Some syzygies of the generators of the ideal of a border basis scheme

TL;DR

This paper investigates the syzygies of the defining ideals of border basis schemes, revealing their structure through identities like Jacobi and trace zero, and proving complete intersection properties in specific cases.

Contribution

It introduces two new families of syzygies related to border basis schemes and demonstrates that these schemes are complete intersections when n=2.

Findings

Identifies syzygies from Jacobi identity and trace properties.

Shows border basis schemes in n=2 are complete intersections.

Provides examples illustrating the syzygies and their implications.

Abstract

A border basis scheme is an affine scheme that can be viewed as an open subscheme of the Hilbert scheme of \mu points of affine n-space. We study syzygies of the generators of a border basis scheme's defining ideal. These generators arise as the entries of the commutators of certain matrices (the "generic multiplication matrices"). We consider two families of syzygies that are closely connected to these matrices: The first arises from the Jacobi identity, and the second from the fact that the trace of a commutator is 0. Several examples of both types of syzygy are presented, including a proof that the border basis schemes in case n = 2 are complete intersections.