Punctual Hilbert schemes of points of $\mathbb{A}^3$ in the Grothendieck group of varieties

TL;DR

This paper provides an explicit stratification of punctual Hilbert schemes of points in affine space, computes their classes in the Grothendieck group, and applies these results to specific cases in three dimensions.

Contribution

It introduces a new stratification method for punctual Hilbert schemes and calculates their classes in the Grothendieck group for low-dimensional cases.

Findings

Explicit stratification of punctual Hilbert schemes of points.

Calculation of classes in the Grothendieck group for $n extless=5$ in $\\mathbb{A}^3$.

Application to understanding the structure of Hilbert schemes in algebraic geometry.

Abstract

We give an explicit stratification of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the Grothendieck group of varieties. As an application, we calculate the classes of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^3$ and the classes of the Hilbert schemes of $n$ points of $\mathbb{A}^3$ in the Grothendieck of varieties for $n\leq 5$.