Question

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.

Answer 1

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 by Polini and Ulrich and the references given there.

Answer 2

Here's a slightly different idea for references.

See On Macaulayfication of Noetherian schemes by Takesi Kawasaki. In particular, Theorem 4.1 gives a criterion for when blow-ups of certain ideals are Cohen-Macaulay.

Macaulayfication is a way of blowing up an ideal on a scheme and obtaining a Cohen-Macaulay scheme. Macaulayfications always exist, even in mixed characteristic, as long as a dualizing complex exists.

The point of Theorem 4.1 is that when you blow-up various things (generated mostly be regular sequences, in other words maybe even close to the the complete intersections you mentioned), you still get a Cohen-Macaulay scheme. If your ambient scheme is already Cohen-Macaulay (for example smooth), then you have a lot more flexibility in how you can choose these parameters, whichwhich sounds potentially useful to you.

I should point out that this may not work at all (ie, nothing interesting may result), it's just an idea.