Isotrivial families of curves on affine surfaces and characterization of the affine plane. (English. Russian original) Zbl 0666.14018

Zajdenberg, M. G.

Math. USSR, Izv. 30, No. 3, 503-532 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 534-567 (1987).

The main result is a characterization of \({\mathbb{C}}^ 2\) 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 \({\mathbb{C}}\). In particular, such curves and families cannot exist on Ramanujam surfaces - topologically contractible smooth algebraic surfaces not isomorphic to \({\mathbb{C}}^ 2\). 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 \({\mathbb{C}}^*\).

MSC:

14J26	Rational and ruled surfaces
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
32Q45	Hyperbolic and Kobayashi hyperbolic manifolds
14J25	Special surfaces
14H10	Families, moduli of curves (algebraic)

Keywords:

characterization of the complex affine plane; simply connected algebraic curves; Ramanujam surfaces; hyperbolic complex analysis; regular actions of the group \({\mathbb{C}}^*\)