Geometry of Lines on Certain Moishezon Threefolds. I. Explicit Description of Families of Twistor Lines

We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP. Here, line means a smooth rational curve whose normal bundle is O(1) and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double covering of CP branched along a singular quartic surface. On these threefolds we find families of real lines in explicit form and prove that which families have to be chosen as twistor lines depend on how we take small resolutions of the double covering. This is a first step for determining the moduli space of self-dual metrics on 3CP of positive scalar curvature, which admit a non-trivial Killing field.