Pull back of ideal sheaf under blow up.

Question

In general if $\varepsilon:\tilde{X}\rightarrow X$ is the blow up of $X$ along the close subvariety $Z\subset X$, ${\cal{I}}_Z$ is the ideal sheaf of $Z$ and $E$ the exceptional divisor, then there is a surjection $$\varepsilon^*{\cal{I}}_Z\rightarrow {\cal{I}}_E$$ with kernel $L^1\varepsilon^*{\cal{O}}_Z$. Can anybody tell me a bit more about this sheaf? I'm interested in the simplest case possible; smooth surface blown up at a point, i.e., $$0\rightarrow L^1\varepsilon^*{\cal{O}}_p\rightarrow \varepsilon^*{\cal{I}}_p\rightarrow {\cal{O}}_{\tilde{X}}(-E)\rightarrow 0.$$ What is the support? Cohomology? Isomorphic to something in terms of $E$?

Answer 1

In the case of the blow up of a point on a smooth surface, $L_1\epsilon^*O_p \cong O_E(-1)$.