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.
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