Sum of ideal sheaves commutes with taking global sections

Question

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ effective divisors intersecting each other at finitely many points. Is it true that $$H^0((\mathcal{I}_C(d)+\mathcal{I}_D(d))/\mathcal{I}_X(d))=(I_d(C)+I_d(D))/I_d(X),$$ where $I_d(-)$ is the degree $d$th graded piece of the respective ideal?

Answer 1

For large $d$ of course this is true. It may be false for small $d$. For example, take $X=\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^3$, the quadric hypersurface. Let $C,D$ be $\mathbb{P}^1\times \{p\},\mathbb{P}^1\times \{q\}$ for points $p\neq q$. You can see that for $d=0$, the map you describe is not onto.

Answer 2

(This is not exactly what you asked, but it's close to it.)

I'm going to start by stating a theorem about $\mathcal{O}$-modules on a projective variety, then I'll apply it to your case.

General Case

Let $X=\operatorname{Proj}(S)$, where $S$ is a graded algebra of finite type over a field $k$. Let $\mathfrak{m}$ be the irrelevant ideal. Given an $\mathcal{O}_X$-module $\mathcal{F}$, set $$\Gamma_*(\mathcal{F}):= \bigoplus_{d\in \mathbb{Z}} \Gamma(X,\mathcal{F}(d)). $$ This is naturally a graded $S$-module.

Theorem. Let $M$ be any graded $S$-module, $\widetilde{M}$ the induced $\mathcal{O}_X$-module. Then there is a (natural) exact sequence of graded modules $$0\to H^0_\mathfrak{m}(M)\to M\to\Gamma_*(\widetilde{M})\to H^1_\mathfrak{m}(M)\to0,$$ where the middle map is the natural one.

Your Case

For you, $S$ is the homogeneous coordinate ring of $X$, and $M=S/(I(C)+I(D))$. Since $S/(I(C)+I(D))$ is a ring, I'll call it $R$ instead of $M$. Since $C\cap D$ is a complete intersection, it's also Cohen-Macaulay, and therefore so is its homogeneous coordinate ring. Therefore, the depth of $R$ (along its irrelevant ideal) is equal to the dimension of $R$, namely $1$. Hence, the first term in the above exact sequence vanishes. However, the third term never vanishes (or else the depth of $R$ would be at least $2$). In summary:

If $C\cap D$ is finite and non-empty, then the natural map $$ S/(I(C)+I(D)) \to \Gamma_*(\mathcal{O}_X/(\mathcal{I}_C+\mathcal{I}_D))$$ is injective but not surjective.