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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 $X$?