Moishezon twistor spaces without effective divisors of degree one (pre-published version)

Abstract

We study simply connected compact twistor spaces Z of positive type. Assuming that the fundamental linear system j \Gamma 1 2 Kj is at least a pencil, we prove the following theorem: the existence of an irreducible curve C ae Z which is invariant under the real structure of Z and has the property C:(\Gamma 1 2 K) ! 0 implies that the twistor space is Moishezon but does not contain effective divisors of degree one. Furthermore, we prove the existence of such twistor spaces with arbitrary Picard number ae(Z) 5. These are the first examples of Moishezon twistor spaces without divisors of degree one.