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In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicitdescription of some new Moishezon twistor spaces. In this paper, developing themethod in the papers much further, we explicitly give projective models of anumber of new Moishezon twistor spaces, as conic bundles over some rationalsurfaces (called minitwistor spaces). These include the twistor spaces studiedin the papers as very special cases.  Our source of the result is a series of self-dual metrics with torus actionconstructed by D. Joyce. Actually, for arbitrary Joyce metrics andU(1)-subgroups of the torus which fixes a torus-invariant 2-sphere, we firstdetermine the associated minitwistor spaces in explicit forms. Next byanalyzing the meromorphic maps from the twistor spaces to the minitwistorspaces, we realize projective models of the twistor spaces of all Joycemetrics, as conic bundles over the minitwistor spaces. Then we prove that forany one of these minitwistor spaces, there exist Moishezon twistor spaces withonly C*-action whose quotient space is the given minitwistor space. This resultgenerates numerous Moishezon twistor spaces which cannot be found in theliterature (including the author's papers), in quite explicit form.

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