TITLE: Blowing-up and direct image sheaf.
QUESTION [7 upvotes]: Let  $\pi:Z=Bl_{Y}(X)\rightarrow X$ be the blowing-up of a smooth projective variety X along a subvariety $Y$, $E$ the closed subscheme defined by $\pi^{−1}I_{Y,X}\cdot O_Z$.
Is it true that (without assume $Y$ smooth) $\pi_{\ast}(O_Z(-E))=I_{Y,X}$?
Thanks for your help.

REPLY [7 votes]: As Sándor's nice answer shows, $\pi_{\ast}(O_Z(-E))=I_{Y}$ is not neccesarily true, even for $Y$ a normal subvariety of $X$. On the other hand, the following statement
$$\pi_{\ast}(O_Z(-mE))=I_{Y}^m \quad\mbox{ for $m\gg 0$}$$
always holds, even without any assumptions on the subscheme $Y$. Here is a short explaination why: 
It suffices to deal with the case $X=\mbox{Spec} A$ is affine and $I=I_Y=(g_1,\ldots,g_n)\subset A$. The $g_i$'s determine a surjection $A^r \to I$, and hence a surjection of Rees algebras $\mbox{Sym}^*(O_X^r) \to \bigoplus_{m \ge 0} I^m$. Taking Proj this means that there is an embedding $Z=Bl_Y X \subset \mathbb{P}=\mathbb{P}(O_X^n)$ where the exceptional divisor $E$ corresponds to $O(-1)\big|_Z$. Now, if $p:\mathbb{P}(O_X^n)\to X$ is the projection, we have $$p_* O_{\mathbb{P}}(m) \to p_*O_Z(-mE)$$ is surjective for $m$ large by relative Serre vanishing. Moreover, since $p_*O_{\mathbb{P}}(m)=Sym^m(O_X^n)$ we can identify the image of this map with $I^m$ and hence we have $\pi_{\ast}(O_Z(-mE))=I^m$. 
In particular, if $\pi_* O_E=O_Y$ holds (as in Sándor's answer) the above map will always be surjective and $\pi_{\ast}(O_Z(-mE))=I^m$ for all $m\ge 0$.