On the algebraic dimension of twistor spaces over the connected sum of four complex projective planes

Abstract

We study the algebraic dimension of twistor spaces of positive type over $4\bbfP^2$. We show that such a twistor space is Moishezon if and only if its anticanonical 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 $|-1/2K|$. This implies, for example, $\dim |-1/2K|\le a(Z)$. We characterize those twistor spaces over $4\bbfP^2$, which contain a pencil of divisors of degree one by the property $\dim |-1/2K|=3$.

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