Twistor lines on algebraic surfaces

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ 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. Afterwards we prove that twistor lines are Zariski dense in the Grassmannian $Gr(2,4)$. Then, for any degree $d\ge 4$, 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.