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.



when is a section of vector bundle determined by its zero locus?
Take a look at [On sections with isolated singularities of twisted bundles and applications to foliations by curves](http://www.mathjournals.org/mrl/2003-010-005/2003-010-005-008.pdf1.2.) by Campillo and Olivares.


There you will find the following result. 



> 
> **Theorem.** Let $X$ be a projective manifold of dimension $n$, $\mathcal L$ be an ample >line-bundle on $X$, and $E$ a rank $n$ vector bundle over $X$. If $k\gg 0$ and $s, s'$ >are sections of $E \otimes \mathcal L^{\otimes k}$ such that 
> 
> 
> * the zero scheme $Z(s)$ of $s$ has dimension zero, and
> * the zero scheme $Z(s')$ of $s'$ contains $Z(s)$
> 
> 
> then there exists $\varphi \in H^0(X,\mathrm{End}(E))$ such that $s' = \varphi(s)$. In particular, if $E$ is simple ( $H^0(X,\mathrm{End}(E))=\mathbb C$ ) then $s = \lambda s'$ for a suitable complex number $\lambda$.
> 
> 
> 


In the particular case $E=T\mathbb P^n$, it suffices to take $\mathcal L= \mathcal O\_{\mathbb P^n}(1)$ and $k\ge 1$.


