Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Assume further that $H^1(\mathcal{O}\_X)=0$. Is $H^0(R^1j\_\*\mathcal{O}\_{X \backslash S})=0$? 



Global sections of higher direct image sheaf
Since $S$ is finite, $X$ being projective is a red herring, so is anything about the (global) cohomology of $\mathscr O\_X$, and 
(to some extent) it being a surface, or regular.


If $X$ is $S\_2$, and $\dim X\geq 2$, then the sheaf version of Exercise III.2.3 in [Hartshorne] (or the exercise itself noting that by $S$ being finite one may start with restricting to affine schemes) implies that there is a long exact sequence
$$
0\to \mathscr H\_S^0(\mathscr O\_X) \to \mathscr O\_X \to j\_\*\mathscr O\_{X\setminus S} \to \mathscr H\_S^1(\mathscr O\_X) \to \\ \to \mathscr H\_{\emptyset}^1(\mathscr O\_X) \to R^1j\_\*\mathscr O\_{X\setminus S} \to \mathscr H\_S^2(\mathscr O\_X) \to \mathscr H\_{\emptyset}^2(\mathscr O\_X) \to \dots
$$
Since $X$ is $S\_2$, $\mathscr H\_S^0(\mathscr O\_X) =\mathscr H\_S^1(\mathscr O\_X) =0$ and $\mathscr H\_{\emptyset}^i(\mathscr O\_X) =0$ for all $i>0$, so this falls apart to the isomorphisms
$$
j\_\*\mathscr O\_{X\setminus S}\simeq \mathscr O\_X
$$
and
$$
R^ij\_\*\mathscr O\_{X\setminus S} \to \mathscr H\_S^{i+1}(\mathscr O\_X)
$$
for all $i>0$. (In particular $R^ij\_\*\mathscr O\_{X\setminus S}$ is supported on $S$). 
Since $S$ is finite, $R^ij\_\*\mathscr O\_{X\setminus S}$ is the sheaf associated ot its own global sections, in particular $R^ij\_\*\mathscr O\_{X\setminus S}\neq 0$ if and only if $H^0(X,R^ij\_\*\mathscr O\_{X\setminus S})\neq 0$. 


By Grothendieck vanishing $H\_x^{d}(\mathscr O\_X)\neq 0$ for $d=\dim \mathscr O\_{X,x}$ for any $x\in X$, which shows that if $\dim X=d$ near $S$ (e.g., $X$ is equidimensional, or at least that the dimension of the union of irreducible components of $X$ containing $S$ is $d$), then $H^0(X,R^{d-1}j\_\*\mathscr O\_{X\setminus S})\neq 0$.


