What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?

Question

Let $X$ be a smooth projective surface over a field $k$. Is there a way to compute $H^1(X - \{x\},\mathcal{O}_{X-\{x\}})$ in terms of similar invariants for $X$? Actually I'd like to remove even more points, finitely many points.

Answer 1

The general reference for what follows is SGA 2, §1 to 4. Let $S\subset X$ be a finite subset, and $U:=X\smallsetminus S$. There is an exact sequence $$H^1_S(X,\mathcal{O}_X)\rightarrow H^1(X,\mathcal{O}_X)\rightarrow H^1(U,\mathcal{O}_U)\rightarrow H^2_S(X,\mathcal{O}_X)\rightarrow H^2(X,\mathcal{O}_X)$$ Now $H^i_S(X,\mathcal{O}_X)=\bigoplus_{x\in S}H^i_{\{x\} }(X,\mathcal{O}_X)$. This is zero for $i<2=\mathrm{depth}(\mathcal{O}_x)$, and $H^2_{\{x\} }(X,\mathcal{O}_X)$ is a dualizing module over $\mathcal{O}_x$, hence infinite-dimensional over $k$. Thus $H^1(U,\mathcal{O}_U)$ is infinite-dimensional over $k$, and can be described as an extension of the kernel of the homomorphism $\bigoplus_{x\in S}H^2_{\{x\} }(X,\mathcal{O}_X)\rightarrow H^2(X,\mathcal{O}_X)\ $ by $\ H^1(X,\mathcal{O}_X)$.