Su un teorema di Chisini

Two general multiple planes having the same branch curve cannot be too different; a central result in the theory of multiple planes first proved by Chisini asserts that two such multiple planes, with some additional hypothesis, are birational. Let S be a complex projective nonsingular algebraic surface, R a net on S, f:S \to P^2 che associated multiple plane. We prove that if the moving divisor of R is ample, then the ramification curve G of f is ample too. So S \setminus G is Stein. Now, let f:S \to P^2, f':S'\to P^2 be two general multiple planes having the same branch curve and such that the moving divisors of the corresponging nets are ample. Then one can extend to S and S' an isomorphism between two tubular neighbourhoods of the ramification curves G and G', whose existence was claimed by Chisini.