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).
Best Answer
An example was given in Section 3 of this paper by Cutkosky: "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 here, or papers by Vasconcelos (he has a book called "Arithmetic of Blow-up algebras" which discussed, among other things, Serre's condition $(S_n)$ on Rees algebras), and the references there.