On the functoriality of marked families

TL;DR

This paper establishes a functorial framework for marked schemes, demonstrating their embedding into Hilbert schemes and generalizing results about Gr"obner strata, thus advancing computational algebra methods for studying Hilbert schemes.

Contribution

It provides a functorial foundation for marked schemes, showing their independence from coefficient rings and their embedding into Hilbert schemes, along with a generalization of Gr"obner strata results.

Findings

Marked schemes can be embedded in Hilbert schemes as locally closed subschemes.

Marked schemes are open under certain conditions on the ideal.

Gr"obner strata of any ideals are locally closed in Hilbert schemes.

Abstract

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $\mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $\mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.