Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3

TL;DR

This paper studies Borel ideals related to ACM curves in projective 3-space, providing criteria, examples, and conjectures about their placement on Hilbert scheme components.

Contribution

It introduces a necessary criterion for Borel ideals to lie on ACM curve components and verifies this through explicit examples and partial proofs.

Findings

Identified Borel ideals on Hilbert scheme components of ACM curves in P^3.

Computed all Borel ideals on these components in several examples.

Formulated and supported conjectures about which Borel ideals belong to such components.

Abstract

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in P^3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.