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

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.

### MSC:

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