Timeline for Do singular fibers determine the elliptic K3 surface, generically?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 12, 2021 at 5:06 | answer | added | HYL | timeline score: 1 | |
| Jun 11, 2021 at 21:22 | answer | added | Evgeny Shinder | timeline score: 6 | |
| Jun 8, 2021 at 12:09 | comment | added | naf | Thanks for the reference: I found an English translation of Shafarevich's ICM talk in his collected works. | |
| Jun 8, 2021 at 8:58 | comment | added | Evgeny Shinder | @naf: thanks, I will try to work out the details of the argument. I've tracked the historical development of this subject: Shafarevich 1962 -> Parshin 1968 -> Arakelov 1971 -> Faltings 1983 -> Deligne 1987. The original paper by Shafarevich is his ICM 1962 talk in Russian mathunion.org/fileadmin/ICM/Proceedings/ICM1962.1/… (page 174), where he says the result is true for hyperelliptic curves of g > 1. | |
| Jun 8, 2021 at 4:10 | comment | added | naf | I haven't been able to find the original reference (for elliptic curves) but the proof is actually quite easy: after replacing $C$ by a finite cover of bounded degree, unramified outside $S$, we may assume that the family of elliptic cuvrves has level $n$ structure for some large integer $n$. This gives a map to the modular curve of level $n$ (which determines the family) and by the de Franchis theorem there are only finitely many such maps (since $n$ is big). | |
| Jun 7, 2021 at 11:09 | comment | added | Evgeny Shinder | @naf: thanks, if you'd like to, please feel free to put this as an answer. | |
| Jun 7, 2021 at 7:42 | comment | added | naf | Finiteness holds for (non-isotrivial) elliptic surfaces with a section (and therefore for all $\mathcal{M}_{d,t}$) much more generally: given any finite set $S$ of points on any smooth projective curve $C$ there are only finitely many non-isotrivial families of elliptic curves on $C - S$. I am not sure who first proved this (maybe Shafarevich) but for more general results see the paper of Faltings "Arakelov's theorem for abelian varieties" and Deligne "Un theoreme de finitude pout la monodromie" | |
| Jun 7, 2021 at 0:20 | history | edited | Evgeny Shinder | CC BY-SA 4.0 |
added 478 characters in body
|
| Jun 7, 2021 at 0:08 | comment | added | Evgeny Shinder | @JasonStarr: I clarified the question, hopefully it's clearer now. I thought about your suggestion of using Vakil's work on rational ellitpic surfaces but I don't see the complete argument: so rational elliptic surfaces are determined by their branch points, and hence possibly elliptic K3s which are double covers of those too, but what if you have more K3s with the same branch points which are not double covers?? | |
| Jun 7, 2021 at 0:04 | history | edited | Evgeny Shinder | CC BY-SA 4.0 |
clarified the question
|
| Jun 6, 2021 at 10:30 | comment | added | Jason Starr | Could you please clarify: are you asking if the discriminant is bounded, i.e., if there are possibly countably infinite different K3 surfaces with specified critical locus in $\mathbb{P}^1$? Or are you asking about positive-dimensional families with specified critical locus? The latter can be studied via infinitesimal deformation theory (pair the 1-dimensional kernel of the infinitesimal “discriminant map” against the first Chern class of a fiber). | |
| Jun 5, 2021 at 13:35 | comment | added | user25309 | A maybe obvious remark: given an elliptic K3 surface with 24 nodal singular fibers and without section, the corresponding Jacobian surface is a different elliptic K3 surface with section and with 24 nodal singularities at the same 24 points. | |
| Jun 5, 2021 at 12:51 | comment | added | Evgeny Shinder | Thanks Jason! Discriminant is arbitrary, I consider rank two elliptic K3s as e.g. in Section 3.2 of arxiv.org/pdf/1907.01335.pdf. | |
| Jun 5, 2021 at 12:48 | comment | added | Jason Starr | Here is a reference for the result about rational elliptic surfaces: arxiv.org/abs/math/9910077 | |
| Jun 5, 2021 at 12:47 | comment | added | Jason Starr | What is the discriminant of your Picard lattice? At least for K3 surfaces that are obtained as deformations of double covers of elliptic rational surfaces, this follows from the corresponding statement for the elliptic rational surfaces (by deforming away from the locus of double covers): the orbit of a line in the $\mathbb{P}^9$ of plane cubics is determined up to finite indeterminacy by its intersection with the discriminant (degree $12$) hypersurface. | |
| Jun 5, 2021 at 9:20 | history | asked | Evgeny Shinder | CC BY-SA 4.0 |