Rank two bundles on the blow-up of \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Rank two bundles on the blow-up of \(\mathbb{C}^2\)
zbMATH Open document ID: 0902.14030
author: Elizabeth Gasparim
published in: Journal of Algebra
publication date: 1998-12-10T00:00:00Z
review text: The author considers the problem of the classification of rank 2 vector bundles with trivial determinant on the blow-up of \(\mathbb{C}^2\) at the origin. For each integer \(j \geq 0\), let \(M_j\) be the set of isomorphism classes of such bundles whose restriction to the exceptional divisor is of the form \({\mathcal O} (j)\oplus {\mathcal O} (-j)\). By elementary arguments, the author shows that the transition matrix of a bundle in \(M_j\) has a canonical form. Using this result, one gets immediately that \(M_0\) and \(M_1\) consist of one point. The author gives a precise description of \(M_2\). In general, \(M_j\) is the quotient of \(\mathbb{C}^N\) by an equivalence relation and looks, generically, like \(\mathbb{P}^{2j-3} \setminus\) a closed subset of codimension \(\geq 2\). Furthermore, \(M_j\) with the quotient topology is not Hausdorff: For example, \({\mathcal O} (j) \oplus {\mathcal O} (-j)\) is arbitrarily closed to any other bundle in \(M_j\).
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1117783
zbMATH Keywords: blow-up of \(\mathbb{C}^2\)
zbMATH Keywords: classification of rank 2 vector bundles
zbMATH Keywords: exceptional divisor
zbMATH Keywords: transition matrix
reviewed by: Iustin Coandă
MaRDI profile type: Publication
full work available at URL: jabr.1997.7156
OpenAlex ID: W2075921490
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DOI: 10.1006/JABR.1997.7156
description: scientific article; zbMATH DE number 1117783