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 −KS is ample and that 2KS is Cartier. The log del Pezzo surfaces of index two are shown to be constructed from data (X,E,∆) called fundamental triplets consisting of a non-singular rational surface X, a simple normal crossing divisor E of X, and an effective Cartier divisor ∆ 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 [3]. The comparison between two classifications is