Doubt about normality and rational singularities - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:57:08Z http://mathoverflow.net/feeds/question/110317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110317/doubt-about-normality-and-rational-singularities Doubt about normality and rational singularities Rurik 2012-10-22T11:05:04Z 2012-10-22T19:09:48Z <p>In M. Reid Canonical 3-folds I found this proposition:</p> <p>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</p> <p>1) if $X$ has canonical singularities so does $Y$</p> <p>2) if $Y$ has canonical singularities and $X$ is Gorenstein, then $X$ has canonical singularities</p> <p>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?</p> <p>Could give me an example of a variety that it is smooth in codimension 1 but not normal?</p> http://mathoverflow.net/questions/110317/doubt-about-normality-and-rational-singularities/110325#110325 Answer by Karl Schwede for Doubt about normality and rational singularities Karl Schwede 2012-10-22T12:12:05Z 2012-10-22T19:09:48Z <p>First, I don't think Miles Reid is dealing with pairs in that paper.</p> <p>Second, I think Miles wants $f$ to be etale in codimension 1 <em>on $Y$</em>. 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.</p> <hr> <p>I'm going to tackle both statements if you just assume etale in codimension 1 on $X$ to show they are false. </p> <p><strong>1.</strong> This is 1.7(II) in Miles Reid's paper.</p> <p>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.</p> <p>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.</p> <p><strong>2.</strong> Even 2. isn't true if you don't assume that $f$ is etale in codimension 1 on $Y$. You can see <a href="http://mathoverflow.net/questions/43336/blowups-of-cohen-macaulay-varieties/43415#43415" rel="nofollow">THIS answer of Hailong Dao</a> 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). </p>