The matroid stratification of the Hilbert scheme of points on P^1

TL;DR

This paper investigates the matroid stratification of the Hilbert scheme of points on P^1, revealing its structure through examples, visualization techniques, and connections to Schur polynomials, with applications to the T-graph problem.

Contribution

It introduces a detailed study of matroid stratifications on Hilbert schemes, especially for points on P^1, and links these structures to Schur polynomials and the T-graph problem.

Findings

Matroid stratification in the Hilbert scheme of points on P^1 is generated by Schur polynomials.

Provided visualization methods for matroid strata.

Established the existence of an infinite class of edges in the T-graph problem.

Abstract

Given a homogeneous ideal $I$ in a polynomial ring over a field, one may record, for each degree $d$ and for each polynomial $f\in I_d$, the set of monomials in $f$ with nonzero coefficients. These data collectively form the tropicalization of $I$. Tropicalizing ideals induces a "matroid stratification" on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points $(\mathbb{P}^1)^{[k]}$ is generated by all Schur polynomials in $k$ variables. We end with an application to the $T$-graph problem of $(\mathbb{A}^2)^{[n]}$; classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges.