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

Abstract

In [K2] Moishezon twistor spaces over the connected sum \(n{\mathbb{CP}}^2\) (\(n\geq 4\)), 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 \(n\geq 4\) by constructing a modification which is a conic bundle over \({\mathbb P}^2\). 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 \({\mathbb P}^3\).