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Logarithmic del Pezzo surfaces of rank one with non-contractible boundaries. (English) Zbl 0596.14024
In this paper and the one announced above the authors continue their study of non-complete algebraic surfaces. The first paper is a collection of technical results, which are necessary for the sequel. In the second paper the authors prove many theorems about surfaces with logarithmic Kodaira dimension $$-\infty$$ [for standard definitions and earlier results see M. Miyanishi, ”Non-complete algebraic surfaces”, Lect. Notes Math. 857 (1981; Zbl 0456.14018)]. - The main theorem is the following: Let (V,D) be a logarithmic del Pezzo surface of rank one with non-contractible boundary, and let $$X=V-D$$. Then either X is affine ruled or X is a platonic $${\mathbb{A}}_*^ 1$$-fibre space.
Affine ruledness by definition is that X contains an open set isomorphic to $$U\times {\mathbb{A}}^ 1$$, U some curve. The ”del Pezzo surface” definition is a bit long winded. Definition of platonic $${\mathbb{A}}_*^ 1$$-fibre space is even more so; but over the complex numbers, the authors prove that these surfaces are nothing but quotients of the affine plane by a non-cyclic small finite subgroup of GL(2,$${\mathbb{C}})$$ and deleting the unique singular point. - They also prove the following theorem: Let X be a non-singular rational surface with logarithmic Kodaira dimension $$-\infty$$. Assume that X is not affine ruled and that for a smooth completion (V,D,X) of X, the intersection matrix of D is not negative definite. Then X is an $${\mathbb{A}}_*^ 1$$-fibre space over $${\mathbb{P}}^ 1$$ ($${\mathbb{A}}_*^ 1 = {\mathbb{A}}^ 1-$$single point) (i.e., there exists a morphism $$\pi : X\to {\mathbb{P}}^ 1$$, surjective and the general fibre is isomorphic to $${\mathbb{A}}^ 1-$${point}. Moreover X is affine uniruled, i.e., there exists a dominant quasi-finite morphism $$p : U\times {\mathbb{A}}^ 1\to X$$ where U is a curve.
Reviewer: N.Mohan-Kumar

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J25 Special surfaces