Let $X$ be a smooth projective variety, and let $|D|$ be a complete linear system of divisors on $X$. For a given codimension-two locus $S$ within $X$, a natural question is: what is the dimension of the subspace of elements of $|D|$ which completely contain $S$? In terms of cohomologies, we are interested in the subset $$ R := \Big\{ \gamma \in H^0 \big(X,\mathcal{O}_X(D) \big) ~ \big| ~ \gamma(p) = 0 ~\,\forall p \in S \Big\} \,. $$ In full generality this is likely to be a very difficult problem. I am interested in the case that $S$ is smooth, connected, and irreducible. Introducing the blow-up $\pi_S: \hat{X} \to X$ of $X$ along $S$, with $E_S$ the exceptional divisor that projects to $S$, my question is whether the following isomorphism always holds, $$ R \stackrel{?}{\cong} H^0\Big(\hat{X},\mathcal{O}_\hat{X}\big(\pi_S^*(D)-E_S \big) \Big) \,. $$
The underlying intution is that, from the point of view of $\mathcal{O}(D)$ as the sheaf of functions $f$ with divisor class $(f) \geq -D$, the term $-E_S$ restricts the functions on $\hat{X}$ to those that vanish over $E_S$, which should be in correspondence with functions on $X$ that vanish over $S$. In the simple examples I've checked, this isomorphism appears to hold. But I'm not sure whether or not this is due to a special simplification in those cases.
Yes, this follows from the exact sequence $$ 0 \to H^0(X,I_S(D)) \to H^0(X,\mathcal{O}_X(D)) \to H^0(S,\mathcal{O}_S(D)) $$ and an isomorphism $$ \pi_{S*}\mathcal{O}_{\hat{X}}(\pi_S^*(D) - E) \cong \mathcal{O}_X(D) \otimes \pi_{S*}\mathcal{O}_{\hat{X}}(- E) \cong \mathcal{O}_X(D) \otimes I_S \cong I_S(D). $$