Computable Bia{\l}ynicki-Birula decomposition of the Hilbert scheme

TL;DR

This paper introduces a Bia{
42}ynicki-Birula decomposition of the Hilbert scheme using Gr"obner schemes, enabling a computable homology formula and insights into smooth points.

Contribution

It presents a novel decomposition of the Hilbert scheme into cells corresponding to Gr"obner schemes, facilitating explicit homology calculations and smoothness criteria.

Findings

Decomposition of Hilbert scheme into Gr"obner scheme cells.

Homology formula for smooth Hilbert schemes.

Smoothness of Gr"obner schemes at smooth points.

Abstract

We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gr\"obner scheme. We introduce a Bia{\l}ynicki-Birula decomposition of the Hilbert scheme $\mathrm{Hilb}^{P}_n$ for any Hilbert polynomial $P$ such that the cells are the Gr\"obner schemes in set-theoretically. Then we obtain a computable homology formula for smooth Hilbert schemes. As a corollary of our argument, we show that the Gr\"obner scheme for a monomial ideal defining a smooth point in the Hilbert scheme is smooth.