Twisted rings and moduli stacks of "fat" point modules in non-commutative projective geometry

TL;DR

This paper extends the study of point modules in non-commutative projective geometry to 'fat' point modules, establishing a map to twisted rings on moduli stacks and providing criteria for non-commutative surfaces to be birationally PI.

Contribution

It introduces a framework for moduli stacks of 'fat' point modules and constructs a map to twisted rings, advancing the understanding of non-commutative surfaces and their classification.

Findings

Established a map from non-commutative algebras to twisted rings on moduli stacks.

Provided a criterion for non-commutative surfaces to be birationally PI.

Linked the structure of 'fat' point modules to birational classification conjectures.

Abstract

The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational classification of non-commutative surfaces.