Orbits in (Pr)n and equivariant quantum cohomology

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 ea...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2020-03, Vol.362, p.106951, Article 106951
Main Authors: Lee, Mitchell, Patel, Anand, Spink, Hunter, Tseng, Dennis
Format: Article
Language:English
Subjects:
Equivariant intersection theory
Matroid
Quantum cohomology
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2019.106951