Regular Paper

We study the algebraic dimension of twistor spaces of positive type over 4CP². We show that such a twistor space is Moishezon if and only if its anti-canonical class is not nef. More precisely, we show the equivalence of being Moishezon with the existence of a smooth rational curve having negative intersection number with the anticanonical class. Furthermore, we give precise information on the dimension and base locus of the fundamental linear system |-1/2|. This implies, for example, dim|-1/2K| a(Z). We characterize those twistor spaces over 4CP², which contain a pencil of divisors of degree one by the property dim|-1/2K| = 3.