On a problem of Chisini
A finite morphism \(f: S\to {\mathbb{P}}^ 2\) from a smooth, compact, complex surface S is called a multiple plane. It is said to be generic if the ramification curve R is the normalization of the branch curve \(B=f(R)\), and if B has only nodes and ordinary cusps as singularities. A problem, posed by Chisini, is to determine when two multiple planes are isomorphic, or, more precisely, when they are strictly isomorphic: given (S,f) and (S',f'), generic multiple planes with the same branch curve, does there exist an isomorphism \(\phi: S\to S'\) such that \(f'\circ \phi =f ?\) The author proves that the answer is yes if and only if a certain 2-torsion marked line bundle on R, associated with f and f', is trivial. This theorem corrects an assertion by \textit{O. Chisini} [Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. 77 (III. Ser. 8), 339-356 (1944; Zbl 0061.353)] which was subsequently used by \textit{A. Lanteri} [Atti Accad. Naz Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 66, 523-532 (1979; Zbl 0529.14009)]. Finally, the author gives an example of three generic multiple planes, all generic projections of the Veronese surface, that are not strictly isomorphic.
scientific article; zbMATH DE number 3985420