GK-dimension of birationally commutative surfaces

TL;DR

This paper investigates the Gelfand-Kirillov dimension of certain graded subalgebras of skew polynomial rings over function fields of surfaces, linking algebraic growth to geometric properties of automorphisms.

Contribution

It establishes the possible GK-dimensions of these subalgebras as 3, 4, 5, or infinite, depending solely on the geometry of the automorphism, using birational surface techniques.

Findings

GKdim of subalgebras is 3, 4, 5, or infinite

Dimension depends only on geometric properties of automorphism

Uses techniques from birational geometry of surfaces

Abstract

Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest.