Fine multidegrees, universal Grobner bases, and matrix Schubert varieties

TL;DR

This paper establishes a criterion for universal Gröbner bases based on multidegrees, introduces fine Schubert polynomials for matrix Schubert varieties, and computes these polynomials for specific permutations to identify universal Gröbner bases.

Contribution

It provides a new multidegree-based criterion for universal Gröbner bases, introduces fine Schubert polynomials, and computes them for certain permutations to find universal Gröbner bases.

Findings

Criterion for universal Gröbner bases using multidegrees

Introduction of fine Schubert polynomials for matrix Schubert varieties

Explicit computation of fine Schubert polynomials for permutations with 0/1 coefficients

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.