Sheaf cohomology of a complement of finitely many points

Question

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $p \subset X$ a closed point in $X$. How do I compute $H^1(\mathcal{O}_{X\backslash p})$? Any reference/idea will be most welcome.

Answer 1

For what follows, I recommend SGA 2 (available on Arxiv). There is an exact sequence $$0\rightarrow H^1(X,\mathcal{O}_X)\rightarrow H^1(X\smallsetminus p,\mathcal{O}_X)\rightarrow H^2_{p}(X,\mathcal{O}_X)\rightarrow H^2(X,\mathcal{O}_X)$$ where $ H^2_{p}(X,\mathcal{O}_X)$ is infinite-dimensional, while $H^i(X,\mathcal{O}_X)$ is finite-dimensional. Therefore $H^1(X\smallsetminus p,\mathcal{O}_X)$ is infinite-dimensional. The computation of $ H^2_{p}(X,\mathcal{O}_X)$ given in SGA 2 and the exact sequence above give an expression for this space, probably not very pleasant.