Twistor lines on algebraic surfaces

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.

arXiv commentWe removed the last section, slightly changed the title and reorganized the proofs

References in corpus (2)

• Twistor transforms of quaternionic functions and orthogonal complex structures
• Twistor interpretation of slice regular functions

Cited by in corpus (5)

• Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in ${\mathbb{CP}}^3$
• Twistor geometry of the Flag manifold
• Algebraic surfaces with infinitely many twistor lines
• Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere
• Surfaces in the flag threefold containing smooth conics and twistor fibers