Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is constant on $Z$.
Is it true that the blow-up $\operatorname{Bl}_Z(X)$ of $X$ along $Z$ is normal ?
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2$\begingroup$ I am not exactly sure how you intend to apply Zariski's main theorem, but the bit in parentheses on the last line cannot be true. If $X_n$ is the isolated 3-fold hypersurface singularity $X_n=(xy = z(z+t^{n}))$ in $\mathbb A^4_{x,y,z,t}$ where $n\geq2$, then $Z$ is a point, and it admits a small resolution with $f^{-1}(Z)\cong \mathbb P^1$ by blowing up the ideal $(x,z)$ (this is one side of a 3-fold flop known as Reid's pagoda). The blowup $Bl_ZX_n$ is normal, but not smooth. $\endgroup$– Tom DucatCommented Mar 29, 2021 at 13:11
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$\begingroup$ Thank you very much for your remark. I have corrected the question. $\endgroup$– pi_1Commented Mar 29, 2021 at 13:25
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