The commuting scheme and generic pipe dreams

Abstract: The space of pairs of commuting matrices is more mysterious than you might think -- in particular, Hochster's 1984 conjecture that it is reduced remains unresolved. I'll explain how to degenerate it to one component of the "lower-upper scheme" {(X,Y) : XY lower triangular, YX upper triangular}, a reduced complete intersection, and how to compute the degree of any component as a sum over "generic pipe dreams". As a consequence, this recovers both the "pipe dream" and "bumpless pipe dream" formulae for double Schubert polynomials. Some of this work is joint with Paul Zinn-Justin.

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar