# Non-isomorphic line bundles detected by sub-curves?

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...

• You might look at the answer to the following MO question: mathoverflow.net/questions/233157/…. That question assumes that $X$ is a projective surface. Assuming $X$ is a normal, quasi-projective variety, there is always a normal projective compactification to which both $\mathcal{L}_1$ and $\mathcal{L}_2$ extend as invertible sheaves. – Jason Starr Aug 3 '16 at 18:01

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.
• Just to compare this proof with the proof in that other MO post, here the choice of sufficiently ample divisor $H$ depends, a priori, on $L$. In that other MO post, the goal was to prove that there exists a single ample hyperplane section (after base change to a bigger algebraically closed field, if the ground field is countable) that simultaneously works for all $L$. But that is not what user84144 asked, and Mohan's proof is definitely simpler. – Jason Starr Aug 3 '16 at 18:08