Twistor lines on algebraic surfaces

TL;DR

This paper investigates the properties and distribution of twistor lines on algebraic surfaces in complex projective space, providing bounds and density results for these special lines.

Contribution

It offers new quantitative and qualitative insights into surfaces with twistor lines, including density in the Grassmannian and bounds on their number.

Findings

Twistor lines are Zariski dense in Gr(2,4).

Provides lower bounds on the number of twistor lines in degree d surfaces.

Analyzes smooth and singular cases, including the j-invariant case.

Abstract

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.