Is there a simple relation between the (global) sections of a line bundle and the (global) sections of its restriction to some subvariety?
Let $X$ be a smooth projective variety and $j\colon C\hookrightarrow X$ some smooth irreducible (codimension 1) subvariety. What is the relation between
$H^0(C,j^\ast\mathcal{O}(D))$ and $H^0(X,\mathcal{O}(D))$,
where $\mathcal{O}(D)$ is some line bundle on $X$ ? (I am considering $\dim X=2$ but I guess this is general)