Global sections of Serre's twisting sheaf

Question

Let $I_1$ and $I_2$ be homogenous ideals in $A:=\mathbb{C}[X_0,\ldots,X_n]$. Assume that $I_1 \subset I_2$. Let $X=\mathrm{Proj} (A/I_1)$ and $Y=\mathrm{Proj}(A/I_2)$. Then,

1) Is it true that the natural morphism $\Gamma(X,\mathcal{O}_X(n)) \to \Gamma(Y,\mathcal{O}_Y(n))$ is surjective?

2) Is it true that $\Gamma(X,\mathcal{O}_X(n))$ is isomorphic to the degree $n$ graded piece in $A/I_1$?

Answer 1

I think when both $I_1$ and $I_2$ are saturated，the answer to the first question is always right.
In the general case, if we regard $C[x_1,x_2,\ldots,x_n]/I_i ,i=1,2$ as a graded module over itself, then according an exercise in GTM 52 Chapter2 section5, one may find that $C[x_1,x_2,\ldots,x_n]/I_i$ and the graded module $M$, which is the direct sum of modules of the form as $\Gamma(X,\mathcal{O}_X(n)),n>0$, have the same degree parts when the degree $n$ is large enough. So when $n$ is large enough, your question is correct.
I hope this can provide some help.