Suppose I have a projective toric variety $X = X(N)$ (over a field $k$) associated to a polytope $N$, and I have $p_1,...,p_n \in D$ generically chosen points in the toric boundary $D$ of $X$.
I am interested in when I can say that the subspace $V \subset H^0(X,-K_X)$ of sections which vanish at all the $p_i$ is of dimension $h^0(X, -K_X) - n$, for generic choices of the $p_i \in D$.
We may write $V = H^{0}(X, -K_X \otimes \mathcal{I}_{p_1} \otimes \cdots \otimes \mathcal{I}_{p_n})$.
The question can be rephrased as follows:
I can consider the morphism $\prod_{i = 1}^k ev_i : H^0(X,-K_X) \rightarrow k^n$ given by evaluating a section at each $p_i$. I am interested in some general situations when such a morphism is surjective (for generic p_i). (Each $ev_i$ is only well-defined up to a non-zero scalar, but asking whether the morphism is surjective or not still makes sense).
Another way again to rephrase this, is whether for each $i$ we can find a divisor $D_i \in |-K_{X}|$ such that $p_i \in D$ and $p_j \not \in D_i \ \forall j \not = i$. (This is equivalent to saying we map on to the $ith$ basis vector of $k^n$ above).
Intuitively this should be true in many nice instances, as vanishing at a point should be a codimension $1$ condition on the space of anti-canonical sections, but of course the subspaces for vanishing at different $p_i$ may not intersect transversely.
Is this true for instance if I assume $X$ is smooth Fano surface ($N$ a Fano polytope of dim 2)?
Any help is appreciated.