Flat families of point schemes for connected graded algebras

TL;DR

This paper investigates families of truncated point schemes of connected graded algebras, establishing flatness over a dense open subset and confirming a conjecture about the number of point modules in specific Artin-Schelter regular algebras.

Contribution

It proves flatness of these families over a dense open locus and verifies a conjecture regarding the number of point modules in certain regular algebras.

Findings

Families are flat over the open dense locus where point schemes are minimal.

Number of points in zero-dimensional schemes is computed via Chow ring methods.

Confirms a conjecture about 17 point modules in specific Artin-Schelter regular algebras.

Abstract

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e. minimal) dimension. When the truncated point scheme is zero-dimensional we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield to the effect that a generic two-generator, two-relator 4-dimensional Artin-Schelter regular algebra has seventeen truncated point modules of length six.