Extremal rays of non-integral $L$-length

Abstract

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant $\tau_L(R):=-K_X\cdot\Gamma/(L \cdot \Gamma)$, where $\Gamma$ is a minimal curve of an extremal ray $R:=\mathbb R_+[\Gamma]$ on $X$ such that $L \cdot R>0$.