an:03959631
Zbl 0596.14024
Miyanishi, Masayoshi; Tsunoda, Shuichiro
Logarithmic del Pezzo surfaces of rank one with non-contractible boundaries
EN
Jap. J. Math., New Ser. 10, 271-319 (1984).
00151644
1984
j
14J10 14J25
platonic fibre space; non-complete algebraic surfaces; surfaces with logarithmic Kodaira dimension \(-\infty\); Affine ruledness
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 \textit{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.
N.Mohan-Kumar
Zbl 0596.14023; Zbl 0456.14018