Explicit construction of new Moishezon twistor spaces

Abstract

In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the author's knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C*-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP^4. It turns out that the structure of the minitwistor space is independent of n. Further we concrelely construct a CP^2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP^1 x CP^1 in LeBrun's case is replaced by our minitwistor spaces found in math.DG/0508088.