Algebraic foliations defined by quasi-lines

TL;DR

This paper explores algebraic foliations generated by quasi-lines in projective manifolds, showing how their properties can lead to rational fibrations under certain conditions, revealing new geometric structures.

Contribution

It introduces a method to construct algebraic foliations from quasi-lines and links their singularities to rational fibrations in projective manifolds.

Findings

Foliations associated to quasi-lines can induce rational fibrations.

The size of the foliation's singular locus affects the existence of fibrations.

General leaves of the foliation map onto quasi-lines in rational varieties.

Abstract

Let X be a projective manifold containing a quasi-line l. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety.