On almost revlex ideals with Hilbert function of complete intersections

TL;DR

This paper provides a new constructive proof for the existence of almost revlex ideals matching the Hilbert function of complete intersections, exploring their properties and singularities in Hilbert schemes.

Contribution

It introduces a novel inductive and constructive method for establishing almost revlex ideals with specific Hilbert functions, differing from previous approaches.

Findings

Constructed almost revlex ideals with Hilbert functions of complete intersections.

Identified cases where these ideals correspond to singular points in Hilbert schemes.

Analyzed tangent space dimensions related to stable ideals.

Abstract

In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field $K$, we give a new constructive proof of the existence of the almost revlex ideal $J\subset K[x_1,\dots,x_n]$, with the same Hilbert function as a complete intersection defined by $n$ forms of degrees $d_1\leq \dots \leq d_n$. Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We also detect several cases in which an almost revlex ideal having the same Hilbert function as a complete intersection corresponds to a singular point in a Hilbert scheme. This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.