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Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does there exist a smooth projective curve $C\subset X$ such that the natural homomorphism $NS(X)\to NS(C)$ is injective?

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    $\begingroup$ If $NS(X) = \mathbb{Z}$ the homomorphism to $NS(C)$ for ANY curve $C \subset X$ is injective. $\endgroup$ – Sasha Feb 1 at 20:35
  • $\begingroup$ I agree. But NS(X) might not be $\mathbb{Z}$. I am only assuming $X$ is projective and integral over $k$. $\endgroup$ – JB_TI Feb 1 at 20:38
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    $\begingroup$ $NS(C)=\mathbb{Z}$ for a smooth projective curve. $\endgroup$ – abx Feb 1 at 20:59

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