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).