A conic bundle description of Moishezon twistor spaces without effective divisors of degree one

Abstract.: In [K2] Moishezon twistor spaces over the connected sum (), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for by constructing a modification which is a conic bundle over . We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and .