A New Series of Compact Minitwistor Spaces and Moishezon Twistor Spaces over Them

In recent papers [8, 9], we gave explicit description of some new Moishezon twistor spaces. In this paper, generalizing the method in the papers, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases. Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce [12]. Actually, for arbitrary Joyce metrics and U(1)-subgroups of the torus which fixes some 2-sphere, we first determine the associated minitwistor spaces in an explicit form. By analyzing meromorphic quotient maps for the naturally induced C-actions on the twistor spaces, we realize projective models of the twistor spaces of all Joyce metrics, as conic bundles over the minitwistor spaces. Then we investigate deformations of these twistor spaces which preserve the C-action but do not preserve the whole of the torus action. We give a sufficient condition for the U(1)-subgroup for which the twistor spaces admit such a deformation whose resulting twistor space still has a (meromorphic) conic bundle structure over the minitwistor spaces.