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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 6, Pages 83–116 (Mi izv267)  

This article is cited in 28 scientific papers (total in 28 papers)

On Chisini's conjecture

Vik. S. Kulikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Chisini's conjecture asserts that if $B\subset\mathbb P^2$ is a cuspidal curve, then a generic morphism $f$, $\deg f\geqslant 5$, of a smooth projective surface to $\mathbb P^2$ branched along $B$ is unique up to isomorphism. In this paper we prove that Chisini's conjecture is true for $B$ if $\deg f$ is greater than the value of some function depending on the degree, genus and the number of cusps of $B$. This inequality holds for almost all generic morphisms. In particular, on a surface with ample canonical class, it holds for generic morphisms defined by a linear subsystem of the $m$-canonical class, $m\in\mathbb N$.
Moreover, we present examples of pairs $B_{1,m},B_{2,m}\subset\mathbb P^2$ ($m\in\mathbb N$, $m\geqslant 5$) of plane cuspidal curves such that
(i) $\deg B_{1,m}=\deg B_{2,m}$, and these curves have homeomorphic tubular neighbourhoods in $\mathbb P^2$, but the pairs $(\mathbb P^2,B_{1,m})$ and $(\mathbb P^2,B_{2,m})$ are not homeomorphic;
(ii) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}\colon S_i\to\mathbb P^2$, $i=1,2$, where $S_i$ are surfaces of general type;
(iii) the surfaces $S_1$ and $S_2$ are homeomorphic (as four-dimensional real manifolds);
(iv) the morphism $f_{i,m}$ is defined by a three-dimensional linear subsystem of the $m$-canonical class of $S_i$.


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English version:
Izvestiya: Mathematics, 1999, 63:6, 1139–1170

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MSC: 14E20
Received: 26.05.1998
Revised: 22.09.1998

Citation: Vik. S. Kulikov, “On Chisini's conjecture”, Izv. RAN. Ser. Mat., 63:6 (1999), 83–116; Izv. Math., 63:6 (1999), 1139–1170

Citation in format AMSBIB
\by Vik.~S.~Kulikov
\paper On Chisini's conjecture
\jour Izv. RAN. Ser. Mat.
\yr 1999
\vol 63
\issue 6
\pages 83--116
\jour Izv. Math.
\yr 1999
\vol 63
\issue 6
\pages 1139--1170

