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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Veröffentlicht in: | Advances in mathematics (New York. 1965) 2020-03, Vol.362, p.106951, Article 106951 |
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Sprache: | eng |
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Zusammenfassung: | 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. |
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ISSN: | 0001-8708 1090-2082 |
DOI: | 10.1016/j.aim.2019.106951 |