Completions of $\C^*$-surfaces

Hubert Flenner (1) , Shulim Kaliman (2) , Mikhail Zaidenberg (3)

1 Fakultät für Mathematik - Fakultät für Mathematik
2 Department of Mathematics [Miami]
3 IF - Institut Fourier

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.

Mots clés: C^{*}-actions; affine surfaces

Domaines: Géométrie algébrique [math.AG]

Hubert Flenner, Shulim Kaliman, Mikhail Zaidenberg. Completions of $\C^*$-surfaces. T. Hibi. Affine algebraic geometry, Osaka University Press, pp.149-201, 2007. ⟨hal-00323513⟩