Su un teorema di Chisini

Lanteri, Antonio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 66 (1979) no. 6, pp. 523-532

Résumé

Two general multiple planes having the same branch curve cannot be too "different". As it is well known, a central result in the theory of multiple planes, first proved by Chisini in [3], asserts that two such multiple planes, with some additional hypothesis, are birational. In this paper we prove, with a different additional hypothesis, that two general multiple planes having the same branch curve are isomorphic. Let S be a complex projective non-singular algebraic surface, R a net on S, $\Phi : S \rightarrow \mathbf {P}^{2}$ the associated multiple plane. Firstly we prove that if the moving divisor of R is ample, then the ramification curve $\Gamma$ of $\Phi$ is ample too. So, $S \setminus \Gamma$ is Stein. Now, let $\Phi : S \rightarrow \mathbf {P}^{2}$ and $\Phi' : S' \rightarrow \mathbf {P}^{2}$ be two general multiple planes having the same branch curve and such that the moving divisors of the corresponding nets are ample. Then the previous result allows us to extend to S and S' an isomorphism $f : U_{\Gamma} \to U'_{\Gamma'}$, between two tubular neighbourhoods of the ramification curves $\Gamma$ and $\Gamma'$.