Refined Hilbert schemes, E-polynomials, and the number of generators of finite colength ideals in the plane

TL;DR

This paper derives an elegant formula for the E-polynomials of strata in the punctual Hilbert scheme of points on the affine plane, related to the number of generators of ideals, advancing understanding of their geometric structure.

Contribution

It provides a new explicit formula for the E-polynomials of strata in the Hilbert scheme based on the number of generators of ideals.

Findings

Derived an explicit formula for E-polynomials of strata

Connected stratification to the number of generators of ideals

Enhanced understanding of the geometric structure of Hilbert schemes

Abstract

The study of the stratification associated to the number of generators of the ideals in the punctual Hilbert scheme of points on the affine plain goes back to the '70s. In this paper, we present an elegant formula for the E-polynomials of these strata.