Abstract. In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on nCP 2 for arbitrary n ≥ 3, which can be regarded as a generalization of the twistor spaces of 'double solid type' on 3CP 2 studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of 'double solid type' on 3CP2 , projective models of the present twistor spaces have a natural structure of double covering of a CP 2 -bundle over CP 1 . We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n ≥ 4 these are new twistor spaces, to the best of the author's knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.

In this paper we explicitly construct Moishezon twistor spaces on nCP 2 for arbitrary n ≥ 2 which admit a holomorphic C * -action. When n = 2, they coincide with Y. Poon's twistor spaces. When n = 3, they coincide with the ones studied by the author in [14]. When n ≥ 4, 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 explicitly 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 ×CP 1 in LeBrun's case is replaced by our minitwistor spaces found in [15].

We study self-dual metrics on 3CP 2 of positive scalar curvature admitting a non-zero Killing field, but which are not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined by the equations always become twistor spaces of self-dual metrics on 3CP 2 of the above kind. As a corollary, we determine a global structure of the moduli spaces of these self-dual metrics; namely we show that the moduli space is naturally identified with an orbifold R 3 /G, where G is an involution of R 3 having two-dimensional fixed locus. Combined with works of LeBrun, this settles a moduli problem of self-dual metrics on 3CP 2 of positive scalar curvature admitting a non-trivial Killing field. In particular, it is shown that any two self-dual metrics on 3CP 3 of positive scalar curvature admitting a non-zero Killing field can be connected by deformation keeping the self-duality. In our proof, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

Abstract. In this article we investigate deformations of a scalar-flat Kähler metric on the total space of complex line bundles over CP 1 constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is not the standard one.

Abstract. We explicitly construct the twistor spaces of some self-dual metrics with torus action given by D. Joyce. Starting from a fiber space over a projective line whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the desired twistor spaces. Especially an important role is played by Atiyah's flop.