Let $V_X$ be a vector bundle of rank $r>1$ on a smooth (connected) projective variety of dimension $r$. Let s be the global holomorphic section, whose zero locus is a zero dimensional subscheme $Z\subset X$. Is there some result that the section is determined (up to scaling) by its zero locus? At least for some nice vector bundles on nice manifolds?
upd. I meant $X_\Bbb C$ and some conditions on some resolution of $V_X$ by some v.a. line bundles on $X$. jvp cites an excellent result below.