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

Abstract

A border basis scheme is an affine scheme that can be viewed as an open subscheme of the Hilbert scheme of μpoints 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.

arXiv commentRevised in accordance with referee comments: exposition improved, typos & minor errors fixed. To appear in Collectanea Mathematica. 25 pages