Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any *a priori* assumption on its positivity). 


Suppose that for each $x,y\in X$, with $x\ne y$, there exists a $k\_0\in\mathbb N$ which depends on $x$ and $y$, such that for all $k\in\mathbb N$, $k\ge k\_0$ there exists a global holomorphic section $\sigma\_k\in H^0(X,L^{\otimes k})$ such that $\sigma\_k(x)=0$ and $\sigma\_k(y)\ne 0$.


**Is it then true that this property is uniform with respect to couples of distinct points of $X$?**


In other words, is it then true that there exists a $k\_1\in\mathbb N$ such that for all $k\in\mathbb N$, $k\ge k\_1$, and for all $x,y\in X$, with $x\ne y$, one can find a global holomorphic section $\tau\_k\in H^0(X,L^{\otimes k})$ such that $\tau\_k(x)=0$ and $\tau\_k(y)\ne 0$?


Thanks in advance!



Uniformity of injectivity for maps associated to linear systems
I think this is true.


The condition implies that for some $n$ the sections of $L^{\otimes n}$ are base point free, so $L^{\otimes n}$ is obtained by pulling back $\mathcal O(1)$ along a map $X \to \mathbb P^N$ for some $N$. This map must be finite, because otherwise there would be a positive dimensional connected subspace of $X$ along which $L^{\otimes n}$ would be trivial, and then its point could not be separated by any power of $L$. By GAGA $X$ is projective, and then $L^{\otimes n}$ is ample (it is a standard result that a pullback of an ample line bundle along a finite map is ample). But this implies that $L$ is ample, so all sufficiently large powers of $L$ must be very ample.


[Addendum]: I guess I am using three facts: that the image $Y$ of $X$ in $\mathbb P^N$ is closed, hence projective, by Chow's theorem; that given a finite map $X \to Y$, if $Y$ is algebraic so is $X$, by GAGA; that the the pullback of an ample line bundle along a finite map is still ample (which implies that $X$ is projective).


