Intersection theory of matroids: variations on a theme

Abstract

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and Allermann--Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh--Katz's formula ${\mu }^k(M)=de{g}_M({\alpha }^{r-k}{\beta }^k)$ for the coefficients of the reduced characteristic polynomial of a matroid $M$ as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes $\alpha$ and $\beta$ in the Chow ring of $M$. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory.

30 pages. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference. Comments are welcome