Q1821155
type: Item
instance of: scholarly article
title: Groupe de Picard des variétés de modules de faisceaux semi-stable sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Picard group of the moduli varieties of semi- stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\)
zbMATH Open document ID: 0616.14006
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74805
review text: The subject of this paper is the Picard group of the moduli variety \(M(r,c_ 1,c_ 2)\) of semi-stable algebraic sheaves on \({\mathbb{P}}_ 2({\mathbb{C}})\), of \(rank\quad r\) and Chern classes \(c_ 1, c_ 2\). The first result is that if \(M(r,c_ 1,c_ 2)\) is locally factorial, so \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to the group of linear equivalence classes of Weil divisors of \(M(r,c_ 1,c_ 2)\). There is a unique map \(\delta: {\mathbb{Q}}\to {\mathbb{Q}}\) such that \(\dim (M(r,c_ 1,c_ 2))>0\) if and only if \((c_ 2-(r-1)c^ 2_ 1/2r)/r\geq \delta (c_ 1/r)\). Then if one has equality, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}\), and if the inequality is strict, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}^ 2\). A description of \(Pic(M(r,c_ 1,c_ 2))\) is given, using a subgroup of the Grothendieck group \(K({\mathbb{P}}_ 2)\) of \({\mathbb{P}}_ 2\). It is then possible to compute the canonical bundle \(\omega _ M\) of \(M(r,c_ 1,c_ 2)\).
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 3997961
zbMATH Keywords: Chern classes
zbMATH Keywords: Weil divisors
zbMATH Keywords: Picard group of moduli variety
zbMATH Keywords: factoriality of moduli variety
author: Jean-Marc Drézet
MaRDI profile type: Publication
OpenAlex ID: W2049465728
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DOI: 10.5802/AIF.1143
published in: Annales de l’institut Fourier
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label: Groupe de Picard des variétés de modules de faisceaux semi-stable sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Picard group of the moduli varieties of semi- stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\)
description: scientific article; zbMATH DE number 3997961