Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$. 



> 
> Under what conditions on $Z$ is $X'$
>  Cohen-Macaulay?
> 
> 
> 


In the case $Z$ is non-singular, the blow-up $X'$ will also be non-singular, so in particular CM. At the other extreme, any birational morphism is the blow-up of some ideal, so if $Z$ is horrible, there is no hope of having Cohen-Macaulayness. 


I'm sure this question has been studied in the literature before and I'd be interested in references for sufficient conditions when $X'$ is CM. The case I find most interesting is when $Z$ is a locally complete intersection.



When is a blow-up Cohen-Macaulay?
Since Cohen-Macauleyness is a local property, we can restrict ourselves to the affine case.


So, let $R$ be a Noetherian ring, $I \subset R$ be an ideal and let us consider the so called *Rees algebra*


$\mathcal{R}:= \oplus\_{n=0}^{\infty} I^n=R[It]\subset R[t]$,


together with the associated graded ring


$\mathcal{G}:=\mathcal{R}/I \mathcal{R}$.


Then $\textrm{Proj}(\mathcal{R})$ is the blow-up of $\textrm{Spec}(R)$ along $V(I)$, and the exceptional divisor is $\textrm{Proj}(\mathcal{G})$.


Then your question is closely related to the following:


When is $\mathcal{R}$ Cohen-Macauley?


This problem was studied by several authors and there are many results. See for instance the paper 


[Necessary and sufficient conditions for the Cohen-macauleyness of blow-up algebras](http://www.nd.edu/~cpolini/pdf.dir/pu2.pdf%20) by Polini and Ulrich and the references given there. 


