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