Compatibly split subvarieties of the Hilbert scheme of points in the plane

TL;DR

This paper investigates the stratification of the Hilbert scheme of points in the plane by compatibly Frobenius split subvarieties, providing explicit classifications for small n and conjectural descriptions for n=5, with algebraic and combinatorial insights.

Contribution

It classifies compatibly Frobenius split subvarieties of the Hilbert scheme for n ≤ 4 and offers a conjectural framework for n=5, including partial proofs and degenerations to Stanley-Reisner schemes.

Findings

Complete classification for n ≤ 4

Conjectural description for n=5 with partial validation

Explicit degenerations to Stanley-Reisner schemes

Abstract

Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert scheme of n points in the plane. In this thesis, we investigate the question, "what is the stratification of the Hilbert scheme of points in the plane by all compatibly Frobenius split subvarieties?" We provide the answer to this question when n is at most 4 and we give a conjectural answer when n=5. We prove that this conjectural answer is correct up to the possible inclusion of one particular one-dimensional subvariety of the Hilbert scheme of 5 points, and we show that this particular one-dimensional subvariety is not compatibly split for at least those primes p between 3 and 23. Next, we restrict the splitting of the Hilbert scheme of n points in the plane (now for arbitrary n) to the affine open patch U_<x,y^n> and describe all compatibly split subvarieties of this patch and their defining ideals. We find degenerations of these subvarieties to Stanley-Reisner schemes, explicitly describe the associated simplicial complexes, and use these complexes to prove that certain compatibly split subvarieties of U_<x,y^n> are Cohen-Macaulay.