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Determinantal varieties and symmetric polynomials. (English. Russian original) Zbl 0633.14029

Funct. Anal. Appl. 21, No. 1-3, 249-250 (1987); translation from Funkts. Anal. Prilozh. 21, No. 3, 89-90 (1987).
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).
Reviewer: P.Pragacz

MSC:

14M12 Determinantal varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C05 Parametrization (Chow and Hilbert schemes)
57R20 Characteristic classes and numbers in differential topology
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References:

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