## Classification of log del~Pezzo surfaces of index two

J. Math. Sci. Univ. Tokyo
Vol. 14 (2007), No. 3, Page 293-498.

Noboru Nakayama
Classification of log del~Pezzo surfaces of index two
In this article, a log del~Pezzo surface of index two means a projective normal non-Gorenstein surface $S$ such that $(S, 0)$ is a log-terminal pair, the anti-canonical divisor $-K_{S}$ is ample and that $2K_{S}$ is Cartier. The log del~Pezzo surfaces of index two are shown to be constructed from data $(X, E, \Delta)$ called fundamental triplets consisting of a non-singular rational surface $X$, a simple normal crossing divisor $E$ of $X$, and an effective Cartier divisor $\Delta$ of $E$ satisfying a suitable condition. A geometric classification of the fundamental triplets gives a classification of the log del~Pezzo surfaces of index two. As a result, any log del~Pezzo surface of index two can be described explicitly as a subvariety of a weighted projective space or of the product of two weighted projective spaces. This classification does not use the theory of K3 lattices, which is essential for the classification by Alexeev--Nikulin \cite{AN}. The comparison between two classifications is also discussed.