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

Buddhadev Hajra

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arXiv · 2608.04214

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

Buddhadev Hajra

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Abstract

We classify smooth complex affine surfaces admitting a finite surjective morphism from $\mathbf{C}^*\times\mathbf{C}^*$. 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 $\mathbf{C}\times\mathbf{C}^*$, while those of logarithmic Kodaira dimension $0$ are precisely $\mathbf{C}^*\times\mathbf{C}^*$ and Fujita's surface $H[-1,0,-1]$. This establishes the classification of smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$ anticipated by M.~Furushima in 1989.

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Buddhadev Hajra. 2026-08-04. Smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$. https://arxiv.org/abs/2608.04214

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