Classification of Moishezon twistor spaces on 4CP^2

In this paper we provide a classification of all Moishezon twistor spaces on the connected sum of four complex projective planes. This is given by means of the anticanonical system of the twistor spaces. In particular, we show that the anticanonical map is birational, two to one over the image, or otherwise the image of the anticanonical map is a rational surface. We also obtain the structure of the images of the anticanonical map in each of the three cases in explicit forms.