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.



What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?
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 $\ H^1(X,\mathcal{O}\_X)$.


