On a problem of Chisini
type: Item
instance of: scholarly article
title: On a problem of Chisini
zbMATH Open document ID: 0609.14031
DOI: 10.1215/S0012-7094-86-05302-0
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 3985420
zbMATH Keywords: monk's belt
zbMATH Keywords: balanced neighborhood
zbMATH Keywords: ramification curve
zbMATH Keywords: branch curve
zbMATH Keywords: multiple planes
author: Fabrizio Catanese
reviewed by: Ragni Piene
MaRDI profile type: Publication
cites work: Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications
cites work: Q5837523
cites work: The ramification divisors for branched coverings of \(P_ k^ n\)
cites work: Q3215513
cites work: Q3310749
cites work: Q3933100
cites work: ON INTERSECTIONS OF QUADRICS
full work available at URL: s0012-7094-86-05302-0
OpenAlex ID: W2001068855
description: scientific article; zbMATH DE number 3985420