Gr\"obner scheme in the Hilbert scheme and complete intersection monomial ideals

TL;DR

This paper explores the structure of Gr"obner schemes within the Hilbert scheme, demonstrating their local closedness for saturated ideals and characterizing those that define complete intersections.

Contribution

It establishes that Gr"obner schemes are locally closed subschemes of the Hilbert scheme for saturated ideals and identifies conditions under which they correspond to complete intersections.

Findings

Gr"obner schemes are locally closed in the Hilbert scheme for saturated ideals.

Complete intersection ideals correspond to Gr"obner schemes of complete intersections.

The paper provides a functorial framework connecting Gr"obner bases and Hilbert schemes.

Abstract

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gr\"obner scheme, which parameterizes all homogeneous reduced Gr\"obner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gr\"obner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gr\"obner scheme consists of complete intersections if $J$ defines a complete intersection.