The versal deformation space of a reflexive module on a rational cone
type: Item
instance of: scholarly article
title: The versal deformation space of a reflexive module on a rational cone
zbMATH Open document ID: 1060.14005
author: Trond Stølen Gustavsen
author: Runar Ile
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
full work available at URL: URN:NBN:no-23783
review text: The versal deformation space \(R\) of a reflexive module on the cone over the rational normal curve of degree m is described. A resolution of each component of this space is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, the components are obtained by projection. In particular equations for the minimal stratum in the Chern class filtration of the versal deformation space are given. The components are classified, and the deformation relations are described combinatorially. A formula for the number of components is given. The interesting case studied in this article is \(\text{rk}M>1\) so that \(M\) is decomposable. Let \(X\) be a rational surface singularity and let \(\pi:\widetilde{X}\rightarrow X\) be a minimal resolution. Then a reflexive module \(M\) on \(X\) corresponds to a full sheaf \(\widetilde{M}\) on \(\widetilde{X}\). For each \(d\in\text{Pic}(\widetilde{X})\), \textit{A. Ishii} [Math. Ann. 317, 239--262 (2000; Zbl 1016.14018)] defined a functor of families parametrised by schemes over \(R_{\text{red}}\), of semi-full sheaves \(\mathcal{E}\) on \(\widetilde{X}\) with an isomorphism. This functor is represented by a regular scheme \(F^d\) which is projective over \(R_{\text{red}}\). As \(d\) varies, a finite stratification \(\coprod S^d\) of \(R_{\text{red}}\) is obtained such that if the fibre of the versal family at \(t\in R\) is the reflexive module \(N\), then \(t\in S^d\) if and only if the full sheaf \(\widetilde{N}\) has Chern class \(d\). Each component in \(R_{\text{red}}\) is given as the closure of an \(S^d\). By the McKay correspondence this gives the stratification by isomorphism classes if \(X\) is a rational double point. When \(X\) is the cone over the rational normal curve of degree \(m\), there are \(m\) isomorphism classes of r…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 2104455
zbMATH Keywords: reflexive modules on surfaces
zbMATH Keywords: deformation theory
zbMATH Keywords: stratification
MaRDI profile type: Publication
OpenAlex ID: W2018979381
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DOI: 10.1016/J.JALGEBRA.2004.03.018
description: scientific article; zbMATH DE number 2104455