Let $X$ be a compact complex manifold, let $M \to X$ be a holomorphic line bundle, and write $M$ also for its sheaf of holomorphic sections. Let $x,y \in X$ be distinct points. Let $\mathcal{I}_{x,y} \subset \mathcal{O}_X$ denote the ideal sheaf of holomorphic functions vanishing at both $x$ and $y$. If
is surjective. In particular, there exists a section $s \in H^0(X,M)$ with $s(x)=0$ and $s(y)\ne 0$, so the complete linear system $|M|$ separates the points $x$ and $y$.