Special rank two vector bundles over Enriques surfaces
type: Item
instance of: scholarly article
title: Special rank two vector bundles over Enriques surfaces
zbMATH Open document ID: 0808.14027
DOI: 10.1007/BF01450488
author: Daniel Naie
published in: Mathematische Annalen
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165254
review text: Let \(\pi : X \to S\) with \(S\) an Enriques surface over \(\mathbb{C}\), \(X\) the K3-universal cover and \(\sigma\) the corresponding involution. A simple rank two vector bundle \(E\) over \(S\) is called special if \(E\simeq E\otimes[K_ S]\), where \([K_ S]\) is the canonical line bundle of \(S\). If \(E\) is special, then there is a line bundle \(L\) over \(X\) which is not \(\sigma\)-invariant such that: \((*)\) \(E = \pi_ * L\). This is the key remark of the paper. In particular the bundle is stable and \(c_ 2 (\text{End} E) = c^ 2_ 1(E) - 4c_ 2(E) = - 2k\), \(k \geq 1\), \(k \in \mathbb{Z}\). For \(k=1\) the bundle is extremal. In the paper it is shown that any extremal rank two vector bundle over an Enriques surface is a Reye bundle, i.e. \(S \subset G(1, \mathbb{P}^ n)\) being homologous to \((3n + 2) \sigma' + n \sigma''\), where \(\sigma'\) and \(\sigma''\) are Schubert cycles, and the extremal bundle is the restriction of the dual of the tautological bundle. The proof is given for special bundles using \((*)\): the line bundle \(L\) modulo tensoring with very ample line bundles, gives an embedding of \(X\) in \(\mathbb{P}^ n\). \(S\) appears as the complex of lines \(\{Sp (x, \sigma (x)) | x \in X \subset \mathbb{P}^ n\}\). The special bundles are shown to represent exactly the singularities of the moduli spaces of stable rank two vector bundles over Enriques surfaces. Furthermore, these singular points are ordinary double points. For the proof one uses again \((*)\). At the end the moduli spaces of dimension one are considered. They are birationally isomorphic with curves of arithmetic genus one.
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 704907
zbMATH Keywords: K3-universal cover
zbMATH Keywords: extremal rank two vector bundle over Enriques surface
zbMATH Keywords: Reye bundle
zbMATH Keywords: moduli spaces
MaRDI profile type: Publication
cites work: Q4852361
cites work: Smooth Enriques surfaces in \(\mathbb{P}^ 4\) and exceptional bundles
cites work: Reye Constructions for Nodal Enriques Surfaces
cites work: Q4195061
cites work: Residues and zero-cycles on algebraic varieties
cites work: Exceptional bundles on nodal Enriques surfaces
cites work: On deformations of complex analytic structures. I
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Chamber structures of algebraic surfaces with Kodaira dimension zero and moduli spaces of stable rank-two bundles
cites work: Projective Models of K - 3 Surfaces
cites work: Stable vector bundles on algebraic surfaces II
cites work: SPECIAL 0-CYCLES ON A POLARIZED K3 SURFACE
description: scientific article; zbMATH DE number 704907