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Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which an isomorphism from $X\setminus D$ to $Y\setminus Z$. Assume that $X$ is normal.

What is an explicit example of such a divisorial contraction such that $Y$ is not normal?

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    $\begingroup$ What is your definition of "divisorial contraction"? If $X$ is normal, then the morphism $f$ always factors through the normalization of the image of $f$. How is the induced morphism from $X$ to this normalization a "different contraction" from $f$? $\endgroup$
    – Jason Starr
    Commented Dec 3, 2020 at 11:19
  • $\begingroup$ Yes, it boils down to construct a birational morphism $g:X\rightarrow W$ contracting $D$ onto $S$, where $\nu:W\rightarrow Y$ is the normalization of $Y$, $\nu$ is $1$-to-$1$ and an isomorphism between $W\setminus S$ and $Y\setminus \nu(S)$. $\endgroup$
    – user114666
    Commented Dec 3, 2020 at 14:23
  • $\begingroup$ "Yes, it boils down to construct a birational morphism ..." I still do not quite understand your definition of "contraction". In the definition in textbooks on the Minimal Model Program, usually there is a hypothesis that the natural morphism from the structure sheaf of the target to the pushforward of the structure sheaf of the domain is an isomorphism of sheaves. Under that hypothesis, using the factorization from my comment, automatically $Y$ is normal. If you are allowing a more general notion of "contraction", please clarify your definition. $\endgroup$
    – Jason Starr
    Commented Dec 3, 2020 at 15:33
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    $\begingroup$ There is a projective surface $Y$ with an isolated non-normal singularity $p$ such that the normalisation $W \rightarrow Y$ is bijective and $W$ is nonsingular. Let $q$ be the unique point of $W$ over $p$, and let $X \rightarrow W$ be the blowup of $q$. Is this the kind of thing you are looking for? $\endgroup$
    – Pop
    Commented Dec 3, 2020 at 16:02
  • $\begingroup$ Yes, an example like this is exactly what I am looking for. With divisorial contraction I do not mean Mori divisorial contraction. Just a morphism mapping a divisor to something of smaller dimension. $\endgroup$
    – user114666
    Commented Dec 3, 2020 at 16:21

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