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Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ whereis an isomorphism, where $i:S\hookrightarrow X$ is the inclusion. Furthermore, assume that there is a curve $C\subset S$ such that $h^0(S,aC) = 1$ for all $a\geq 0$. Then $C$ generates an extremal ray of the Mori cone of $S$. Does $C$ generate an extremal ray of the Mori cone $\overline{\text{NE}}(X)$ of $X$ as well or might it lie in the interior of $\overline{\text{NE}}(X)$?

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ where $i:S\hookrightarrow X$ is the inclusion. Furthermore, assume that there is a curve $C\subset S$ such that $h^0(S,aC) = 1$ for all $a\geq 0$. Then $C$ generates an extremal ray of the Mori cone of $S$. Does $C$ generate an extremal ray of the Mori cone $\overline{\text{NE}}(X)$ of $X$ as well or might it lie in the interior of $\overline{\text{NE}}(X)$?

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ is an isomorphism, where $i:S\hookrightarrow X$ is the inclusion. Furthermore, assume that there is a curve $C\subset S$ such that $h^0(S,aC) = 1$ for all $a\geq 0$. Then $C$ generates an extremal ray of the Mori cone of $S$. Does $C$ generate an extremal ray of the Mori cone $\overline{\text{NE}}(X)$ of $X$ as well or might it lie in the interior of $\overline{\text{NE}}(X)$?

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Mori cone of Picard rank two varieties

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ where $i:S\hookrightarrow X$ is the inclusion. Furthermore, assume that there is a curve $C\subset S$ such that $h^0(S,aC) = 1$ for all $a\geq 0$. Then $C$ generates an extremal ray of the Mori cone of $S$. Does $C$ generate an extremal ray of the Mori cone $\overline{\text{NE}}(X)$ of $X$ as well or might it lie in the interior of $\overline{\text{NE}}(X)$?