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

This article is cited in 6 scientific papers (total in 6 papers)

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
Received: 19.03.1985

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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. R V Gurjar, “A new proof of Suzuki's formula”, Proc Math Sci, 107:3 (1997), 237  crossref  mathscinet  zmath
    3. T. BANDMAN, A. LIBGOBER, “COUNTING RATIONAL MAPS ONTO SURFACES AND FUNDAMENTAL GROUPS”, Int. J. Math, 15:07 (2004), 673  crossref
    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  isi
    5. E. A. Rogozinnikov, “O svyazi geometricheskikh svoistv krivykh so svoistvami ikh grupp dvizhenii”, Tr. IMM UrO RAN, 16, no. 3, 2010, 227–233  mathnet  elib
    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  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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