When is the morphism between global sections surjective

Question

Let $C$ be a smooth projective irreducible curve. Let $Z$ be a closed subscheme of $C$ consisting of a finite set of points. Denote by $i:Z \to C$ the closed immersion. Note that this is a proper morphism. Let $\mathcal{F}$ be a locally free sheaf on $C$. So, $i_*i^* \mathcal{F}$ is a coherent sheaf. Hence, $H^0(i_*i^* \mathcal{F})$ is finite dimensional. If $\mathcal{F}$ is globally generated or ample, can we expect that the natural morphism $$H^0(\mathcal{F}) \to H^0(i_*i^* \mathcal{F})$$ is surjective?

Answer 1

The sources has some fixed dimension, whereas the target is equal to the direct sum, over $z \in Z$, of the fibre of $\mathcal F$ at $z$. So if the number of points in $Z$ is sufficienly large, the map can't be surjective.