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Open surfaces with non-positive Euler characteristic. (English) Zbl 0849.14013
This paper is concerned with a non-complete algebraic surface $$S$$ with non-positive topological Euler number $$e(S) \leq 0$$ of the form $$S = X - D$$, where $$X$$ is a smooth complex projective surface and $$D$$ is a connected curve on $$X$$. The authors first show that, if $$e(S) < - 1$$, then there is a morphism $$f : S \to B$$ onto a curve $$B$$ with $$\overline \kappa (B) = 1$$ such that any general fiber is $$\mathbb{P}^1$$ or $$\mathbb{C}$$, where $$\overline \kappa$$ denotes the log Kodaira dimension. Next, in case $$e(S) = 0$$ or $$-1$$, they show that $$X$$ is birationally a ruled surface, hyperelliptic surface, abelian surface or elliptic surface with $$\kappa (X) = 1$$. Moreover, the possible types of $$D$$ are precisely described in case $$\kappa (X) \geq 0$$. From the proofs they get also the following non-complete version of Castelnuovo’s criterion:
Let $$S$$ be a smooth quasi-projective surface connected at infinity. Suppose that $$e(S) < - 1$$ or $$S$$ is affine and $$e(S) < 0$$. Then there is a morphism $$S \to B$$ onto a curve $$B$$ with $$\overline \kappa (B) = 1$$ such that any general fiber is $$\mathbb{P}^1$$ or $$\mathbb{C}$$.
The argument for this is similar to the classical one due to Castelnuovo, with the help of Deligne’s Hodge theory for open varieties. In another important step of the proof the authors rule out the case $$\overline \kappa (S) = 2$$ by using R. Kobayashi’s inequality of Miyaoka-Yau type. – Finally, by examining the case in which $$e(S) \leq 0$$ and $$D$$ has an irrational component, they show that if $$C \subset \mathbb{P}^2$$ is a reduced plane curve with $$e (\mathbb{P}^2 - C) \leq 0$$, then every irreducible component of $$C$$ is rational.
[Reviewer’s remark: The authors give no precise proof of this fact, and the reader may wonder how to rule out the case in theorem 3 (2) (iv). But this can be done by a precise analysis of the structure of singular fibers of $$\mathbb{C}^*$$-fibrations. The reader may find troubles in other places, e.g., in proposition 1, $$V$$ is asserted to be a subset of $$Y - C$$, which does not seem to be fully correct, since in the proof one needs some birational contractions. But this is harmless for proving proposition 2].
Reviewer: T.Fujita (Tokyo)

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14F45 Topological properties in algebraic geometry 14J25 Special surfaces
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##### References:
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