Smooth Affine $\mathbb{G}_m$-surfaces with Finite Picard Groups and Trivial Units

Abstract

We determine the smooth affine $\mathbb{G}_m$-surfaces of logarithmic Kodaira dimension $\leq 0$ with finite Picard groups and trivial units defined over an algebraically closed field. Furthermore, we prove that a smooth factorial affine surface with trivial units is isomorphic to the affine plane if and only if it admits an effective $\mathbb{G}_m$-action.