Stability of the restriction of a rank 2 fibre bundle on an algebraic surface to a smooth curve
type: Item
instance of: scholarly article
title: Stability of the restriction of a rank 2 fibre bundle on an algebraic surface to a smooth curve
zbMATH Open document ID: 0834.14023
DOI: 10.1007/BF01446284
author: Cristian Anghel
published in: Mathematische Annalen
publication date: 1996-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/165394
review text: The aim of this work is to find a criterion for the stability of the restriction of a rank 2 vector bundle to a smooth curve embedded in an algebraic surface \(S\). If the bundle \(E\) is given as a nontrivial extension \[ 0 \to {\mathcal O}_S \to E \to L {\mathcal J}_Z \to 0, \tag{1} \] and if \(E |_C\) is unstable, with \(C\) satisfying certain numerical conditions, we obtain the existence of a curve \(F\) containing a subscheme of \(Z\) such that \(F^2\) and \(F \cdot C\) are bounded by two explicit quantities depending on \(E\). Then this result is applied to an abelian surface with \(\text{NS} (S) = \mathbb{Z} H\): Let \({\mathcal M}_m\) be the moduli space of rank 2 stable vector bundles over \(S\) with \(c_1 = 0\) and \(c_2 = m\). The restriction to a smooth curve in \(|nH |\) of a generic bundle from a component satisfying a property expressed in terms of the extension (1), is stable if \(n \geq 2 \sqrt {{m \over H^2}} + 5\).
reviewed by: Cristian Anghel
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 840213
zbMATH Keywords: stability of the restriction of a rank 2 vector bundle
zbMATH Keywords: abelian surface
MaRDI profile type: Publication
cites work: Q3944702
cites work: Residues and zero-cycles on algebraic varieties
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Vector bundles of rank 2 and linear systems on algebraic surfaces
OpenAlex ID: W2071999890
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description: scientific article; zbMATH DE number 840213