Gorenstein log del Pezzo surfaces of rank one

Abstract

Let S be a projective normal surface defined over the field C of complex numbers. We say that S is a log de1 Pezzo surface provided S has only quotient singularities and the anticanonical divisor -X, is ample. Note that if S has only quotient singularities, a multiple XA of every Weil divisor A on S becomes a Cartier divisor for some integer N determined by S and hence the intersection theory of Weil divisors is available (cf. Miyanishi and Tsunoda [S]). The rank of S is the Picard rank p(S) = dimo Pit(S) @ Q. Moreover, S is Gorenstein, i.e., K, is a Cartier divisor, if and only if S has only rational double points provided S is a log de1 Pezzo surface. Let p: Y-+ S be the minimal resolution of singularities and let D be the exceptional locus, which we identify with an effective reduced divisor with support D. According to the terminology of [8], a pair (V, D) is a log del Pezzo surface of rank one with contractible boundary iff S is a log dei Pezzo surface of rank one in the above sense. En the present article, we are mainly interested in the case where S is a Gorenstein log de1 Pezzo surface. As for the construction of such a surface, Demazure [j] has shown the existence of a birational morphism G: V+ P*, unless S is P’ x P’ or a quadric cone, which is a composite of at most 8 blowing-ups (cf. Hidaka and Watanabe [7]). They have also shown that 1 --I&I admits a smooth member. On the other hand, Brenton [3] and Bindschadler and Brenton [2] observed topological properties of S and Gorenstein compactification of C*. Though the subject treated in the present article is related to theirs, our approach is more algebro-geometric and different from their topological ones. We shall prove that: