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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 $Bl_Z(X)$ of $X$ along $Z$ is normal ?

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 ?

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 $Bl_Z(X)$ of $X$ along $Z$ is normal (then it would be even smooth by Zariski main theorem)?

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 $Bl_Z(X)$ of $X$ along $Z$ is normal ?