Extremal rays of non-integral $L$-length

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 $τ_L(R):=-K_X\cdotΓ/(L \cdot Γ)$, where $Γ$ is a minimal curve of an extremal ray $R:=\mathbb R_+[Γ]$ on $X$ such that $L \cdot R>0$.

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