Isotrivial families of curves on affine surfaces and characterization of the affine plane

M. G. Zaidenberg

Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 503-532

Résumé

The main result is a characterization of $\mathbf C^2$ as a smooth acyclic algebraic surface on which there exist simply connected algebraic curves (possibly singular and reducible) or isotrivial (nonexceptional) families of curves with base $\mathbf C$. In particular, such curves and families cannot exist on Ramanujam surfaces – topologically contractible smooth algebraic surfaces not isomorphic to $\mathbf C^2$. The proof is based on a structure theorem which describes the degenerate fibers of families of curves whose geometric monodromy has finite order. Techniques of hyperbolic complex analysis are used; an important role is played by regular actions of the group $\mathbf C^*$. Bibliography: 40 titles.