Let $X$ be a projective variety of Picard rank two. We may assume that $X$ is $\mathbb{Q}$-factorial. Then the Mori cone $NE(X)$ has two extremal rays $R_1,R_2$.

Assume that $R_i$ is generated by the class of an effective irreducible curve in $X$, and let $Z_i$ be the locus covered by the effective curves in $X$ numerically equivalent to $C_i$.

Does there exist an example of a variety $X$ with the above properties where the $Z_i$ are reducible subvarieties of $X$?


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