# Is the set where $B\otimes\mathfrak{m}_y$ is globally generated, with $B$ a globally generated ample line bundle, non-empty?

Let $$X$$ be a complex projective variety, and let $$B$$ be an ample and globally generated line bundle on $$X.$$ By Example 1.2.9 in Lazarsfeld's book, the set $$U$$ of points $$y\in X$$ such that $$B\otimes\mathfrak{m}_y$$ is globally generated is an open subset, $$\mathfrak{m}_y$$ being the ideal sheaf of the closed point $$y.$$ However I do not understand if this set is also non-empty.

If $$B$$ is not very ample, in which case $$U=X,$$ I don't see why $$U\neq\emptyset$$: denoting by $$\phi\colon X\rightarrow\mathbb{P}^N_{\mathbb{C}}$$ the morphism determined by $$B,$$ (which is finite by ampleness,) if $$y\in U,$$ then $$\phi$$ separates points from $$y$$ and is unramified at $$y.$$ However $$\phi$$ is not injective (actually there exists a non-empty open subset in $$\mathbb{P}^N_{\mathbb{C}}$$ where fibers contain exactly $$\deg(\phi)$$ points), and the fibers containing a single point are those containing a point of total ramification.

Any help is greatly appreciated.

This set can be empty. Let $$X$$ be an elliptic curve, let $$B=\mathcal{O}_X(2p)$$ for some point $$p\in X$$. Then $$B$$ is ample and globally generated, but $$\mathcal{O}_X(2p-q)\cong \mathcal{O}_X(2p)\otimes \mathfrak{m}_q$$ is not globally generated for any $$q\in X$$: the space of global sections has dimension one by Riemann-Roch, so global generation would imply that there is a surjection $$\mathcal{O}_X\to\mathcal{O}_X(2p-q)$$. But a surjection of line bundles is an isomorphism, contradiction.