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Let $X$ be a smooth complex projective variety and let $L$ be a line bundleon it. We describe the structure of the pre-polarized manifold $(X,L)$ for nonintegral 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 $L \cdot R>0$.

Extremal rays of non-integral $L$-length