Classifying birationally commutative projective surfaces

TL;DR

This paper classifies a specific class of noncommutative algebraic surfaces called birationally commutative projective surfaces, revealing they belong to four geometric families, thus extending previous classification results.

Contribution

It provides a new classification of birationally commutative projective surfaces into four families based on geometric data, generalizing earlier work with different methods.

Findings

Four families of birationally commutative projective surfaces identified

Classification extends previous degree 1 generation results

New proof techniques for classifying noncommutative surfaces

Abstract

Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3 over an uncountable algebraically closed field. Suppose that the graded quotient ring of R is a skew-Laurent ring over a field; we say that R is a birationally commutative projective surface. We classify birationally commutative projective surfaces and show that they fall into four families, parameterized by geometric data. This generalizes work of Rogalski and Stafford on birationally commutative projective surfaces generated in degree 1; our proof techniques are quite different.