Algebraic surfaces with infinitely many twistor lines

TL;DR

This paper investigates algebraic surfaces in complex projective space that contain infinitely many twistor lines, establishing degree restrictions and providing constructive existence results for surfaces of even degree using quaternionic slice regularity.

Contribution

It proves that such surfaces cannot have odd degree and offers constructive methods for even degrees via quaternionic slice regularity and normalization techniques.

Findings

Surfaces with infinitely many twistor lines must have even degree.

Constructive existence results are provided for surfaces of even degree.

Odd degree surfaces with infinitely many twistor lines do not exist.

Abstract

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalization map of a surface, we give constructive existence results for even degrees.