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 Tr. Mat. Inst. Steklova, 2003, Volume 241, Pages 122–131 (Mi tm392)

Generalized Chisini's Conjecture

Vik. S. Kulikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Chisini's Conjecture claims that a generic covering of the plane of degree $\geq 5$ is determined uniquely by its branch curve. A generalization (to the case of normal surfaces) of Chisini's Conjecture is formulated and considered. The generalized conjecture is checked in the following two cases: when the maximum of degrees of two generic coverings $\geq 12$ and when it $\leq 4$. Conditions on the number of singular points of a cuspidal curve $B$ necessary for $B$ to be the branch curve of a generic covering of given degree are found. In particular, it is shown that, if $B$ is a pure cuspidal curve (i.e. all its singular points are ordinary cusps), then $B$ can be the branch curve only of a generic covering of degree $\leq 5$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 241, 110–119

Bibliographic databases:

Document Type: Article
UDC: 512.7

Citation: Vik. S. Kulikov, “Generalized Chisini's Conjecture”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 122–131; Proc. Steklov Inst. Math., 241 (2003), 110–119

Citation in format AMSBIB
\Bibitem{Kul03} \by Vik.~S.~Kulikov \paper Generalized Chisini's Conjecture \inbook Number theory, algebra, and algebraic geometry \bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 241 \pages 122--131 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm392} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2024048} \zmath{https://zbmath.org/?q=an:1078.14017} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 241 \pages 110--119 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Vik. S. Kulikov, “A factorization formula for the full twist of double the number of strings”, Izv. Math., 68:1 (2004), 125–158
2. Labs O., “Dessins d'Enfants and hypersurfaces with many A(j)–singularities”, Journal of the London Mathematical Society–Second Series, 74:3 (2006), 607–622
3. Vik. S. Kulikov, V. M. Kharlamov, “Automorphisms of Galois coverings of generic $m$-canonical projections”, Izv. Math., 73:1 (2009), 121–150
4. Calabri A., Paccagnan D., Stagnaro E., “Plane Algebraic Curves With Many Cusps, With An Appendix By Eugenii Shustin”, Ann. Mat. Pura Appl., 193:3 (2014), 909–921
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