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.


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.

  • $\begingroup$ Thanks, yes, I've seen that paper. I guess $\mathcal{R}$ CM implies that $X'$ is CM. Do you know if these papers say something in particular about complete intersections? $\endgroup$ – J.C. Ottem Apr 20 '11 at 12:33
  • $\begingroup$ I think there are some results of Cutkosky and Herzog dealing with the case where $I$ is a locally complete intersection ideal. Look at their paper "Cohen-Macauley coordinate rings of blow-up schemes", Comm. Math. Helv. 72 (1997) $\endgroup$ – Francesco Polizzi Apr 20 '11 at 13:29

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.)

  • $\begingroup$ Dear Prof. Kov\'acs, I have a small question when applying this theorem. Here in you theorem, do we assume $Z$ is equidimensional? Thanks for your reading! $\endgroup$ – user42804 Sep 1 '19 at 15:32
  • $\begingroup$ Also, if $Z$ is local complete intersection, is $Z$ always lci subscheme in $X$? $\endgroup$ – user42804 Sep 1 '19 at 16:17
  • $\begingroup$ Dear @user42804: $Z$ is automatically locally equidimensional for being CM. If $X$ is disconnected, then it can have different dimensional connected components and then $Z$ could also, but I would expect that for all intents and purposes that's as good as being equidimensional. I am not sure about what you are asking in your second question... $\endgroup$ – Sándor Kovács Sep 2 '19 at 17:26
  • $\begingroup$ Dear Professor Kov\'acs, thanks for your answer to the first question. Regarding my second question, we have the definition of local complete intersection ring as a quotient ring of a regular ring cutting down by a regular sequence; also the definition is independent of the choice of the regular ring. My question is $Z$ is locally a spectrum of a local complete intersection ring is it enough to show that $Z$ is a lci subscheme in $X$, especially here $X$ is not smooth in general. $\endgroup$ – user42804 Sep 4 '19 at 1:54

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.