Moduli spaces of vector bundles on higher-dimensional varieties.
type: Item
instance of: scholarly article
title: Moduli spaces of vector bundles on higher-dimensional varieties.
zbMATH Open document ID: 1085.14513
author: Laura Costa
author: Rosa M. Miró-Roig
published in: Michigan Mathematical Journal
publication date: 2002-05-20T00:00:00Z
review text: Let \(X\) be a \(\mathbb{P}^d\)-bundle over a smooth projective curve \(C\). The authors study the behaviour of moduli of \(\mu\)-stable vector bundles of rank two on \(X\) with increasing second Chern class. In contrast to the case of surfaces, where large \(c_2\) guarantees irreducibility and generical smoothness of such moduli spaces, the situation is usually much more complicated in higher dimensions, involving certain relations between the polarization and the Chern classes. However, for projective bundles there exist polarizations closely related to the second Chern class, providing sufficient assumptions to prove the desired results. More precisely, for second Chern classes fulfilling a certain condition, and a suitably chosen polarization on \(X\), a general point in the moduli space is an extension of two line bundles. By using this and the technically important result that for a suitable twist of a rank two vector bundle on \(X\) there exists a section with zero locus of codimension two, the authors prove the irreducibility and smoothness of the moduli spaces for such invariants, and additionally the rationality of these spaces if \(C\) is rational. In the last section, an example for a change of the polarization is discussed.
reviewed by: Olaf Teschke
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 1742564
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1012409973
OpenAlex ID: W1972161873
DOI: 10.1307/MMJ/1012409973
description: scientific article; zbMATH DE number 1742564