1. Fine multidegrees of matrix Schubert varieties
- Author
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Huang, Daoji and Larson, Matt
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Commutative Algebra ,Mathematics - Combinatorics ,14M15, 13P10, 13C40, 14M12, 05E14 - Abstract
We introduce fine Schubert polynomials, which record the multidegrees of the closures of matrix Schubert varieties in $(\mathbb{P}^1)^{n^2}$. We compute the fine Schubert polynomials of permutations $w$ where the coefficients of the Schubert polynomials of $w$ and $w^{-1}$ are all either 0 or 1. We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^n$. This criterion gives simple proofs of several existing results on universal Gr\"{o}bner bases. We use this criterion to give a universal Gr\"{o}bner basis for the ideal of a matrix Schubert variety of a permutation $w$ where the coefficients of the Schubert polynomial of $w$ and $w^{-1}$ are all either 0 or 1.
- Published
- 2024