When are multidegrees positive?

Abstract: Multidegrees of multiprojective varieties extend the notion of degree of projective varieties. These invariants can be defined via intersection theory, or algebraically as the leading coefficients of multivariate Hilbert polynomials. It follows that multidegrees are nonnegative integers, so a fundamental question is: When are multidegrees positive? In the first part of the talk, I will survey definitions and key properties of degrees and multidegrees, including some examples.

In the second part, I will present a complete characterization of the positivity of multidegrees, and establish a combinatorial description using convex geometry. I will also show applications of our result to mixed multiplicities of ideals and to the support of Schubert polynomials. The talk is based on joint work with F. Castillo, Y. Cid-Ruiz, B. Li, and N. Zhang.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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