Log del Pezzo surfaces with not small fractional indices (Q2786724)

A log del Pezzo surface \(S\) is a normal projective variety of dimension 2 with log terminal singularities such that the anticanonical divisor \(-K_S\) is ample. The index of \(S\) is NEWLINE\[NEWLINEr(S)=\sup\{r\in\mathbb{Q}|-K_S\equiv rL\text{ for }L\text{ Cartier on }S\}.NEWLINE\]NEWLINE The log del Pezzo surfaces with \(r(S)\in[1,+\infty)\) have been classified in a series of works [\textit{L. Brenton}, Math. Ann. 248, 117--124 (1980; Zbl 0407.14013)], [\textit{M. Demazure}, Lect. Notes Math. 777, 21--69 (1980; Zbl 0444.14024)], [\textit{T. Fujita}, J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 103--115 (1975; Zbl 0333.14004)], [\textit{F. Hidaka} and \textit{K-i. Watanabe}, Tokyo J. Math. 4, 319--330 (1981; Zbl 0496.14023)].NEWLINENEWLINEBuilding on a construction by Nakayama, who treated the case \(r(S)=2\), the author classifies log del Pezzo surfaces with \(r(S)\in[1/2,1)\). Following [\textit{N. Nakayama}, J. Math. Sci., Tokyo 14, No. 3, 293--498 (2007; Zbl 1175.14029)], to a log del Pezzo surface \(S\) one can associate a triplet \((X,E,\Delta)\) where \(X\) is a smooth surface, \(E\subseteq X\) is a divisor and \(\Delta\subseteq X\) is a zero-dimensional subscheme. Such a triplet is associated to \(S\) in the following way: there exists a pair \((M,E_M)\) and two maps \(\alpha,\beta\) such that \(\beta: M\rightarrow S\) is the minimal resolution of \(S\), \(E_M\) is a multiple of the exceptional divisor, \(\alpha: M\rightarrow X\) is the composition of the minimal resolution of \(\Delta\) and its minimal resolution and \(E_M\) is a linear combination of the pullback of \(E\) and the exceptional divisor of \(\alpha\). The author proves that \(X\) can be either \(\mathbb{P}^2\) or a Hirzebruch surface \(\mathbb{F}_n\). Thus, log del Pezzo surfaces are classified, if \(X=\mathbb{P}^2\), in terms of \([(a,b),k]\) where \(a/b=r\) and \(k=\deg E\) and if \(X=\mathbb{F}_n\), in terms of \([(a,b),n,k_1,k_2]\) where \(a/b=r\) and \(E\equiv k_1\sigma+ k_2 l\) where \(\sigma\) is a section of minimal selfintersection and \(l\) is a fibre of the ruling.