Timeline for Is the moduli space of curves defined over the field with one element?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 4, 2012 at 0:31 | answer | added | JBorger | timeline score: 18 | |
| Jan 3, 2012 at 8:59 | vote | accept | Jeffrey Giansiracusa | ||
| Jan 3, 2012 at 6:20 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
Changed link to arXiv article, so as to link to abstract page
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| Jan 3, 2012 at 5:51 | comment | added | S. Carnahan♦ | Others have mentioned that it seems unlikely that you can define the whole moduli space over $\mathbb{F}_1$. However, it looks like you can define a formal neighborhood of any maximally degenerate point of the boundary, since such loci describe gluings of three-pointed genus zero curves. This suggests that objects like the Tate curve, $n$-gons, and Néron models of maximally degenerating Jacobians are definable over $\mathbb{F}_1$. | |
| Jan 3, 2012 at 3:57 | answer | added | Dan Petersen | timeline score: 34 | |
| Jan 2, 2012 at 22:08 | comment | added | Jason Starr | Just to add to what Marty said, it seems to me that any scheme glued from schemes of the form $\mathbb{Z}[M]$ will be geometrically rational when you base change to $\overline{\mathbb{Q}}$. Since $M_g$ is definitely irrational when $g$ is large, it seems to me it cannot be of this form. | |
| Jan 2, 2012 at 21:11 | history | edited | Jeffrey Giansiracusa | CC BY-SA 3.0 |
added more details and refined question
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| Jan 2, 2012 at 21:07 | comment | added | Harry Altman | Could you fix the link to go to the actual arXiv page rather than directly to the PDF? | |
| Jan 2, 2012 at 20:50 | history | edited | Jeffrey Giansiracusa | CC BY-SA 3.0 |
added reference to toen-vaquie
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| Jan 2, 2012 at 19:36 | comment | added | Chris Brav | Having read the question and answers to which Marty links (worth reading), I think my comment about To\"en-Vaquie maybe way off. Please everyone ignore it. I'll do some reading. | |
| Jan 2, 2012 at 18:44 | comment | added | Jeffrey Giansiracusa | @Chris - in low dimensional topology there is an abundance of interesting sets on which the mapping class group acts, such as the n-simplices of the curve complex and all kinds of related objects. In the examples that spring to mind there isn't a fixed basepoint, but one can of course always add a fixed basepoint. So of course, a family of interesting questions is whether any of these sorts of objects admit algebraic descriptions. This wasn't my original motivation for asking the question, but it could have been. | |
| Jan 2, 2012 at 18:18 | comment | added | Marty | Did you look at mathoverflow.net/questions/8190/elliptic-curves-over-f-1? Outside of $M_{0,n}$ (moduli of curves of genus zero with $n$ marked points), it seems doubtful to me that any of the other $M_{g,n}$ (except a few small g,n cases?) would be defined over $F_1$, whatever that means. For example, don't the L-functions of schemes over $F_1$ (after base change to schemes over $Z$) coincide with L-functions of mixed Tate motives? Maybe $M_{1,1}$ would be interesting, since the orbifold points could be definable over $F_1$ (or at least something "below" $Z$). | |
| Jan 2, 2012 at 16:52 | comment | added | Chris Brav | I think that all the various categories of $\mathbb{F}_{1}$-scheme are known to (fully?) embed into that of To\"en-Vaquie, so the question seems fairly unambiguous. But let me add some different ambiguity: you would expect at least that some $D$-modules over $M_{g}/\mathbb{F}_{1}$ should be equivalent to representations of the genus $g$ mapping class group in $\mathbb{F}_{1}$-vector spaces, which would be just pointed sest with action of the mapping class group. Are the latter known to be of any interest? | |
| