Smooth affine surfaces properly dominated by ${\mathbf{C}}^{\ast }\times {\mathbf{C}}^{\ast }$

Abstract

We classify smooth complex affine surfaces admitting a finite surjective morphism from ${\mathbf{C}}^{\ast }\times {\mathbf{C}}^{\ast }$. Using the classification theory of smooth affine surfaces according to logarithmic Kodaira dimension, we show that every such surface has logarithmic Kodaira dimension either $-\infty$ or $0$, and determine all possibilities. More precisely, the only surfaces of logarithmic Kodaira dimension $-\infty$ are ${\mathbf{C}}^2$ and $\mathbf{C}\times {\mathbf{C}}^{\ast }$, while those of logarithmic Kodaira dimension $0$ are precisely ${\mathbf{C}}^{\ast }\times {\mathbf{C}}^{\ast }$ and Fujita's surface $H[-1,0,-1]$. This establishes the classification of smooth affine surfaces properly dominated by ${\mathbf{C}}^{\ast }\times {\mathbf{C}}^{\ast }$ anticipated by M.~Furushima in 1989.

14 pages. Comments are most welcome!