Segments and Hilbert schemes of points

TL;DR

This paper investigates the singularity of certain segment ideals in Hilbert schemes of points, explores properties of segment ideals, and introduces an algorithm for computing saturated Borel ideals with a specified Hilbert polynomial.

Contribution

It proves the singularity of revlex segment ideals in Hilbert schemes and develops an algorithm to compute all saturated Borel ideals for a given Hilbert polynomial.

Findings

Revlex segment ideals are singular points in Hilbert schemes.

Properties of various segment ideals are analyzed and compared.

An algorithm for computing saturated Borel ideals with a given Hilbert polynomial is provided.

Abstract

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $\mathbb{P}^n$. In this context, we look inside properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.