Timeline for Codimension of non-flat locus
Current License: CC BY-SA 4.0
15 events
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| Apr 8, 2019 at 15:37 | comment | added | user74900 | @SándorKovács thank you for your clarification. I meant, if you have a surjective non-birational morphism between connected smooth $\mathbb{C}$-schemes, can non-flat locus be non-empty of codimension$\geq 2$? | |
| Apr 8, 2019 at 15:27 | comment | added | Sándor Kovács | @AknazarKazhymurat: if the target is normal, then by Zariski's Main Theorem a birational morphism is flat exactly where it is an isomorphism. Same with "equidimensional" or "étale" replacing "flat". | |
| Apr 8, 2019 at 13:41 | comment | added | user74900 | @SándorKovács in the link they talk about etale morphisms. Is the following reasoning correct? A surjective morphism between connected smooth schemes of finite type over an algebraically closed field of characteristic 0 is smooth on a non-empty open set of the source (I am not sure if this is true). Then smooth morphisms are etale locally affine spaces (stacks.math.columbia.edu/tag/054L). | |
| Apr 8, 2019 at 4:25 | comment | added | Sándor Kovács | @AknazarKazhymurat: yes, you are right. (See stacks.math.columbia.edu/tag/0EA4). However, in these examples only one of the players is smooth. :) | |
| Apr 8, 2019 at 4:24 | comment | added | Sándor Kovács | @PiotrAchinger: I changed the notation to make it clearer. In the first step the target is singular and the source is smooth and in the next the other way around. Cheers. | |
| Apr 8, 2019 at 1:31 | comment | added | Piotr Achinger | @SándorKovács Sorry, it must have been a typo in an earlier version. | |
| Apr 7, 2019 at 8:44 | comment | added | user74900 | if I understand correctly, for a surjective birational morphism between connected smooth schemes of finite type over $\mathbb{C}$, the non-flat locus can not be non-empty of codimension $\geq 2$. Is it true that the non-flat locus of a surjective morphism between connected smooth schemes of finite type over $\mathbb{C}$ can not be non-empty of codimension $\geq 2$? | |
| Apr 7, 2019 at 3:39 | history | edited | Sándor Kovács | CC BY-SA 4.0 |
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| Apr 7, 2019 at 3:36 | comment | added | Sándor Kovács | @PiotrAchinger: I didn't claim that $X$ was smooth. | |
| Apr 7, 2019 at 3:00 | comment | added | Piotr Achinger | The second example ($X$ and $Y$ smooth, $f$ iso away from a curve in $X$) seems to contradict van den Waerden's theorem. What am I missing? | |
| Apr 7, 2019 at 2:15 | comment | added | Sándor Kovács | @StepanBanach: I added an example where $Y$ is smooth | |
| Apr 7, 2019 at 2:15 | history | edited | Sándor Kovács | CC BY-SA 4.0 |
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| Apr 5, 2019 at 18:03 | comment | added | user137767 | what happens if $Y$ is smooth? | |
| Apr 5, 2019 at 18:02 | vote | accept | CommunityBot | ||
| Apr 5, 2019 at 17:59 | history | answered | Sándor Kovács | CC BY-SA 4.0 |