Timeline for Existence of a reduced fiber implies generically reduced (Exercise III-74 from Geometry of Schemes)
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2023 at 13:52 | vote | accept | user267839 | ||
| Aug 29, 2023 at 18:25 | answer | added | Libli | timeline score: 3 | |
| Aug 29, 2023 at 15:40 | comment | added | user267839 | @JasonStarr: what I not completely got is where in this argument I exploit the observation that the canonical projection $p_1$ has a reduced fiber? Using Jacobian criterion we deduce that the smooth locus in $B$ with resp to $p_1$ is open. Then one knows that the preimage of it is regular. So if we can show that the smooth locus is not empty, we win. Note the base field $K$ was assumed to be arbitrary (see III.3.5 in the linked book). So it seems that this condition that there exist a reduced fiber, should somehow assure that the smooth locus cannot be empty. Do you see how to fill this gap? | |
| Aug 29, 2023 at 11:06 | comment | added | Jason Starr | Actually Grothendieck proves this after his discussion of eliminating Noetherian hypotheses (thus he does not need to prove it twice -- once with Noetherian hypothesis and once without). It is in Section 9.8 of EGA IV_3. Anyway, your exercise is much easier: just use the Jacobian criterion to prove that the critical locus of your morphism is a proper closed subset of $B$. | |
| Aug 29, 2023 at 6:26 | history | edited | user267839 | CC BY-SA 4.0 |
added 22 characters in body
|
| Aug 28, 2023 at 23:44 | comment | added | Jason Starr | These type of things are discussed in EGA IV_2 somewhere near Section 6.11 (where Nagata’s criterion is discussed). | |
| Aug 28, 2023 at 18:18 | comment | added | Karl Schwede | I didn't think carefully about your example, but I might refer you to the following paper of Murayama for an overview of this and some some related questions. Note, at least for reducedness, you want to work with geometric fibers. Anyways, here are the references: arxiv.org/abs/2004.06737 and cambridge.org/core/journals/compositio-mathematica/article/… | |
| Aug 28, 2023 at 17:34 | history | edited | user267839 | CC BY-SA 4.0 |
deleted 1 character in body
|
| Aug 28, 2023 at 15:00 | history | edited | user267839 | CC BY-SA 4.0 |
added 17 characters in body
|
| Aug 28, 2023 at 14:49 | history | edited | user267839 | CC BY-SA 4.0 |
added 35 characters in body
|
| Aug 28, 2023 at 13:45 | history | edited | user267839 | CC BY-SA 4.0 |
added 447 characters in body
|
| Aug 28, 2023 at 13:31 | history | asked | user267839 | CC BY-SA 4.0 |