Porteous's formula for maps between coherent sheaves
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.
scientific article; zbMATH DE number 2141487