Completions of $\C^*$-surfaces

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.

37 pages

References in corpus (2)

• Birational transformations of weighted graphs
• On the uniqueness of ${\mathbf{C}}^{\ast }$-actions on affine surfaces

Cited by in corpus (2)

• Exceptional Sequences on Rational C*-Surfaces
• Algebraic density property of homogeneous spaces