Schubert Varieties and Degeneracy Loci

William Fulton, Piotr Pragacz (auth.)3540645381, 9783540645382

Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, from a summer school in Thurnau, aim to give an introduction to these topics, and to describe recent progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersection theory and algebraic geometry.

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Table of contents :
Introduction to degeneracy loci and schubert polynomials….Pages 1-13
Modern formulation; Grassmannians, flag varieties, schubert varieties….Pages 14-25
Symmetric polynomials useful in geometry….Pages 26-39
Polynomials supported on degeneracy loci….Pages 40-52
The Euler characteristic of degeneracy loci….Pages 53-64
Flag bundles and determinantal formulas for the other classical groups….Pages 65-78
$$tilde P{text{ – }}$$ and $$tilde Q{text{ – }}$$ polynomial formulas for other classical groups….Pages 79-91
The classes of Brill-Noether loci in Prym varieties….Pages 92-96
Applications and open problems….Pages 97-103