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 Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 3, Pages 534–567 (Mi izv1308)

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

M. G. Zaidenberg

Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 30:3, 503–532

Bibliographic databases:

UDC: 517.5+512.7
MSC: Primary 14J26, 14J25; Secondary 14D05

Citation: M. G. Zaidenberg, “Isotrivial families of curves on affine surfaces and characterization of the affine plane”, Izv. Akad. Nauk SSSR Ser. Mat., 51:3 (1987), 534–567; Math. USSR-Izv., 30:3 (1988), 503–532

Citation in format AMSBIB
\Bibitem{Zai87} \by M.~G.~Zaidenberg \paper Isotrivial families of curves on affine surfaces and characterization of the affine plane \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1987 \vol 51 \issue 3 \pages 534--567 \mathnet{http://mi.mathnet.ru/izv1308} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=903623} \zmath{https://zbmath.org/?q=an:0666.14018} \transl \jour Math. USSR-Izv. \yr 1988 \vol 30 \issue 3 \pages 503--532 \crossref{https://doi.org/10.1070/IM1988v030n03ABEH001027} 

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Erratum

This publication is cited in the following articles:
1. S. Yu. Orevkov, M. G. Zaidenberg, “On rigid rational cuspidal plane curves”, Russian Math. Surveys, 51:1 (1996), 179–180
2. R V Gurjar, “A new proof of Suzuki's formula”, Proc Math Sci, 107:3 (1997), 237
3. T. BANDMAN, A. LIBGOBER, “COUNTING RATIONAL MAPS ONTO SURFACES AND FUNDAMENTAL GROUPS”, Int. J. Math, 15:07 (2004), 673
4. Kaliman Sh., “Actions of C* and C+ on affine algebraic varieties”, Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 80, no. 1- 2, 2009, 629–654
5. E. A. Rogozinnikov, “O svyazi geometricheskikh svoistv krivykh so svoistvami ikh grupp dvizhenii”, Tr. IMM UrO RAN, 16, no. 3, 2010, 227–233
6. A. J. Parameswaran, Mihai Tibăr, “On the geometry of regular maps from a quasi-projective surface to a curve”, European Journal of Mathematics, 2015
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