Timeline for Do morphisms locally decompose into finite surjective followed by smooth? (update: Is every projective variety over a finite field a finite cover of $\mathbb{P}^d$ for some $d$?)
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
|
|
| May 6, 2010 at 10:16 | vote | accept | Matthew Morrow | ||
| May 6, 2010 at 0:33 | answer | added | David E Speyer | timeline score: 6 | |
| May 5, 2010 at 22:32 | comment | added | Qing Liu | Over finite fields, see Kedlaya's paper arxiv.org/abs/math/0303382. | |
| May 5, 2010 at 19:52 | comment | added | Matthew Morrow | @BCnrd Thank you very much for all your comments on my question. I sure hope that the answer to the finite field question is yes, because otherwise my original question remains open under what you have shown to be the only reasonable hypotheses, namely $X\to S$ being flat and projective. Poonen's paper certainly looks relevant, so I will wait hear what he says. | |
| May 5, 2010 at 18:47 | comment | added | BCnrd | The interesting question of projective Noether normalization over finite fields has Bjorn "Bertini over finite fields" Poonen's name written all over it. I eagerly await his clever solution. | |
| May 5, 2010 at 17:44 | history | edited | Matthew Morrow | CC BY-SA 2.5 |
added 1198 characters in body; edited title
|
| May 5, 2010 at 14:36 | comment | added | BCnrd | For the 2nd question, again assume $\pi$ flat to avoid dumb counterexamples. Then by fiber-dimension reasons, any map $X \rightarrow \mathbf{P}^d_ A$ over $A$ that is finite on special fiber is finite surjective. As you expect, by fixing projective embedding for $X$, using linear projections away from some suitable linear space disjoint from $X$ (on special fiber, thus over $A$ by properness) thereby does the job when residue field is infinite. I don't remember the specific formulas (in Red Book?) used to handle finite field case, but is there a problem lifting those formulas? | |
| May 5, 2010 at 14:03 | comment | added | BCnrd | For first question must avoid $\pi$ a closed immersion. But false even for $\pi$ flat . Let $K$ be non-Galois cubic number field and $p$ rat'l prime over which there is ramified $P$ and another prime $Q$. (Trivial to make $K$, $p$.) Let $S={\rm{Spec}}(\mathbf{Z}_ {(p)})$ and $X = {\rm{Spec}}(O_ {K,P})$. The "problem" is gap between quasi-finite and "locally finite" for Zariski top. No such gap in analytic theory, and likewise if use etale top.then for flat $\pi$ it is affirmative (target an affine space), via Noether normalization on a fiber and Zariski's Main Theorem (in EGA form). | |
| May 5, 2010 at 11:45 | history | asked | Matthew Morrow | CC BY-SA 2.5 |