zbMATH — the first resource for mathematics

Hyperbolicity of the complements of plane algebraic curves. (English) Zbl 0842.32021
For generic hypersurfaces \(\Gamma_1, \dots, \Gamma_k\) in \(\mathbb{P}^n\) with sum of degrees exceeding \(2n\) the complement of their union is conjectured to be complete hyperbolic and hyperbolically embedded, a problem which seems the more difficult the smaller \(k\) is. This paper is concerned with plane curves, and the main result is the proof of the conjecture for three quadrics.
The proof has two parts. Using the Second Main Theorem of Value Distribution Theory and a certain configuration of 18 lines it is shown that a non constant entire curve in the complement has to be algebraically degenerate. This case is excluded by simple geometric arguments.
Furthermore there are some partial results on the case of two quadrics and a line.

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI