Logarithmic del Pezzo surfaces with rational double and triple singular points
A normal projective surface \(\bar V\) is a log (=logarithmic) del Pezzo surface if (i) \(-K_{\bar V}\) is ample, and (ii) every singularity of \(\bar V\) is a quotient singularity. We consider three problems: (1) Construct all these \(\bar V.\) (2) Given \(\bar V,\) when \(\bar V\cong {\mathbb{P}}^ 2/G\) with a finite subgroup G of PGL(2,C)? (3) The smooth part \(V^ 0=\bar V-Sing(\bar V)\supseteq {\mathbb{C}}^ 2\) if and only if the fundamental group \(\pi_ 1(V^ 0)=(1)?\) The problems were completely solved by \textit{M. Miyanishi} and the author [J. Algebra 118, No.1, 63-84 (1988; Zbl 0664.14019)] when the Picard number \(\rho(\bar V)=1\) and \(\bar V\) is Gorenstein, i.e., every singularity of \(\bar V\) is a rational double singular point. Let \(U^ 0\) be the universal covering of \(V^ 0\) and \(\bar U\) the normalization of \(\bar V\) in \(C(U^ 0)\). Applying the author's classification theory [in Osaka J. Math. 25, 461-497 (1988)] we obtain the following theorem: Let \(\bar V(\neq {\mathbb{P}}^ 2)\) be a log del Pezzo surface with \(\rho(\bar V)=1\) and with exactly one rational triple and several double singular points. Then we have the following: (1) There are altogether 97 possible combinations of singularities of V (see appendix). (2) Suppose \(\pi_ 1(V^ 0)\neq (1)\). Then \(\bar V\cong {\mathbb{P}}^ 2/G\) if and only if \(\rho(\bar U)=1.\) (3) If \(\pi_ 1(V^ 0)=(1)\) then \(V^ 0\) contains \({\mathbb{C}}\times ({\mathbb{C}}-0)\) as a Zariski open set.
scientific article; zbMATH DE number 4089711