Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?



Divisor on variety determined by its restriction to curves
If you assume projective and smooth, this is not hard. Induct on dimension, $\dim X=1$ being the hypothesis. If $\dim X=n\geq 2$ and result proved for smaller dimensions, if $Y\in \mathcal{O}(mH)$, $m>>0$, $H$ a hyperplane section, then we may assume $Y$ is smooth and by induction, for the line bundle $L$ in question, we may assume $L\_{|Y}$ is trivial. Since we may assume $H^1(L(-Y))=0$ by choosing $m$ large, we see that the section $1\in H^0(L\_{|Y})$ lifts and thus $L$ is effective, but disjoint from a hypersurface $Y$. This is impossible unless $L$ is trivial, since $n\geq 2$.


