# Computation of cohomology with ideal sheaf involved.

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

• 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. – Sasha Oct 16 '18 at 5:59
• @Sasha This surjectivity and injectivity is what I want with the vanishing... – Alan Muniz Oct 16 '18 at 11:47
• Which is your surface and do you know anything about $c_1(E)$? – Sasha Oct 17 '18 at 9:04
• @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, $C_o^2 = -e$. – Alan Muniz Oct 17 '18 at 15:03
• 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. – Alan Muniz Oct 17 '18 at 15:05