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We prove the existence of twistor spaces of algebraic dimension two over the connected sum of four complex projective planes      4 C P 2 4 \mathbb {C}\mathbb {P}^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.

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