2018/05/21 by Mitchell Lee, Lee, Mitchell, Anand Patel +5
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1805.08181
65 pages, exposition heavily revised, to appear in Advances in Mathematics
openalex publication_date 2018/05/21 · arxiv created 2019/12/12 · arxiv updated 2019/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.