On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case

Details On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case On the Hyperbolicity of the Complements of Curves in Algebraic Surfaces: The Three Component Case

1993-12-14

arxiv.org/abs/alg-geom/9312008v1

Mathematics

Algebraic Geometry

26 pages, LaTeX

Scientific paper

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a line and three quadrics. The main results are Let C be the union of three curves in P_2 whose degrees are at least two, one of which is at least three. Then for generic such configurations the complement of C is hyperbolic and hyperbolically embedded. The same statement holds for complements of curves in generic hypersurfaces X of degree at least five and curves which are intersections of X with hypersurfaces of degree at least five. Furthermore results are shown for curves on surfaces with picard number one.

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