Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units

TL;DR

This paper shows that there are infinitely many non-isomorphic smooth affine factorial surfaces of logarithmic Kodaira dimension zero with trivial units over an algebraically closed field, contradicting previous classifications over the complex numbers.

Contribution

It proves that the family of such surfaces has at least countably infinite isomorphism classes, challenging earlier classification results.

Findings

Number of isomorphism classes is at least countably infinite

Contradicts earlier classification asserting at most two classes over complex numbers

Open problem remains for classification over ield ield

Abstract

This paper considers the family $\mathscr{S}_0$ of smooth affine factorial surfaces of logarithmic Kodaira dimension 0 with trivial units over an algebraically closed field $k$. Our main result (Theorem 4.1) is that the number of isomorphism classes represented in $\mathscr{S}_0$ is at least countably infinite. This contradicts the earlier classification of Gurjar and Miyanishi [5] which asserted that $\mathscr{S}_0$ has at most two elements up to isomorphism when $k=\mathbb{C}$. Thus, the classification of surfaces in $\mathscr{S}_0$ for the field $\mathbb{C}$, long thought to have been settled, is an open problem.