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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 6, Pages 65–106 (Mi izv312)  

This article is cited in 5 scientific papers (total in 5 papers)

Generic coverings of the plane with $A$-$D$-$E$-singularities

V. S. Kulikova, Vik. S. Kulikovb

a Moscow State Academy of Printing Arts
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We investigate representations of an algebraic surface $X$ with $A$-$D$-$E$-singularities as a generic covering $f\colon X\to\mathbb{P}^2$, that is, a finite morphism which has at most folds and pleats apart from singular points and is isomorphic to the projection of the surface $z^2=h(x,y)$ onto the plane $x$$y$ near each singular point, and whose branch curve $B\subset\mathbb{P}^2$ has only nodes and ordinary cusps except for singularities originating from the singularities of $X$. It is regarded as folklore that a generic projection of a non-singular surface $X\subset\mathbb{P}^r$ is of this form. In this paper we prove this result in the case when the embedding of a surface $X$ with $A$-$D$-$E$-singularities is the composite of the original one and a Veronese embedding. We generalize the results of [6], which considers Chisini's conjecture on the unique reconstruction of $f$ from the curve $B$. To do this, we study fibre products of generic coverings. We get the main inequality bounding the degree of the covering in the case when there are two inequivalent coverings with branch curve $B$. This inequality is used to prove Chisini's conjecture for $m$-canonical coverings of surfaces of general type for $m\geqslant 5$.

DOI: https://doi.org/10.4213/im312

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English version:
Izvestiya: Mathematics, 2000, 64:6, 1153–1195

Bibliographic databases:

MSC: 14E20
Received: 27.07.1999

Citation: V. S. Kulikov, Vik. S. Kulikov, “Generic coverings of the plane with $A$-$D$-$E$-singularities”, Izv. RAN. Ser. Mat., 64:6 (2000), 65–106; Izv. Math., 64:6 (2000), 1153–1195

Citation in format AMSBIB
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\by V.~S.~Kulikov, Vik.~S.~Kulikov
\paper Generic coverings of the plane with $A$-$D$-$E$-singularities
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 6
\pages 65--106
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\crossref{https://doi.org/10.4213/im312}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1817250}
\zmath{https://zbmath.org/?q=an:1012.14004}
\transl
\jour Izv. Math.
\yr 2000
\vol 64
\issue 6
\pages 1153--1195
\crossref{https://doi.org/10.1070/IM2000v064n06ABEH000312}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Vik. S. Kulikov, “Generalized Chisini's Conjecture”, Proc. Steklov Inst. Math., 241 (2003), 110–119  mathnet  mathscinet  zmath
    2. Vik. S. Kulikov, “A factorization formula for the full twist of double the number of strings”, Izv. Math., 68:1 (2004), 125–158  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Vik. S. Kulikov, V. M. Kharlamov, “Automorphisms of Galois coverings of generic $m$-canonical projections”, Izv. Math., 73:1 (2009), 121–150  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Ishida H., Tokunaga H.-o., “Triple Covers of Algebraic Surfaces and a Generalization of Zariski's Example”, Singularities - Niigata - Toyama 2007, Advanced Studies in Pure Mathematics, 56, eds. Brasselet J., Ishi S., Suwa T., Vaquie M., Math Soc Japan, 2009, 169–185  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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