Fine multidegrees, universal Grobner bases, and matrix Schubert varieties

Abstract: 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 can be used to give simple proofs of several existing results on universal Gr\"{o}bner 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, and we use this to give a universal Gr\"{o}bner basis for the ideal of the matrix Schubert variety of such a permutation.