On the moduli of reflexive sheaves on a surface with rational double points
type: Item
instance of: scholarly article
title: On the moduli of reflexive sheaves on a surface with rational double points
zbMATH Open document ID: 0786.14023
DOI: 10.1007/BF01934318
author: Akira Ishii
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164994
review text: The author studies stable reflexive sheaves on a surface \(X\) with (at most) rational double points. As motivation for this one should think of a surface of general type: The canonical bundle of such a surface is ample on its canonical model, which is a surface with (at most) rational double points, and hence it is natural to compare stable sheaves on the canonical model with sheaves on the surface stable with respect to an ample divisor ``close'' to the canonical divisor. The paper contains two main results. First the author shows that there exists a stable reflexive sheaf with given rank, first Chern class and type of reflexive module at each singular point, provided the second Chern class is sufficiently large. Secondly he analyzes the relation between the moduli space on \(X\) and that on its minimal desingularization \(\tilde X\). Given a reflexive sheaf \(E\) on \(X\) one associates to \(E\) a locally-free sheaf \(\tilde E\) on \(\tilde X\) whose push-forward is isomorphic to \(E\). The author analyzes the moduli spaces of sheaves on \(X\) and \(\tilde X\) near \([E]\) and \([\tilde E]\) respectively, under the assumption that the rank is two, that the moduli space of sheaves on \(\tilde X\) is smooth near \([\tilde E]\), and that \(c_ 1(\tilde E|_ U)=-Z\), where \(Z\) is the fundamental cycle of one singular point \(P\), and \(U\) is a neighborhood of \(Z\). In this case the moduli space on \(X\) has a singularity at \([E]\) which looks just like the singularity of \(X\) at \(P\).
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 126361
zbMATH Keywords: large second Chern class
zbMATH Keywords: sheaves on a surface
zbMATH Keywords: rational double points
zbMATH Keywords: canonical model
zbMATH Keywords: moduli space
zbMATH Keywords: minimal desingularization
reviewed by: Kieran G. O'Grady
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126361