 492563