# Is there always a curve inside $X$ with larger Neron-Severi group?

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?

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