Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\subset \widetilde X$ of codimension $>1$ and that $n_{|Y}:Y\rightarrow n(Y)$ has degree $2$ (the quotient of $Y$ by an involution).
Are the blow-ups $Bl_Y(\widetilde X)$ and $Bl_{n(Y)}(X)$ necessarily isomorphic?

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    $\begingroup$ That fails if $X$ is a tacnodal curve. $\endgroup$ – Jason Starr May 4 at 17:33
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    $\begingroup$ Oops! A tacnodal curve fails the "codimension $>1$" hypothesis for $Y$. Instead, consider a "tacnodal surface", e.g., the zero scheme in affine $4$-space $\text{Spec}\ \mathbb{C}[s,t,u,v]$ of the ideal $\langle u,v\rangle \cap \langle u-s^2,v-t^2 \rangle$. $\endgroup$ – Jason Starr May 4 at 18:15
  • $\begingroup$ Thank you very much for your counter-example. I edited the question to be closer to the situation I am interested in. $\endgroup$ – pi_1 May 4 at 18:16
  • $\begingroup$ Thanks. The codimension $>1$ does not add anything: it does not guarantee that the projectivized normal is disconnected. $\endgroup$ – pi_1 May 4 at 18:32

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