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.

Su un teorema di Chisini / A. Lanteri. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI DELLA CLASSE DI SCIENZE FISICHE, MATEMATICHE E NATURALI. - ISSN 0392-7881. - 66:6(1979), pp. 523-532.

### Su un teorema di Chisini

#### Abstract

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.
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multiple plane ; ramification divisor ; Stein manifold ; birational map
Settore MAT/03 - Geometria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/191323
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