Manifolds covered by lines and extremal rays

TL;DR

This paper studies the structure of smooth complex projective varieties covered by certain rational curves, proving the existence of extremal rays in the cone of curves spanned by these curves under specific degree conditions.

Contribution

It establishes that varieties covered by rational curves with specified degrees have a covering family whose numerical class spans an extremal ray in the cone of curves.

Findings

Existence of extremal rays spanned by rational curves under degree conditions.

Coverage of the variety by a family of rational curves with extremal properties.

Conditions relating anticanonical degree and covering by rational curves.

Abstract

Let $X$ be a smooth complex projective variety and let $H \in \pic(X)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\dim X -1)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\cone(X)$.