Manifolds Covered by Lines and Extremal Rays

http://dx.doi.org/10.4153/CMB-2011-119-7
Canad. Math. Bull. 55(2012), 799-814

  Published:2011-06-14
 Printed: Dec 2012

Let $X$ be a smooth complex projective variety, and let $H \in \operatorname{Pic}(X)$ 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 $\operatorname{NE}(X)$.