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On Chisini’s conjecture. (English. Russian original) Zbl 0962.14005

Izv. Math. 63, No. 6, 1139-1170 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 83-116 (1999).
Summary: Chisini’s conjecture asserts that if \(B\subset \mathbb{P}^2\) is a cuspidal curve, then a generic morphism \(f\), \(\deg f\geq 5\), 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 \(B_{1,m},B_{2,m} \subset\mathbb{P}^2\) \((m\in\mathbb{N},\;m\geq 5)\) 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 \(f_{i,m}:S_i \to\mathbb{P}^2\) \((i=1,2)\), 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 \(S_i\).

MSC:

14E22 Ramification problems in algebraic geometry
14H50 Plane and space curves
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