Global sections of higher direct image sheaf

Question

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 $H^0(R^1j_*\mathcal{O}_{X \backslash S})=0$?

Answer 1

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 $H^0(X,R^{d-1}j_*\mathscr O_{X\setminus S})\neq 0$.