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It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP 2 , whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP 2  for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP 2 . The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.

Geometry of some twistor spaces of algebraic dimension one