Chisini's conjecture asserts that if B is contained in P~2 is a cuspidal curve, then a generic morphism f, deg f >= 5, of a smooth projective surface to 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 implied by N. Moreover, we present examples of pairs B_(1,m), B_(2,m) is contained in P~2 (m implied by N, m >= 5) of plane cuspidal curves such that (i) deg B_(1,m)= deg B_(2,m), and these curves have homeomorphic tubular neighbourhoods in P~2, but the pairs (P~2,B_(1,m)) and (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 -> 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.
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