Some components of the moduli space of Koszul Artin-Schelter regular algebras of dimension four

TL;DR

This paper investigates the structure of the moduli space of four-dimensional Koszul Artin-Schelter regular algebras by analyzing Hochschild cohomology and the Kodaira-Spencer map, identifying components and their properties.

Contribution

It introduces a method to identify components of the moduli stack using the Kodaira-Spencer map's properties for these algebras.

Findings

When the Kodaira-Spencer map is surjective, the family forms a moduli component.

Bijective Kodaira-Spencer map implies a generically finite map to the moduli stack.

Identified specific components of the moduli stack for these algebras.

Abstract

We compute the Hochschild cohomology and the Kodaira spencer map for known families of Koszul Artin-Schelter regular algebras of dimension four. We show that when the Kodaira Spencer map at a point is a surjection, the image of the family is a component of the moduli stack of such algebras, and when the Kodaira Spencer map is a bijection, the map to the moduli stack is generically finite. We use this to identify some components of the moduli stack.