Almost minimal embeddings of quotient singular points into rational surfaces
Let \(V\) be a normal algebraic surface over an algebraically closed field of characteristic zero. Suppose that the singular locus of \(V\) consists of the only point \(P \in V\) which is a quotient singularity. Let \(f \colon X \rightarrow V\) be a minimal resolution and let \(D\) be the reduced exceptional divisor with respect to \(f.\) The singular point \(P\) is called almost minimal when the pair \((X,D)\) is [see \textit{M. Miyanishi} and \textit{S. Tsunoda}, Jap. J. Math., New Ser. 10, 195-242 (1984; Zbl 0596.14023)]. Making use of Mori theory [\textit{S. Mori}, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], the author studies such singularities with logarithmic Kodaira dimension \(\overline{\kappa}(X-D) = -\infty\) and obtains a classification theorem when the support of \(D\) is contained in a fiber of a certain \({\mathbb P}^1\)-fibration. Then the theory of rational surfaces with almost minimal singularities is considered. In fact, the author gives a classification of almost minimal pairs \((X,D)\) in the cases where \(\overline{\kappa}(X-D) = 1\) or \(D\) is irreducible. In conclusion some intersecting examples with \(\overline{\kappa}(X-D) = 0\) are described in details.
scientific article; zbMATH DE number 1295043