Speaker:  Daoji Huang, IAS
Title: Fine multidegree, universal Gröbner basis, and matrix Schubert varieties
Time: 2:30 PM, Monday, November 25, 2024
Place:  Malott 206

Abstract: A universal Gröbner basis of an ideal is a Gröbner basis under any term order. While general results guarantees the existence of universal Gröbner basis of an ideal, few examples are known. We give a criterion for a collection of polynomials to be a universal Gröbner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give simple proofs of several existing results on universal Gröbner bases. 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 use our criterion to give a universal Gröbner basis for the ideal of the matrix Schubert variety of such a permutation. This is based on joint work with Matt Larson.


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