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Izvestiya: Mathematics, 1999, Volume 63, Issue 6, Pages 1139–1170
DOI: https://doi.org/10.1070/im1999v063n06ABEH000267
(Mi im267)
 

This article is cited in 44 scientific papers (total in 44 papers)

On Chisini's conjecture

Vik. S. Kulikov

Steklov Mathematical Institute, Russian Academy of Sciences
References:
Abstract: Chisini's conjecture asserts that if $B\subset\mathbb P^2$ is a cuspidal curve, then a generic morphism $f$, $\deg f\geqslant 5$, of a smooth projective surface to $\mathbb P^2$ branched along $B$ is unique up to isomorphism. In this paper we prove that Chisini's conjecture is true for $B$ if $\deg f$ is greater than the value of some function depending on the degree, genus and the number of cusps of $B$. This inequality holds for almost all generic morphisms. In particular, on a surface with ample canonical class, it holds for generic morphisms defined by a linear subsystem of the $m$-canonical class, $m\in\mathbb N$.
Moreover, we present examples of pairs $B_{1,m},B_{2,m}\subset\mathbb P^2$ ($m\in\mathbb N$, $m\geqslant 5$) of plane cuspidal curves such that
(i) $\deg B_{1,m}=\deg B_{2,m}$, and these curves have homeomorphic tubular neighbourhoods in $\mathbb P^2$, but the pairs $(\mathbb P^2,B_{1,m})$ and $(\mathbb P^2,B_{2,m})$ are not homeomorphic;
(ii) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}\colon S_i\to\mathbb P^2$, $i=1,2$, where $S_i$ are surfaces of general type;
(iii) the surfaces $S_1$ and $S_2$ are homeomorphic (as four-dimensional real manifolds);
(iv) the morphism $f_{i,m}$ is defined by a three-dimensional linear subsystem of the $m$-canonical class of $S_i$.
Received: 26.05.1998
Revised: 22.09.1998
Bibliographic databases:
Document Type: Article
MSC: 14E20
Language: English
Original paper language: Russian
Citation: Vik. S. Kulikov, “On Chisini's conjecture”, Izv. Math., 63:6 (1999), 1139–1170
Citation in format AMSBIB
\Bibitem{Kul99}
\by Vik.~S.~Kulikov
\paper On Chisini's conjecture
\jour Izv. Math.
\yr 1999
\vol 63
\issue 6
\pages 1139--1170
\mathnet{http://mi.mathnet.ru//eng/im267}
\crossref{https://doi.org/10.1070/im1999v063n06ABEH000267}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1748562}
\zmath{https://zbmath.org/?q=an:0962.14005}
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\elib{https://elibrary.ru/item.asp?id=13305812}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0010564432}
Linking options:
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  • https://www.mathnet.ru/eng/im/v63/i6/p83
    Cycle of papers
    This publication is cited in the following 44 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    References:100
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