This seems like it should be easy, but unfortunately I don't see how to do it. 


Let $X$ be a variety; I'm happy to assume that $X$ is quasiprojective. If $L\_1$ and $L\_2$ are two non-isomorphic line bundles on $X$, then can we find a curve $C$ in $X$ such that $L\_1$ and $L\_2$ restrict to non-isomorphic bundles on $C$? (In other words, is non-isomorphism of line bundles detected by curves?) 


I think that for a general hyperplane section $H$ of $X$, the map $Pic(X) \rightarrow Pic(H)$ should be injective for some range of dimensions, which maybe breaks down for surfaces. So this reduces immediately to the case of $\dim X \approx 2$, but then... 



Non-isomorphic line bundles detected by sub-curves?
Here is a proof for $\dim X=2$, projective and smooth. We may replace $L\_1, L\_2$ by $L\_1\otimes L\_2^{-1}=L$ and thus suffices to prove that if $L$ is not trivial, it is not trivial restricted to some curve. Take $H$ a large hypersurface section. Then $H^1(L-H)$ can be assumed to be zero and so if $L\_{|H}$ is trivial,, we can assume that $H^0(L)\neq 0$ and thus $L=D$ for some effective curve and since $L\neq 0$, $D>0$. Now, restricted to $H$, it is clear that this line bundle has positive degree and can not be trivial.


