The moduli spaces of some rank-2 stable vector bundles over algebraic \(K\)3-surfaces
type: Item
instance of: scholarly article
title: The moduli spaces of some rank-2 stable vector bundles over algebraic \(K\)3-surfaces
zbMATH Open document ID: 0774.14031
DOI: 10.1215/S0012-7094-91-06420-3
author: Kang Zuo
published in: Duke Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: The author proves that certain moduli spaces of stable rank-two vector- bundles on a \(K3\)-surface \(S\) are birational to symmetric products of \(S\). The conditions on the vector bundles \(F\) whose moduli are being considered, are the following: 1. \(\text{det} F\cong{\mathcal O}_ S\), and 2. \(c_ 2(F)=H^ 2n^ 2+3\) for some positive integer \(n\), where \(H\) is the polarization of \(S\). The result is proved as follows. By Riemann-Roch and a dimension count, if \(F\) is the generic \(H\)-stable rank-two vector bundle as above, then there exists a unique exact sequence \[ 0\to{\mathcal O}_ S\to F(nH)\to I_ Z(2nH)\to 0, \] where \(Z\) is a zero-dimensional subscheme of \(S\) of length \(2H^ 2n^ 2+3\). Conversely, if \(Z\) is a generic such subscheme, there exists a unique exact sequence as above. This gives the result.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 58554
zbMATH Keywords: moduli spaces of stable rank-two vector-bundles on a \(K3\)-surface
reviewed by: Kieran G. O'Grady
MaRDI profile type: Publication
cites work: Q3322802
cites work: Q5749404
cites work: Global moduli for surfaces of general type
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Cycles, curves and vector bundles on an algebraic surface
cites work: Weyl-type theorems and \(k\)-quasi-\(M\)-hyponormal operators
full work available at URL: s0012-7094-91-06420-3
OpenAlex ID: W1505756847
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description: scientific article; zbMATH DE number 58554