On Chisini's conjecture

Vik. S. Kulikov

Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1139-1170

Résumé

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 $S_i$.