Let $X$ be a closed subscheme of $\mathbb{P}^n_k$, where $k$ is a field. Let $\mathcal{O}(1)$ be the standard line bundle on projective space, and $\mathcal{O}(m)$ its tensor powers. We know that for $m \gg 0$, the natural map $H^0(\mathbb{P}^n_k, \mathcal{O}(m)) \to H^0(X, \mathcal{O}(m))$ is surjective (this is "Theorem B"). Are there general techniques for showing that this is a surjection for $m$ in a certain interval $[n_0, \infty)$?
This boils down to showing that $H^1(\mathbb{P}^n_k, \mathcal{I}(m)) = 0$ where $\mathcal{I}$ is the sheaf of ideals of $X$. Consequently, when $X$ is a divisor, I know what to do: $\mathcal{I}$ is a line bundle and the cohomology of line bundles is known. But when $X$ is not a divisor, I don't really know any techniques for showing this vanishing fact. I suspect Castelnuovo-Mumford regularity might be relevant, but I don't know how to apply it here. For instance, one can figure out how regular $\mathcal{O}_{\mathbb{P}^n}$ is, and with luck how regular $\mathcal{O}/\mathcal{I}$ is (e.g. if you know what the subvariety $X$ looks like), but I'm not sure why this implies (if it does imply) that $\mathcal{I}$ will satisfy similar regularity conditions.
(Note: this is motivated by certain homework problems I have. I'm just curious about general techniques though, and possibly references.)