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

We prove the existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes 4CP(2). These are the first examples of twistor spaces of algebraic dimension two over a simply connected Riemannian four-manifold with positive scalar curvature. For this purpose we develop a method to distinguish between twistor spaces of algebraic dimension one and two by looking at the order of a certain point in the Picard group of an elliptic curve.