Geometry of some twistor spaces of algebraic dimension one

It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n{\mathbb{C}\mathbb{P}}^2$, 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 $n{\mathbb{C}\mathbb{P}}^2$ for any $n\ge 5$, while the latter kind of example is constructed over $5{\mathbb{C}\mathbb{P}}^2$. Both of these seem to be the first such example on $n{\mathbb{C}\mathbb{P}}^2$. 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.