Non-moishezon twistor spaces of 4CP² with non-trivial automorphism group

We show that a twistor space of a self-dual metric on 4CP² 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 Z₂. We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.