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    This publication is cited in the following articles:
    1. Vik. S. Kulikov, M. Teicher, “Braid monodromy factorizations and diffeomorphism types”, Izv. Math., 64:2 (2000), 311–341  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. S. Kulikov, Vik. S. Kulikov, “Generic coverings of the plane with $A$-$D$-$E$-singularities”, Izv. Math., 64:6 (2000), 1153–1195  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Auroux D., Katzarkov L., “Branched coverings of $C\mathrm{P}^2$ and invariants of symplectic 4-manifolds”, Invent. Math., 142:3 (2000), 631–673  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. S. Yu. Nemirovski, “Kulikov's theorem on the Chisini conjecture”, Izv. Math., 65:1 (2001), 71–74  mathnet  crossref  crossref  mathscinet  zmath
    5. Kharlamov V., Kulikov V., “Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves”, C. R. Acad. Sci. Paris Sér. I Math., 333:9 (2001), 855–859  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Manfredini S., Pignatelli R., “Chisini's conjecture for curves with singularities of type $x^n=y^m$”, Michigan Math. J., 50:2 (2002), 287–312  crossref  mathscinet  zmath  isi  elib  scopus
    7. Garber D., “Plane curves and their fundamental groups: Generalizations of Uludaǧ's construction”, Algebr. Geom. Topol., 3 (2003), 593  crossref  mathscinet  zmath
    8. Vik. S. Kulikov, “Generalized Chisini's Conjecture”, Proc. Steklov Inst. Math., 241 (2003), 110–119  mathnet  mathscinet  zmath
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    10. Artal Bartolo E., Tokunaga H., “Zariski $k$-plets of rational curve arrangements and dihedral covers”, Topology Appl., 142:1-3 (2004), 227–233  crossref  mathscinet  zmath  isi  scopus
    11. Auroux D., “Some open questions about symplectic 4-manifolds, singular plane curves and braid group factorizations”, European Congress of Mathematics, 2005, 23–40  mathscinet  zmath  isi
    12. Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. Libgober A., “Lectures on topology of complements and fundamental groups”, Singularity Theory, 2007, 71–137  crossref  mathscinet  zmath  isi
    14. Vik. S. Kulikov, “On Chisini's conjecture. II”, Izv. Math., 72:5 (2008), 901–913  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Auroux D., Smith I., “Lefschetz pencils, branched covers and symplectic invariants”, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., 1938, Springer, Berlin, 2008, 1–53  crossref  mathscinet  zmath  isi
    16. Catanese F., “Differentiable and deformation type of algebraic surfaces, real and symplectic structures”, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., 1938, Springer, Berlin, 2008, 55–167  crossref  mathscinet  zmath  adsnasa  isi
    17. Vik. S. Kulikov, V. M. Kharlamov, “Automorphisms of Galois coverings of generic $m$-canonical projections”, Izv. Math., 73:1 (2009), 121–150  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. V. S. Kulikov, Vik. S. Kulikov, “On complete degenerations of surfaces with ordinary singularities in $\mathbb P^3$”, Sb. Math., 201:1 (2010), 129–158  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. Eliyahu M., Garber D., Teicher M., “A conjugation-free geometric presentation of fundamental groups of arrangements”, Manuscripta Mathematica, 133:1–2 (2010), 247–271  crossref  mathscinet  zmath  isi  scopus
    20. Eliyahu M., Garber D., Teicher M., “A Conjugation-Free Geometric Presentation of Fundamental Groups of Arrangements II: Expansion and Some Properties”, Internat J Algebra Comput, 21:5 (2011), 775–792  crossref  mathscinet  zmath  isi  elib  scopus
    21. Friedman M., Leyenson M., Shustin E., “On Ramified Covers of the Projective Plane i: Interpreting Segre'S Theory (With An Appendix By Eugenii Shustin)”, Internat J Math, 22:5 (2011), 619–653  crossref  mathscinet  zmath  isi  elib  scopus
    22. Fedor Bogomolov, Viktor S. Kulikov, “On the irreducibility of Hilbert scheme of surfaces of minimal degree”, centr.eur.j.math, 2012  crossref  mathscinet  isi  scopus
    23. Friedman M., Lehman R., Leyenson M., Teicher M., “On Ramified Covers of the Projective Plane II: Generalizing Segre's Theory”, J. Eur. Math. Soc., 14:3 (2012), 971–996  crossref  mathscinet  zmath  isi  elib  scopus
    24. Friedman M., Teicher M., “On Fundamental Groups Related to Degeneratable Surfaces: Conjectures and Examples”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 11:3 (2012), 565–603  mathscinet  zmath  isi
    25. Vik. S. Kulikov, “Dualizing coverings of the plane”, Izv. Math., 79:5 (2015), 1013–1042  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    26. Yu. Burman, Serge Lvovski, “On projections of smooth and nodal plane curves”, Mosc. Math. J., 15:1 (2015), 31–48  mathnet  mathscinet
    27. Finashin S. Kharlamov V., “Apparent Contours of Nonsingular Real Cubic Surfaces”, Trans. Am. Math. Soc., 367:10 (2015), PII S0002-9947(2015)06286-2, 7221–7289  crossref  mathscinet  zmath  isi  scopus
    28. Oba T., “Compact Stein Surfaces as Branched Covers With Same Branch Sets”, Algebr. Geom. Topol., 18:3 (2018), 1733–1751  crossref  mathscinet  zmath  isi  scopus
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