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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
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
Citation:
Vik. S. Kulikov, “On Chisini's conjecture”, Izv. Math., 63:6 (1999), 1139–1170
Linking options:
https://www.mathnet.ru/eng/im267https://doi.org/10.1070/im1999v063n06ABEH000267 https://www.mathnet.ru/eng/im/v63/i6/p83
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Abstract page: | 580 | Russian version PDF: | 249 | English version PDF: | 24 | References: | 100 | First page: | 1 |
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