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We show that a twistor space of a self-dual metric on 4CP 2 with U(1)-isometry is not Moishezon iff there is a C*-orbit biholomorphic to a smooth elliptic curve, where the C*-action is the complexification of the U(1)-action on the twistor space. It follows that the U(1)-isometry has a two-sphere whose isotropy group is Z 2 . We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreusler is affirmative even though a twistor space is required to have a non-trivial automorphism group.

Non-Moishezon twistor spaces of $4{\mathbf {CP}}^2$ with non-trivial automorphism group