A network of rational curves on the Hilbert scheme

TL;DR

This paper presents a method to construct rational curves on the Hilbert scheme via deformations of Borel-fixed ideals, providing new insights into its connectedness and component structure.

Contribution

It introduces an effective algorithmic approach to generate rational deformations between Borel-fixed ideals, aiding in understanding the Hilbert scheme's geometry.

Findings

Proves the connectedness of the Hilbert scheme using Borel-fixed ideals.

Provides a new criterion for identifying points on the same component.

Offers a detailed, implementable algorithm for constructing rational curves.

Abstract

In this paper we introduce an effective method to construct rational deformations between couples of Borel-fixed ideals. These deformations are governed by flat families, so that they correspond to rational curves on the Hilbert scheme. Looking globally at all the deformations among Borel-fixed ideals defining points on the same Hilbert scheme, we are able to give a new proof of the connectedness of the Hilbert scheme and to introduce a new criterion to establish whenever a set of points defined by Borel ideals lies on a common component of the Hilbert scheme. The paper contains a detailed algorithmic description of the technique and all the algorithms are made available.