Rationally cubic connected manifolds I: manifolds covered by lines

TL;DR

This paper investigates smooth complex projective varieties with specific rational curve families, establishing bounds on their Picard number and characterizing their structure when the maximum is achieved.

Contribution

It proves that such varieties have Picard number at most three and describes their structure as an adjunction theoretic scroll when this maximum is attained.

Findings

Picard number of these manifolds is at most three

If Picard number equals three, the manifold has a scroll structure

The study links rational curve families to the manifold's geometric structure

Abstract

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n \ge 2 which admit a dominating family V of rational curves of H-degree 3, such that two general points of X may be joined by a curve parametrized by V, and such that there is a covering family of rational curves of H-degree one. Our main result is that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjuction theoretic scroll structure over a smooth variety.