  In recent papers math.DG/0701278 and arXiv:0705.0060, 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.
  Our source of the result is a series of self-dual metrics with torus action
constructed by D. Joyce. Actually, for arbitrary Joyce metrics and
U(1)-subgroups of the torus which fixes a torus-invariant 2-sphere, we first
determine the associated minitwistor spaces in explicit forms. Next by
analyzing the meromorphic maps from the twistor spaces to the minitwistor
spaces, we realize projective models of the twistor spaces of all Joyce
metrics, as conic bundles over the minitwistor spaces. Then we prove that for
any one of these minitwistor spaces, there exist Moishezon twistor spaces with
only C*-action whose quotient space is the given minitwistor space. This result
generates numerous Moishezon twistor spaces which cannot be found in the
literature (including the author's papers), in quite explicit form.
