A simple proof of the formula for the Betti numbers of the quasihomogeneous Hilbert schemes

TL;DR

This paper provides a concise geometric proof for the formula describing the Betti numbers of quasihomogeneous Hilbert schemes, simplifying previous complex derivations and confirming the generating series' infinite product structure.

Contribution

It offers a new, straightforward geometric proof of the Betti number formula for quasihomogeneous Hilbert schemes, enhancing understanding and accessibility.

Findings

Confirmed the infinite product structure of the generating series

Provided a simplified geometric proof of the Betti number formula

Validated previous results with a more direct approach

Abstract

In a recent paper the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper we give a very short geometrical proof of that formula.