Rational Surfaces With an Anti-Canonical Cycle

Abstract

A curve D on a compact analytic surface Y is called an anti-canonical cycle if Y is smooth near D,-D is a canonical divisor, and if either D is an irreducible rational curve with an ordinary node or D is reducible and the distinct irreducible components Dow ..., D-I of D are smooth rational curves forming a polygon. This paper deals with rational surfaces Y (with at worst rational double points) endowed with an anti-canonical cycle D such that D contains no exceptional curve and the intersection matrix (Di.Dj) is negative. This amounts to D. D ? 0 if D is irreducible and Di. Di -2 for all i if D is reducible. The Picard group of these surfaces is studied in Chapter I. One of our main findings is that for s ? 5 the Picard group of such a surface contains a naturally defined infinite root system which enables us to give a precise description of the classes which represent exceptional curves. The restriction to s Md for rational surfaces Y endowed with an anti-canonical divisor D = Do + + D, with a given cycle of self-intersection numbers d = (-Do.Do,..., -Ds-1.Ds-1) of length s ? 5. In order to get a reasonable space we must discard some pairs (Y, D), however. The base Md is entirely given in terms of the (abstract) infinite root system defined in Chapter I. In fact, it is nothing but the orbit space M introduced and investigated in [19]. In that paper we proved that the global holomorphic functions separate the points of Md and we gave a precise description of the holomorphic hull Md of Md. This holomorphic hull is a Stein manifold and the added material Md Md is naturally stratified.