Gr\"obner strata in the Hilbert scheme of points

TL;DR

This paper develops a framework for studying Gr"obner bases over arbitrary rings, representing associated functors as schemes within the Hilbert scheme of points, and analyzes their geometric properties.

Contribution

It introduces a scheme-theoretic approach to Gr"obner bases over rings, showing representability, connectedness, and the structure of strata within the Hilbert scheme.

Findings

The functor for reduced Gr"obner bases is representable.

Schemes parametrizing Gr"obner bases are connected.

The Hilbert scheme decomposition into Gr"obner strata is not a stratification.

Abstract

The present paper shall provide a framework for working with Gr\"obner bases over arbitrary rings $k$ with a prescribed finite standard set $\Delta$. We show that the functor associating to a $k$-algebra $B$ the set of all reduced Gr\"obner bases with standard set $\Delta$ is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a $k$-algebra $B$ the set of all border bases with standard set $\Delta$ and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gr\"obner bases are connected; give a connectedness criterion for the schemes parametrizing border bases; and prove that the decomposition of the Hilbert scheme of points into the locally closed strata parametrizing Gr\"obner bases is not a stratification.