Global sections of projective schemes

Question

Let $Y$ be a closed subscheme of $X = \operatorname{Proj} S$, where $S = k[x_0,\dots,x_n]$, $k$ algebraically closed. Then $Y = \operatorname{Proj} (S/I)$ for a homegenous ideal $I$ of $S$. How can we understand the global sections $\Gamma(Y,\mathcal{O}_Y(d))$ of the twisted sheaf $\mathcal{O}_Y(d)$ on $Y$? By Exercise II.5.9(b) in Hartshorne, we know that for large $d$ these sections coincide with the degree $d$ component of the ring $S/I$. How large does $d$ need to be for this to be true? Also, what about for smaller $d$?

Answer 1

Since $(S/I)^\sim = \mathcal O_Y$, there is a natural map $S/I\to \Gamma_*(\mathcal O_Y) = \bigoplus_{d\in\mathbb Z}H^0(Y,\mathcal O_Y(d))$ which is an isomorphism in large degrees and whose kernel and co-kernel are given by the local cohomology of $S/I$ with respect to the irrelevant ideal. More precisely, the above map sits in an exact sequence $$0\to H^0_{\mathfrak m}(S/I)\to S/I\to \Gamma_*(\mathcal O_Y)\to H_{\mathfrak m}^1(S/I)\to 0.$$ This can be found in Eisenbud's Geometry of syzygies, Appendix 1 section A1B. In there, you can find more information on (the vanishing of) these local cohomology groups.