Normal del Pezzo surfaces of rank one with log canonical singularities
Let \(X\) be a complex \textit{normal del Pezzo surface of rank one}, i.e., a normal complete algebraic surface defined over the field of complex numbers, such that \(-K_X\) is an ample \(\mathbb{Q}\)-Cartier divisor and the Picard rank of \(X\) is one. These surfaces appear naturally in the log-minimal model program as models of quasiprojective surfaces of negative logarithmic Kodaira dimension, which are not \(\mathbb{C}^1\)-ruled (see \S 2.5 in [\textit{M. Miyanishi}, Open algebraic surfaces. CRM Monograph Series. 12. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0964.14030)]). The authors show (Theorem 1.1) that if \(X\) has only rational log-canonical singularities then \(X\) has at most five singular points. (Under the assumption of rationality of singularities one can in fact replace the assumptions on \(K_X\) being \(\mathbb{Q}\)-Cartier and ample with the conditions \((-K_X)^2>0\) and \(K\cdot C>0\) for every curve \(C\) on \(X\), in which case \(X\) is called a \textit{numerical del-Pezzo surface}.) This is done as follows. It is known that such \(X\) is rational and \(\mathbb{Q}\)-factorial. Let \(\pi:(V,D)\to X\) be the minimal resolution of singularities. There exists an irreducible curve \(C\) such that \(-C\cdot \pi^*K_X\) attains the smallest positive value. By the results of \textit{G. N. Belousov} [Math. Notes 83, No. 2, 152--161 (2008; Zbl 1148.14018)] one may assume that \(X\) contains a non-log-terminal singular point, which essentially implies that every such \(C\) is a \((-1)\)-curve such that \(|K_V+D+C|=\emptyset\). By the results of \textit{F. Hidaka} and \textit{K. Watanabe} [Tokyo J. Math. 4, 319--330 (1981; Zbl 0496.14023)], \textit{M. Miyanishi} and \textit{D. Q. Zhang} [J. Algebra 118, No. 1, 63--84 (1988; Zbl 0664.14019)], \textit{Q. Ye} [Jap. J. Math., New Ser. 28, No. 1, 87--136 (2002; Zbl 1053.14044)] etc., one may also assume that \(X\) is not Gorenstein. Now if \(C\) meets two \((-2)\)-curves \(D_1, D_2\) contained in \(D\) then the theorem is obtained by studying the \(\mathbb{P}^1\)-ruling of \(V\) induced by \(|D_1+2C+D_2|\) (Theorem 4.1). In the remaining cases the proof is reduced to the case when \(X\setminus\pi(C)\) is a singular \(\mathbb{Q}\)-homology plane and then it is a consequence of a result by the reviewer (Proposition 3.3 in [``Classification of singular \(\mathbb{Q}\)-homology planes. I. Structure and singularities'', to appear in Isr. J. Math, (2013); \url{arxiv:0806.3110}]).
scientific article; zbMATH DE number 6103629
scientific article