Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases

TL;DR

This paper develops computational and theoretical tools using relative marked bases to study Hilbert schemes over quotient rings, providing explicit open covers and analyzing lex-points, with applications to Cohen-Macaulay and Macaulay-Lex cases.

Contribution

It introduces the concept of relative marked bases over quasi-stable ideals and applies them to explicitly cover Hilbert schemes and analyze lex-points in quotient rings.

Findings

Explicit open cover of Hilbert schemes for Cohen-Macaulay quotients.

Identification of smooth and singular lex-points in Macaulay-Lex quotients.

Development of computational methods for Hilbert schemes over quotient rings.

Abstract

We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points.