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Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.

How can we detect when $C$ is an extremal ray of the cone of moving curves $Mov_1(X)$, that is the dual of the effective cone of $X$?

For instance, if we have a morphism $f:X\rightarrow Y$ of relative dimension one having $C$ as a fiber may we conclude that $C$ is an extremal ray of $Mov_1(X)$?

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  • $\begingroup$ This doesn't quite answer your question, but [Lehmann–Xiao, Thm. 1.11, Lem. 3.9, Cor. 3.14] describe $\alpha \in \partial\operatorname{Mov}_1(X)$ as either being of the form $\alpha = \langle L^{n-1} \rangle$ for a big and movable divisor class $L \in \partial\operatorname{Mov}^1(X)$, or $\alpha \cdot M = 0$ for a movable divisor class $M$. $\endgroup$
    – Takumi Murayama
    Commented Jan 27, 2017 at 5:23

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