Determinantal varieties and symmetric polynomials
type: Item
instance of: scholarly article
title: Determinantal varieties and symmetric polynomials
zbMATH Open document ID: 0633.14029
DOI: 10.1007/BF02577147
author: Piotr Pragacz
publication date: 1987-01-01T00:00:00Z
review text: The paper is a research announcement of some of the author's recent results concerning degeneracy loci. Let X be a scheme over a field and \(\phi: F\to E\) a morphism of vector bundles over X. For every \(r\geq 0\) the degeneracy locus of rank r associated with \(\phi\) is defined as \(D_ r(\phi)=\{x\in X,\quad rk(\phi (x))\leq r\}.\) In analogy with the general case it is interesting to consider the situation: \(F=E^{\vee}\) and \(\phi\) is symmetric (resp. antisymmetric). Using the classical Schur S- and Q-polynomials, we describe the ideal of all polynomials in the Chern classes of E and F which describe in a universal way all the cycles supported in \(D_ r(\phi)\). As an application we calculate the Chow groups and Chern numbers of determinantal varieties. The ideals that we construct yield also a generalization of the resultant of two polynomials in elimination theory. For a detailed account see ``Enumerative geometry of degeneracy loci'' (to appear in Ann. Sci.Éc. Norm. Supér.) and ``Algebra-geometric applications of Schur S- and Q-polynomials'' (to appear in Sém. Algèbre, Dubreil-Malliavin).
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4029750
zbMATH Keywords: degeneracy loci
zbMATH Keywords: Chow groups
zbMATH Keywords: Chern numbers of determinantal varieties
zbMATH Keywords: resultant
MaRDI profile type: Publication
cites work: Q3328651
cites work: Chern numbers of kernel and cokernel bundles. Appendix to ''On the Kodaira dimension of the moduli space of curves. II. The even-genus case'' by J. Harris
cites work: Q3947818
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 4029750