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Completions of $\C^*$-surfaces

Hubert Flenner·Shulim Kaliman·Mikhail Zaidenberg
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Abstract

Following an approach of Dolgachev, Pinkham and Demazure, we classified in math.AG/0210153 normal affine surfaces with hyperbolic $\C^{*}$-actions in terms of pairs of $\Q$-divisors $(D_+,D_-)$ on a smooth affine curve. In the present paper we show how to obtain from this description a natural equivariant completion of these $\C^*$-surfaces. Using elementary transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of math.AG/0511063 and show in certain cases their uniqueness. This description is especially precise in the case of normal affine surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normal $\C^*$-surfaces.

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Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg. 2005-11-10. Completions of $\C^*$-surfaces. https://arxiv.org/abs/math/0511282

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