Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal. 


**Question:** Is $Y$ also Cohen-Macaulay? Are there common conditions which imply it is? 


If $Y$ is not normal I know of several ways to show that the answer to the first question is no. 


There are obvious spectral sequences but I don't see how to deduce what I want from them, perhaps I'm being dumb (or maybe there is an obvious example).



Blowups of Cohen-Macaulay varieties
An example was given in Section 3 of this [paper by Cutkosky](https://doi.org/10.1007/BF01233425 "Invent Math 102, 157–177 (1990). zbMATH review at https://zbmath.org/?q=an:0718.14025"): "A new characterization of rational surface singularities." (The scheme $Z$ in the last page, which is a blow up of some $m$-primary ideal of a regular local ring of dimension $3$, is normal but not Cohen–Macaulay.)


The algebraic side of this example has been studied quite a bit, so perhaps more explicit examples are known. I am not an expert here, but you can check out a paper by [Huckaba–Huneke](https://doi.org/10.1515/crll.1999.049 "Normal ideals in regular rings. J. Reine Angew. Math. 510, 63–82 (1999). zbMATH review at https://zbmath.org/?q=an:0923.13005") ([Wayback Machine](https://web.archive.org/web/20070117005728/http://www.math.fsu.edu/%7Ehuckaba/normalideals-jour-vers.pdf)), or papers by Vasconcelos (he has a [book](https://doi.org/10.1017/CBO9780511574726 "London Mathematical Society Lecture Note Series. 195. Cambridge: Cambridge University Press. 329 p. (1994). zbMATH review at https://zbmath.org/?q=an:0813.13008") called "Arithmetic of Blowup Algebras" which discussed, among other things, Serre's condition $(S\_n)$ on Rees algebras), and the references there.


