Orbits in $({\mathbb{P}}^r{)}^n$ and equivariant quantum cohomology

Abstract

We compute the $G{L}_{r+1}$-equivariant Chow class of the $G{L}_{r+1}$-orbit closure of any point $(x_1,\dots ,x_n)\in ({\mathbb{P}}^r{)}^n$ in terms of the rank polytope of the matroid represented by $x_1,\dots ,x_n\in {\mathbb{P}}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $d\times n$ matrix $M$ an $n$-ary operation $[M{]}_{\hslash }$ on the small equivariant quantum cohomology ring of ${\mathbb{P}}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M\mapsto [M{]}_{\hslash }$ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree $2r+1$ hypersurface in ${\mathbb{P}}^r$, generalizing the known case of quintic plane curves.

65 pages, exposition heavily revised, to appear in Advances in Mathematics