Let $\nu:\tilde{X}\rightarrow X$ be the normalization of an integral scheme $X$. Let $Y$ be the closed subset of $X$ where $\nu$ fails to be an isomorphism, endowed with its reduced subscheme structure.
I'm trying to better understand $\nu^{-1}(Y)$. In particular, I have the following question:
Is it possible for there to be singular points of $\nu^{-1}(Y)$ that map, by $\nu$, to regular points of $Y$? If so, what is an example?