It has been known that twistor spaces provide nice examples of compact complex 3-fold whose algebraic dimension takes all values from zero to three.
Most compact twistor spaces are of algebraic dimension zero, and also a lot of examples are already known of twistor spaces of algebraic dimension three. Also, twistor spaces of K3 surfaces, complex tori (and also some Hopf surfaces) form a good class of twistor spaces whose algebraic dimension is one.
In this talk, I will present twistor spaces of algebraic dimension one with a different flavor; namely I will present a series of simply connected twistor spaces of algebraic dimension one whose general fiber of the algebraic reduction is birational to an elliptic ruled surface. In these examples, a pair of Hopf surfaces are contained as a reducible fiber of the algebraic reduction.
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