# Smooth normalization and blow-up of the exceptional locus

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?

• That fails if $X$ is a tacnodal curve. – Jason Starr May 4 at 17:33
• 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 $\langle u,v\rangle \cap \langle u-s^2,v-t^2 \rangle$. – Jason Starr May 4 at 18:15
• Thank you very much for your counter-example. I edited the question to be closer to the situation I am interested in. – pi_1 May 4 at 18:16
• Thanks. The codimension $>1$ does not add anything: it does not guarantee that the projectivized normal is disconnected. – pi_1 May 4 at 18:32