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Let $f:X\to Y$ be a finite map of varieties and let $BL_Z(Y)$ be the blow-up of a subscheme $Z\subset Y$. Is there a map $$\phi:BL_{f^{-1}(Z)}(X)\to BL_{Z}(Y)?$$ If so, what can be said about $\phi$? Is it also finite?

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up vote 6 down vote accepted

You need more hypotheses for such a map to exist. For example, suppose that $X = pt$ and $Y = \mathbb A^2$ (over a field $k$, say) and $f$ is the map sending the point to the origin. Now take $Z$ to be the origin of $Y$. The preimage of $Z$ is all of $X$, and so the blow-up of $X$ at $f^{-1}(Z)$ is just $Z$ again. On the other hand, there is no natural map from $X$ to the blow-up of $Y$ along $Z$. (If there were we could pick out a distinguished point on the exceptional divisor, which we can't.)

If $f$ is finite and flat, then unless I am blundering the blow-up of $X$ along $f^{-1}(Z)$ will precisely equal the fibre product over $Y$ of the blow-up of $Y$ along $Z$ with $X$. So in this case, there is a map $\phi$, it is just a base-change, and in particular it is again finite and flat.

If you like, the problem is that Proj (which is what appears in the definition of blow-ups) is not functorial in the most naive way.

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