Here is the setup:
Let $X(\Sigma)$ be a normal projective toric variety over an algebraically closed $k$ with fan $\Sigma \subseteq N$
Let $Y$ be a projective variety over $k$ with a closed immersion $\iota: Y \hookrightarrow X(\Sigma)$.
Let $D$ be a semi-ample $\mathbb{Q}$-Cartier $T_N$-invariant divisor on $X(\Sigma)$.
When can one hope that the map given by restriction of sections $$H^0(X(\Sigma), \mathcal{O}(D)) \to H^0(\iota(Y), \mathcal{O}(D\vert_{\iota(Y)}))$$ is surjective? I know that in general it is not.
However, one can show that if $\iota : \mathbb{P}^m \to \mathbb{P}^n$ is a linear embedding (whose image is not contained in any hyperplane $\{x_i = 0\}$), then for any such divisor $D$ on $\mathbb{P}^n$, the map of sections is surjective. I am really interested in finding slightly more exotic examples besides just regular projective spaces. I don't need (or really want) fully worked out examples; I am wondering if anyone has any other suggestions of embeddings $\iota : Y \to X(\Sigma)$ for which this setup might work.