Geometry of some twistor spaces of algebraic dimension one

TL;DR

This paper constructs new examples of twistor spaces on connected sums of complex projective planes with algebraic dimension one, where the general fiber is birational to either an elliptic ruled surface or a K3 surface, using the anti-canonical system.

Contribution

It provides the first known examples of such twistor spaces on nCP2 with specified fibers and describes their geometric properties, including the presence of non-normal Hopf surfaces.

Findings

Existence of twistor spaces with algebraic dimension one on nCP2 for n≥5.

Construction of twistor spaces with fibers birational to elliptic ruled surfaces or K3 surfaces.

Identification of non-normal Hopf surfaces within these twistor spaces.

Abstract

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 $n\ge 5$, 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.