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Let $X$ be a smooth projective variety. $Z$ be a closed subscheme of codimension $1$ (potentially with embedded points). $Z_{red}$ is a Cartier divisor and $Bl_{Z_{red}}X$ is just $X$. What about the blow up $Bl_ZX$ of $X$ along $Z$?

I am aware of the discussion in both Blow up along codimension one closed subscheme and Blow-up along a subscheme and along its associated reduced closed subscheme and understand in general the blow up can exhibit wild behavior. I am wondering if the seemingly strong restrictions in my case changes the situation.

Thanks!

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    $\begingroup$ Did you compute any examples, e.g. $Z = V(xy, y^2)$ in $\mathbf{A}^2$? $\endgroup$
    – Piotr Achinger
    Commented Sep 15, 2020 at 21:14
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    $\begingroup$ @PiotrAchinger Right, that's the first thing I should have tried before posting this...Sorry. Seems to be just the blow up of the origin in this case, no? $\endgroup$
    – Xuqiang QIN
    Commented Sep 15, 2020 at 22:03

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