A question inspired by Is the Euler characteristic a birational invariant:

As remarked in Mike Roth's answer to the above linked question, if $X$ and $Y$ are smooth projective varieties in characteristic zero that are birational, then there is a smooth $Z$ with morphisms $p: Z \rightarrow X$ and $q: Z \rightarrow Y$ that are both birational isomorphisms.

Question: Is it possible to arrange for at least one of $p$ and $q$ that the locus in $Z$ where the morphism fails to be an isomorphism is of codimension at least $2$?

Here it might be useful to assume that ${\rm dim}X={\dim Y} \geq 3$.

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    $\begingroup$ I'm pretty sure this can basically never happen (I know there are a bunch of results for smooth 3-folds saying that small contractions like you describe can't exist). I also think they don't exist in general (as long as you demand the varieties to be smooth), but I don't recall the argument. If I remember, I'll post, or someone else... $\endgroup$ – Karl Schwede May 28 '10 at 3:20

No it is not possible (see for instance Thm II:2.4 of Shafarevich: Basic Algebraic Geometry) which says that the exceptional locus is always of codimension $1$ provided the target is smooth. The crucial property is that the target variety be $\mathbb Q$-factorial. The example of Dmitri shows that this condition is necessary as the quadric cone is non-$\mathbb Q$-factorial at its apex. Note, however that in his example a $Z$ mapping to both $X$ and $Y$ will contract a divisor. An example is the blowup of the singularity whose exceptional locus is $\mathbb P^1\times \mathbb P^1$ and the mappings to $X$ and $Y$ restrict to projections on the two factors (and indeed any common map factors through that blowing up).

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