Double solid twistor spaces: the case of arbitrary signature

Abstract

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.

A new series of compact minitwistor spaces and Moishezon twistor spaces over them

“…Similarly to the notation in [11], let V m be a subspace of H 0 ðZ; mF Þ generated by the image of the natural map…”

“…Then if n f 3 there exist at least dðn À 1Þ e¤ective Uð1Þ-actions on nCP 2 for which there exist non-Joyce self-dual metrics invariant under the Uð1Þ-action whose twistor spaces are Moishezon (Corollary 5.2). We note that the number of previously known Uð1Þ-actions satisfying these properties was only a few for each n ( [17], [10], [11]), and that this result is not known even if we do not require for the twistor spaces to be Moishezon. We also remark that thanks to Proposition 4.4 mentioned above, this is not just the existence theorem, but we have their detailed structure of the twistor spaces in hand.…”

“…These include the twistor spaces in [10], [11] as very special cases. (However much more investigations are required to reach the detailed results about the structure and the construction given in [10] and [11].) A resource of the present construction is a series of self-dual metrics with 2-dimensional torus action constructed by D. Joyce [15], and their associated twistor spaces investigated by A. Fujiki [5].…”

On pluri-half-anticanonical systems of LeBrun twistor spaces

“…They can be chosen in such a way that all irreducible components of the generators are degree 1 divisors. It might be interesting to remark that the situation is almost the same for the twistor spaces studied in [3]; namely, LeBrun twistor spaces and the twistor spaces in [3] Recall that by a result of Pedersen-Poon [7], on any twistor spaces on nCP 2 , all real irreducible members of the system |F | are non-singular. Corollary 2.6 shows that this is true only for the system |F | itself.…”

“…Recently a large number of new Moishezon twistor spaces was obtained [2,3,4]. An important common property of these twistor spaces is that they all admit a C * -action.…”