Geometric algebras on projective surfaces

TL;DR

This paper investigates subrings of twisted homogeneous coordinate rings on projective surfaces, establishing conditions for noetherianity, exploring their homological properties, and constructing new examples of maximal orders with specific grading features.

Contribution

It provides necessary and sufficient conditions for subrings to be noetherian and introduces new examples of maximal orders with unique grading properties.

Findings

Identified conditions for noetherianity of subrings.

Studied homological properties of these subrings.

Constructed new graded maximal orders with non-standard generation properties.

Abstract

Let X be a projective surface, let \sigma be an automorphism of X, and let L be a \sigma-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate ring B(X, L, \sigma). In particular, we find necessary and sufficient conditions for these subrings to be noetherian. We also study their homological properties, their associated noncommutative projective schemes, and when they are maximal orders. In the process, we produce new examples of maximal orders; these are graded and have the property that no Veronese subring is generated in degree 1. Our results are used in a companion paper to give defining data for a large class of noncommutative projective surfaces.