An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme

TL;DR

This paper presents an efficient implementation of an algorithm to compute all saturated Borel-fixed ideals for given variables and Hilbert polynomial, aiding the study of Hilbert schemes.

Contribution

It provides a practical, optimized implementation of a theoretical algorithm for Borel-fixed ideals in Hilbert scheme analysis.

Findings

Successfully computed all saturated Borel-fixed ideals for specified parameters.

Enhanced algorithm efficiency for handling larger variable sets.

Facilitated deeper understanding of Hilbert scheme components.

Abstract

Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borel-fixed point, i.e. a point corresponding to a subscheme defined by a Borel-fixed ideal. Moreover Borel-fixed ideals have good combinatorial properties, which make them very interesting in an algorithmic perspective. In this paper, we propose an implementation of the algorithm computing all the saturated Borel-fixed ideals with number of variables and Hilbert polynomial assigned, introduced from a theoretical point of view in the paper "Segment ideals and Hilbert schemes of points", Discrete Mathematics 311 (2011).