TITLE: When is a blow-up Cohen-Macaulay?
QUESTION [10 upvotes]: 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.

REPLY [7 votes]: The local complete intersection case is actually straightforward:

Claim
  Let $X$ be CM and $Z$ an lci subscheme. Then $Bl_ZX$, the blow-up of $X$ along $Z$ is also CM.

Proof: 
Let $\pi:Bl_ZX\to X$ denote the blow-up. Clearly, $X\setminus \pi^{-1}Z\simeq X\setminus Z$ is CM.
Since $X$ is CM, so is $Z$. 
Since $\mathscr I$, the ideal sheaf of $Z$ is locally generated by a regular sequence, the sheaf of rings $\oplus \mathscr I^n/\mathscr I^{n+1}$ is locally a polynomial ring and hence $E=\pi^{-1}Z\to Z$ is a $\mathbb P^r$-bundle where $r={\rm codim}_X Z-1$.
Since $Z$ is CM, so is $E$. Finally, since $E$ is a Cartier divisor which is CM, so is $Bl_ZX$ along $E$.   $\square$
(For more on this see $\S$5 of this paper.)