Murphy's Law for Hilbert function strata in the Hilbert scheme of points

TL;DR

This paper demonstrates that loci in the Hilbert scheme of points with a fixed Hilbert function satisfy Murphy's Law, indicating their complex and unpredictable deformation behavior.

Contribution

It shows that the Hilbert function strata in the Hilbert scheme of points exhibit Murphy's Law, revealing their intricate deformation properties, which was previously unknown.

Findings

Hilbert function strata satisfy Murphy's Law

Loci parametrizing punctual schemes are highly singular

Results extend to equivariant deformations of curve singularities

Abstract

An open question is whether the Hilbert scheme of points of a high dimensional affine space satisfies Murphy's Law, as formulated by Vakil. In this short note, we instead consider the loci in the Hilbert scheme parametrizing punctual schemes with a given Hilbert function, and we show that these loci satisfy Murphy's Law. We also prove a related result for equivariant deformations of curve singularities with $\mathbb G_m$-action.