On singular Hilbert schemes of points: Local structures and tautological sheaves

TL;DR

This paper investigates the local structure of Hilbert schemes of up to 7 points in three-dimensional affine space, analyzing singularities and tautological sheaves, and verifies a related conjecture for small cases.

Contribution

It provides an intrinsic version of Thomason's fixed-point theorem and determines singularity types and Euler characteristics for small Hilbert schemes.

Findings

Identified singularity types for points with the same extra dimension.

Computed equivariant Hilbert functions at singularities.

Verified Zhou's conjecture for up to 6 points.

Abstract

We show an intrinsic version of Thomason's fixed-point theorem. Then we determine the local structure of the Hilbert scheme of at most $7$ points in $\mathbb{A}^3$. In particular, we show that in these cases, the points with the same extra dimension have the same singularity type. Using these results, we compute the equivariant Hilbert functions at the singularities and verify a conjecture of Zhou on the Euler characteristics of tautological sheaves on Hilbert schemes of points on $\mathbb{P}^3$ for at most $6$ points.