On a Chisini Conjecture

Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if $\deg f$ is greater than the value of some function depending on the degree, genus, and number of cusps of $B$, then the Chisini conjecture holds for $B$. This inequality holds for many different generic morphisms. In particular, it holds for a generic morphism given by a linear subsystem of the $m$th canonical class for almost all surfaces with ample canonical class.