Porteous's formula for maps between coherent sheaves
type: Item
instance of: scholarly article
title: Porteous's formula for maps between coherent sheaves
zbMATH Open document ID: 1078.14072
author: Steven P. Diaz
published in: Michigan Mathematical Journal
publication date: 2005-03-07T00:00:00Z
review text: Let \(\sigma:E\to F\) be a homomorphism of coherent sheaves on an algebraic variety \(X\) and let \(Y\) be the locus where either \(E\) or \(F\) fails to be locally free. Blowing up the variety \(X\setminus Y\) twice in Fitting ideals the author obtains a morphism \(g:Z\to X\) that gives an isomorphism between \(X\setminus Y\) and \(g^{-1}(X\setminus Y)\), and such that the restrictions of \(E\) and \(F\) to \(X\setminus Y\) extend to vector bundles \(E'\) and \(F'\) on \(Z\), and \(\sigma\) extends to a homomorphism \(\sigma':E'\to F'\). The goal is to use the Porteous formula for \(\sigma'\) and the explicit form of the map \(g\) to study the degeneracy of the homomorphism \(\sigma\). The construction is used to treat an example of moduli spaces of curves of genus \(3\) from the book of \textit{J. Harris} and \textit{I. Morrison} [``Moduli of curves'' (1998; Zbl 0913.14005)], which is the origin of the question of the existence of a Porteous type formula for coherent sheaves.
reviewed by: Dan Laksov
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 2141487
zbMATH Keywords: Thom-Porteous formula
zbMATH Keywords: coherent sheaves
zbMATH Keywords: degeneracy class
zbMATH Keywords: Chow group
zbMATH Keywords: residual intersection
zbMATH Keywords: Deligne-Mumford compactification
zbMATH Keywords: Fitting ideals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1100623410
OpenAlex ID: W2028547035
cites work: Q4317713
cites work: Q3328651
cites work: Q4398861
cites work: On the Jacobian Ideal of the Module of Differentials
DOI: 10.1307/MMJ/1100623410
description: scientific article; zbMATH DE number 2141487