Geodesic
Fractional indices of log Del Pezzo surfaces
Izvestiya. Mathematics , Tome 33 (1989) no. 3, pp. 613-629

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The fractional index of a (possibly singular) $Q$-Gorenstein del Pezzo surface $X$ is the greatest rational number $r$ such that $-K_X\equiv rH$, where $H$ is a primitive Cartier divisor. This paper describes the set of values taken by fractional indices of del Pezzo surfaces with log terminal singularities. Bibliography: 7 titles. 
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     author = {V. A. Alexeev},
     title = {Fractional indices of log {Del} {Pezzo} surfaces},
     journal = {Izvestiya. Mathematics },
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     language = {en},
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V. A. Alexeev. Fractional indices of log Del Pezzo surfaces. Izvestiya. Mathematics , Tome 33 (1989) no. 3, pp. 613-629. http://geodesic.mathdoc.fr/item/IM2_1989_33_3_a7/