Twistor lines on algebraic surfaces

We give quantitative and qualitative results on the family of surfaces in CP3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterward we prove that twistor lines are Zariski dense in the Grassmannian Gr(2, 4). Then, for any degree d≥4, we give lower bounds on the maximum number of twistor lines contained in a degree d surface. The smooth and singular cases are studied as well as the j-invariant one.