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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 $h^0(Z,\mathcal{O}_Z) = c_2(E)$.

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    $\begingroup$ It vanishes if the map $H^0(E) \to H^0(E\vert_Z)$ is surjective and the map $H^1(E) \to H^1(E\vert_Z)$ is injective. $\endgroup$ – Sasha Oct 16 '18 at 5:59
  • $\begingroup$ @Sasha This surjectivity and injectivity is what I want with the vanishing... $\endgroup$ – Alan Muniz Oct 16 '18 at 11:47
  • $\begingroup$ Which is your surface and do you know anything about $c_1(E)$? $\endgroup$ – Sasha Oct 17 '18 at 9:04
  • $\begingroup$ @Sasha I'm trying to calculate this for $X$ a ruled surface over a curve of genus $g$ with invariant $e$ and $E= TX\otimes L $ where $L$ is numerically equivalent to $(n + 2g-2)f$, $f$ the class of a fiber and $n>0$. Hence $c_2(E) = 2n$ and $c_1(E) = 2C_0 + (2n+e-2g-2)f$ where $C_0$ is a minimal section, $C_o^2 = -e$. $\endgroup$ – Alan Muniz Oct 17 '18 at 15:03
  • $\begingroup$ I'd like to know how to work with these cohomologies in general but I'll be happy to know what happens in this particular case. $\endgroup$ – Alan Muniz Oct 17 '18 at 15:05

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