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Plane curves with hyperbolic and \(C\)-hyperbolic complements. (English) Zbl 0919.32015
A complex space \(X\) is said to be \(C-\)hyperbolic if there exists a non-ramified covering \(Y\rightarrow X\) such that \(Y\) is Carathéodory hyperbolic, i.e. the points in \(Y\) are separated by bounded holomorphic functions. If there exists a covering \(Y\) of \(X\) such that for any point \( p\in Y\) there exists only finitely many points \(q\in Y\) which cannot be separated from \(p\) by bounded holomorphic functions on \(Y\), then say that \(X\) is almost \(C-\)hyperbolic.
Authors study plane projective curves whose complements are \(C-\) hyperbolic.
Theorem 1.1 Let \(C\subset \mathbb P^2\) be an irreducible curve of geometric genus \(g\). Assume that its dual curve \(C^{*}\) is an immersed curve of degree \(n\).
a) If \(g\geq 1\), then \(\mathbb P^2\backslash C\) is \(C-\)hyperbolic.
b) If \(g=0\), \(n\geq 5\) and \(C^{*}\) is a generic rational nodal curve, then \( \mathbb P^2\backslash C\) is almost \(C\)-hyperbolic.
c) In both cases \(\mathbb P^2\backslash C\) is Kobayashi complete hyberbolic and hyperbolically embedded into \(\mathbb P^2\).
They also give explicit examples of such irreducible curves of any even degree \(d\geq 6\).
MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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References:
[1] P. Aluffi - C. Faber , Linear orbits of d-tuples of points in P1 , J. Reine Angew. Math . 445 ( 1993 ), 205 - 220 . MR 1244973 | Zbl 0781.14036 · Zbl 0781.14036
[2] A.B. Aure , Plücker conditions on plane rational curves , Math. Scand 55 ( 1984 ), 47 - 58 , with Appendix by S.A. Strömme, ibid. 59 - 61 . MR 769024 | Zbl 0578.14029 · Zbl 0578.14029
[3] K. Azukawa - M. Suzuki , Some examples of algebraic degeneracy and hyperbolic manifolds , Rocky Mountain J. Math 10 ( 1980 ), 655 - 659 . MR 590228 | Zbl 0416.32012 · Zbl 0416.32012
[4] J.A. Carlson - M. Green , Holomorphic curves in the plane , Duke Math. J. 43 ( 1976 ), 1 - 9 . Article | MR 397026 | Zbl 0333.32022 · Zbl 0333.32022
[5] A.I. Degtyarev , Topology of plane projective algebraic curves , PhD Thesis, Leningrad State University , 1987 (in Russian).
[6] A. Degtyarev , Quintics in CP2 with nonabelianfundamental group , preprint, Max-Planck-Institut für Mathematik , Bonn , 1995 , 21 p.
[7] P. Deligne , Le groupe fondamental du complement d’une courbe plane n’ayant que des points doubles ordinaries est abélien, Sem. Bourbaki, 1979/1980 , Lecture Notes in Math . 842 , Springer-Verlag , N.Y ., 1981 , 1 - 25 . Numdam | MR 636513 | Zbl 0478.14008 · Zbl 0478.14008
[8] G. Dethloff - G. Schumacher - P.-M. Wong , Hyperbolicity of the complements of plane algebraic curves , Amer. J. Math. 117 ( 1995 ), 573 - 599 . MR 1333937 | Zbl 0842.32021 · Zbl 0842.32021
[9] G. Dethloff - G. Schumacher - P.-M. Wong , On the hyperbolicity of the complement of curves in algebraic surfaces: The three component case , Duke Math. J. 78 ( 1995 ), 193 - 212 . Article | MR 1328756 | Zbl 0847.32028 · Zbl 0847.32028
[10] G. Dethloff - M. Zaidenberg , Examples of plane curves of low degrees with hyperbolic and C-hyperbolic complements, Proc. Conf. ”Geometric Complex Analysis” , Hayama , March 19-29, 1995 , J. Noguchi a.e. eds., World Scientific Singapore e.a., 1996 , 147 - 161 . MR 1453597 | Zbl 0943.32016 · Zbl 0943.32016
[11] I. Dolgachev - A. Libgober , On the fundamental group of the complement to a discriminant variety, in: Algebraic Geometry , Lecture Notes in Math . 862 Springer N.Y . 1981 , 1 - 25 . MR 644816 | Zbl 0475.14011 · Zbl 0475.14011
[12] H. Flenner - M. Zaidenberg , On a class of rational cuspidal plane curves , Manuscripta Math. 89 ( 1996 ), 439 - 459 . Article | MR 1383524 | Zbl 0868.14014 · Zbl 0868.14014
[13] W. Fulton , On the fundamental group of the complement to a node curve , Ann. of Math. 11 ( 1980 ), 407 - 409 . MR 569076 | Zbl 0406.14008 · Zbl 0406.14008
[14] I.M. Gelfand - M.M. Kapranov - A.V. Zelevinsky , Discriminants, Resultants and Multidimensional Determinants , Bikhäuser , Boston , 1994 . MR 1264417 | Zbl 0827.14036 · Zbl 0827.14036
