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In M. Reid Canonical 3-folds I found this proposition:

If $\phi:Y\rightarrow X$ is a proper morphism with both $X$ an $Y$ normal and such that $f$ is étale in codimension 1 then

1) if $X$ has canonical singularities so does $Y$

2) if $Y$ has canonical singularities and $X$ is Gorenstein, then $X$ has canonical singularities

Do I interpret it correctly if I say that it implies that if I have a normal divisor $D\subset Y$ with $Y$ smooth projective, then $D$ has canonical (and hence rational since it is Gorenstein) singularities? (I take a log-resolution $(Y', D')$ with $D'$ smooth, thus I have a proper map étale in codimension 1 $D'\rightarrow D$ with both varieties normal an $D'$ with only canonical sing, furthermore the dualizing sheaf of $D$ is a line bundle. Then I use the proposition). Why does this sound so weird to me? What do I miss?

Could give me an example of a variety that it is smooth in codimension 1 but not normal?

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An example of variety which is smooth in codimension $1$ but not normal is given by the union of two planes in $\mathbb{P}^4$ intersecting in a single point. One can also find irreducible examples with the same kind of singularity (a "non normal double point"). –  Francesco Polizzi Oct 22 '12 at 11:11
    
Thank you very much! –  Rurik Oct 22 '12 at 12:00

1 Answer 1

up vote 3 down vote accepted

First, I don't think Miles Reid is dealing with pairs in that paper.

Second, I think Miles wants $f$ to be etale in codimension 1 on $Y$. Thus a blowup is not allowed unless the blowup is a small map. You need every divisor on $Y$ to really have image of as divisor on $X$. You can see Miles Reid using this in his proofs.


I'm going to tackle both statements if you just assume etale in codimension 1 on $X$ to show they are false.

1. This is 1.7(II) in Miles Reid's paper.

Obviously this is false for a blowup. Take for example $X = \text{Spec } k[x,y,z]/(x^n+y^n+z^n)$ with $n \geq 3$ and let $\pi : Y \to X$ be the blowup of the origin. That's an isomorphism in codimension 1 on $X$, but not on $Y$. Certainly $X$ is not canonical but $Y$ is not.

On the other hand, the case when $f$ is finite (and etale in codimension 1) is fairly standard. See for example 5.20 in Koll\'ar-Mori, where the etale in codimension 1 assumption implies that the ramification divisor is zero.

2. Even 2. isn't true if you don't assume that $f$ is etale in codimension 1 on $Y$. You can see THIS answer of Hailong Dao which links to an example of Dale Cutkosky of a normal blowup of $\mathbb{C}[x,y,z]$ which is not Cohen-Macaulay (and thus not Canonical).

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I used divisors because they are the first example of something alway gorenstein I had... Now I understand! I was pretty sure that there were something not right in how I understood the proposition but I could not see what point I was missing. Thank you very much –  Rurik Oct 22 '12 at 15:08

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