Logarithmic del Pezzo surfaces of rank one with non-contractible boundaries.

*(English)*Zbl 0596.14024In 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.

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