Preimage by birational maps

Question

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$):

Let $i: X \longrightarrow M$ be an embedding of a closed subvariety $X$ in a nonsingular variety (algebraic) $M$ and let $\pi :\tilde{M} \longrightarrow M$ be a proper birational map with $\tilde{M}$ a nonsingular variety, such that $\pi^{-1}(X)= \tilde{X}$ (total transform) is a hypersurface with its singular scheme, denoted by $\tilde{Y}$ has the condition $codim_{\tilde{M}}(\tilde{Y})\geq 3$ and $\pi|_{\tilde{M}\setminus \tilde{X}}$ is an isomorphism.

Thank you so much!

Answer 1

I think there are no such non trivial examples. Indeed, since $M$ is smooth and $\pi$ non trivial, the exceptional locus, say $E$, of $\pi$ in $\tilde{M}$ has codimension $1$. Now, as $\tilde{X}$ is assumed to be an hypersurface, it implies that the strict transform of $X$, say $\pi^{-1}(X)$ is a hypersurface in $\tilde{M}$.

The ambient variety $\tilde{M}$ being smooth, the intersection of $\pi^{-1}(X)$ and $E$, if non-empty, must have codimension $1$ in $\pi^{-1}(X)$. The hypothesis $\pi_{\tilde{M}\backslash \tilde{X}}$ is an isomorphism guarantees that the image of $E$ is included in $X$, which implies that the intersection of $\pi^{-1}(X)$ and $E$ is non empty. Hence $E$ and $\pi^{-1}(X)$ are unions of irreducible components of $\tilde{X}$ and they meet in codimension $1$ in $\pi^{-1}(X)$, so that the singular locus of $\tilde{X}$ has codimension $1$ in $\pi^{-1}(X)$. And thus codimension $2$ in $\tilde{M}$ because $\tilde{X}$ is a hypersurface in $\tilde{M}$.