Twistor lines on algebraic surfaces

We give quantitative and qualitative results on the family of surfaces in CP 3 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