[15] H. Grauert - U. Peternell , Hyperbolicity of the complements of plane curves , Manuscripta Math. 50 ( 1985 ), 429 - 441 . Article | MR 784151 | Zbl 0581.32031 · Zbl 0581.32031
[16] M. Green , The complement of the dual of a plane curve and some new hyperbolic manifolds, in: Value Distribution Theory , Kujala and Vitter eds., Marcel Dekker , N.Y ., 1974 , 119 - 131 . MR 352541 | Zbl 0289.32015 · Zbl 0289.32015
[17] M. Green , Some examples and counterexamples in value distribution theory , Compositio Math. 30 ( 1975 ), 317 - 322 . Numdam | MR 374498 | Zbl 0307.32022 · Zbl 0307.32022
[18] D. Hilbert , Theory of Algebraic Invariants , Cambridge University Press , Cambridge , 1993 . MR 1266168 | Zbl 0801.13001 · Zbl 0801.13001
[19] Sh.I. Kaliman , The holomorphic universal covers of space of polynomials without multiple roots , Selecta Mathem. form. Sovietica 12 ( 1993 ), 395 - 405 . MR 1254030 | Zbl 0789.32012 · Zbl 0789.32012
[20] J. Kaneko , On the fundamental group of the complement to a maximal cuspidal plane curve , Mem. Fac. Sci. Kyushu Univ. Ser. A. 39 ( 1985 ), 133 - 146 . MR 783228 | Zbl 0604.14007 · Zbl 0604.14007
[21] SH. Kobayashi , Hyperbolic Manifolds and Holomorphic Mappings , Marcel Dekker , N.Y ., 1970 . MR 277770 | Zbl 0207.37902 · Zbl 0207.37902
[22] V.Ja. LIN , Liouville coverings of complex spaces, and amenable groups , Math.USSR-Sb. 60 ( 1988 ), 197 - 216 . MR 882834 | Zbl 0655.32025 · Zbl 0655.32025
[23] K. Masuda - J. Noguchi , A construction of hyperbolic hypersurfaces of Pn (C) , Math. Ann. 304 ( 1996 ), 339 - 362 . MR 1371771 | Zbl 0844.32018 · Zbl 0844.32018
[24] A. Nadel , Hyperbolic surface in P3 , Duke Math. J. 58 ( 1989 ), 749 - 771 . Article | MR 1016444 | Zbl 0686.32015 · Zbl 0686.32015
[25] M. Namba , Geometry of projective algebraic curves , Marcell Dekker , N.Y. , 1984 . MR 768929 | Zbl 0556.14012 · Zbl 0556.14012
[26] M. Oka , Symmetric plane curves with nodes and cusps , J. Math. Soc. Japan 44 ( 1992 ), 375 - 414 . Article | MR 1167373 | Zbl 0767.14011 · Zbl 0767.14011
[27] A.Yu. Ol’shanskiy - A.L. Shmel’kin , Infinite groups, in: Algebra IV, Encyclopaeadia of Math. Sci . 37 Springer Berlin , 1993 , 3 - 95 .
[28] R. Piene , Cuspidal projections of space curves , Math. Ann. 256 ( 1981 ), 95 - 119 . MR 620126 | Zbl 0468.14010 · Zbl 0468.14010
[29] V. Popov - E. Vinberg , Invariant theory, in: Algebraic Geometry IV , Parshin and Shafarevich, eds., 123 - 278 , Springer Berlin , 1994 . MR 1309681 | Zbl 0789.14008 · Zbl 0789.14008
[30] H. Royden , Automorphisms and isometries of Teichmüller space, in: Advantances in the Theory of Riemann Surface, 1969 Stony Brook Conf. Ann. of Math. St. 66 , Princeton Univ. Press , 1971 , Princeton, N.J ., 369 - 383 . MR 288254 | Zbl 0218.32011 · Zbl 0218.32011
[31] F. Severi , Vorlesungen über algebraische Geometrie , Leipzig : Teubner , 1921 . JFM 48.0687.01 · JFM 48.0687.01
[32] Y.-T. Siu - S.-K. Yeung , Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane , Invent. Math. 124 ( 1996 ), 573 - 618 . MR 1369429 | Zbl 0856.32017 · Zbl 0856.32017
[33] G. Veronese , Behandlung der projectivischen Verhältnisse der Räume von verschiedenen Dimensionen durch das Princip des Projicirens und Schneidens , Math. Ann. 19 ( 1882 ), 193 - 234 . Article | JFM 13.0485.01 · JFM 13.0485.01
[34] R.J. Walker , Algebraic Curves, Princeton Math . Series 13 Princeton University Press , Princeton, N.J ., 1950 . MR 33083 | Zbl 0039.37701 · Zbl 0039.37701
[35] M. Zaidenberg , On hyperbolic embedding of complements of divisors and the limiting behavior of the Kobayashi-Royden metric , Math. USSR-Sb . 55 ( 1986 ), 55 - 70 . MR 791317 | Zbl 0593.32016 · Zbl 0593.32016
[36] M. Zaidenberg , Stability of hyperbolic imbeddedness and construction of examples , Math. USSR-Sb. 63 ( 1989 ), 351 - 361 . MR 937646 | Zbl 0668.32023 · Zbl 0668.32023
[37] M. Zaidenberg , Hyperbolicity in projective spaces , Proc. Conf. on Hyperbolic and Diophantine Analysis, RIMS , Kyoto , Oct. 26-30, 1992 , TIT , Tokyo , 1992 , 136 - 156 . MR 1247074
[38] O. Zariski , Collected Papers. Vol III: Topology of curves and surfaces, and special topics in the theory of algebraic varieties , The MIT Press , Cambridge Mass. , 1978 . MR 505104 | Zbl 0446.14001 · Zbl 0446.14001
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