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Geometry of some twistor spaces of algebraic dimension one. (English) Zbl 1326.32034
Summary: It is shown that there exists a twistor space on the \(n\)-fold connected sum of complex projective planes \(n\mathbb{CP}^2\), 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 \(n\mathbb{CP}^2\) for any \(n \geq 5\), while the latter kind of example is constructed over \(5\mathbb{CP}^2\). Both of these seem to be the first such example on \(n\mathbb{CP}^2\). 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.
32L25 Twistor theory, double fibrations (complex-analytic aspects)
53C28 Twistor methods in differential geometry
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