Let,$S$ be a smooth projective surface and $C \subset S$ be a smooth irreducible curve over an algebraically closed field $\mathbb K$ of characteristic $0$.Let, $j:C \to S$ be the inclusion morphism.
Let, $L$ be a line bundle on $C$ which is globally generated (meaning there are global sections of $L$ whose images in the stalk generates $L_x$ as $\mathcal O_x$ module and this happens for every $x$).
At this point my question is that "Is $j_*L$ also a globally generated sheaf on $S$?"
My attempt:$L_c = \sum (f_i)_c .\mathcal O_c$ for all except finitely many $i$.I hope this can be rewritten as $ (j_*L)_c= \sum (f_i)_c .\mathcal O_c$, where now the structure sheaf is the structure sheaf on surface and this happens for all points of the curve.But how to show the same equality for points of $S-C$?
Is the more general version also true that "pushforward of any globally generated vector bundle is globally generated"?
Any help from anyone is welcome.