The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX \rightarrow X$ is flat?
I’m mainly concerned with $X$ being a smooth projective variety but a general answer would be appreciated.
My idea was to work affine locally and recover something from $Bl_0 \mathbb{A}^n \rightarrow \mathbb{A}^n$ which it seems to me to be flat (but I might be wrong), but I wasn’t able to get far.
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.