Completions of $\C^*$-surfaces

TL;DR

This paper develops a method to obtain natural equivariant completions of hyperbolic $ ext{C}^*$-surfaces classified via $ ext{Q}$-divisors, and classifies boundary zigzags for these surfaces.

Contribution

It provides a systematic way to construct equivariant completions of $ ext{C}^*$-surfaces and classifies boundary configurations in these completions.

Findings

Constructed natural equivariant completions of $ ext{C}^*$-surfaces.

Classified boundary zigzags for smooth or normal $ ext{C}^*$-surfaces.

Showed uniqueness of certain boundary completions.

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.