Orbits in [formula omitted] and equivariant quantum cohomology

Abstract

We compute the GLr+1-equivariant Chow class of the GLr+1-orbit closure of any point (x1,…,xn)∈(Pr)n in terms of the rank polytope of the matroid represented by x1,…,xn∈Pr. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each d×n matrix M an n-ary operation [M]ħ on the small equivariant quantum cohomology ring of Pr, which is the n-ary quantum product when M is an invertible matrix. We prove that M↦[M]ħ 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 Pr, generalizing the known case of quintic plane curves.