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

In recent papers [Honda, J. Diff. Geom 82: 411–444, 2009], [Honda, Invent. Math 174, 463–504, 2008], we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, 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.