Worst Unstable Points of a Hilbert Scheme

TL;DR

This paper characterizes the most unstable points in Hilbert schemes for specific polynomials and spaces, analyzing their geometric properties and stability conditions.

Contribution

It introduces a description of worst unstable points in Hilbert schemes using Gotzmann monomial sets, linking stability and geometric features.

Findings

Worst unstable points often fail to be K-stable.

These points can have maximal regularity.

The geometric structure of these points is explicitly described.

Abstract

In this paper, we describe the worst unstable points of a Hilbert scheme for some special Hilbert polynomials and ambient spaces using Murai's work on Gotzmann monomial sets. We investigate the geometry of the projective schemes represented by worst unstable Hilbert points and see that in certain cases that they fail to be $K$-stable or attain maximal regularity.