Complements of plane curves with logarithmic Kodaira dimension zero. (English) Zbl 1018.14010

Kojima, Hideo

J. Math. Soc. Japan 52, No. 4, 793-806 (2000).

Let \(B\) be a reduced plane curve in the complex plane \(\mathbb P^2\) with \(b\) (\(\geq 1\)) irreducible component(s). The author studies the case where the logarithmic Kodaira dimension \(\overline\kappa (\mathbb P^2-B)\) is zero, and proves the following assertions:
(1) The logarithmic geometric genus of \(\mathbb P^2-B\) is 1.
(2) Each component of \(B\) is a rational curve and the topological Euler characteristic of \(\mathbb P^2-B\) is \(3-b\), unless \(B\) is a smooth cubic.
(3) \(b\leq 3\) always, and \(b=3\) if and only if \(\mathbb P^2-B\cong \mathbb C^* \times \mathbb C^*\).
(4) If \(b=1\) and \(B\) is a rational curve, then \(B\) has a unique singular point \(P\), and \(B\) has exactly two analytic branches at \(P\).
(5) The topological fundamental group of \(\mathbb P^2-B\) is abelian.
The proof depends heavily on the classification theory of affine surfaces with \(\overline\kappa=0\) by the author and on the theory of non-complete surfaces developed by Miyanishi and others. The author further studies the above case (4) and gives some normal forms of the defining equation of \(B\) in the case in which the two analytic branches at \(P\) are both smooth. He also presents examples where the analytic branches are singular.

Reviewer: Takao Fujita (Tokyo)

MSC:

14H50	Plane and space curves
14H20	Singularities of curves, local rings
14J26	Rational and ruled surfaces

Keywords:

reduced plane curve; logarithmic Kodaira dimension; fundamental group; classification theory of affine surfaces