On Chisini's conjecture

Abstract

Chisini's conjecture asserts that if is a cuspidal curve, then a generic morphism , , of a smooth projective surface to branched along is unique up to isomorphism. In this paper we prove that Chisini's conjecture is true for if is greater than the value of some function depending on the degree, genus and the number of cusps of . 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 -canonical class, .Moreover, we present examples of pairs (, ) of plane cuspidal curves such that (i) , and these curves have homeomorphic tubular neighbourhoods in , but the pairs and are not homeomorphic;(ii) is the discriminant curve of a generic morphism , , where are surfaces of general type;(iii) the surfaces and are homeomorphic (as four-dimensional real manifolds);(iv) the morphism is defined by a three-dimensional linear subsystem of the -canonical class of .