Let $X$ be a complex projective surface an $Z\subset X$ be a finite set of points (reduced closed subscheme of dimension zero).
Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$. Let $E$ be a vector bundle over $X$.
Under which conditions does the cohomology group $$ \mbox{H}^1(X,E\otimes\mathcal{I}_Z) $$ vanish?
I am aware of some results for $E$ a line bundle but none in higher rank. Any reference will be appreciated.
Added: We have the long exact sequence $$ 0 \to \mbox{H}^0(X,E\otimes\mathcal{I}_Z) \to \mbox{H}^0(X,E) \to \mbox{H}^0(Z,E|_Z) \to \mbox{H}^1(X,E\otimes\mathcal{I}_Z) \to \mbox{H}^1(X,E) \to 0 $$ and the vanishing of $\mbox{H}^1(X,E\otimes\mathcal{I}_Z)$ implies the vanishing of $\mbox{H}^1(X,E)$. Thus $\mbox{H}^1(X,E) = \{0\}$ is a necessary condition. What can be imposed to $E$ for this condition also be sufficient?
The case that interests me is when $E$ has rank two and $Z$ is the zero scheme of a global section of $E$. Hence $h^0(Z,\mathcal{O}_Z) = c_2(E)$.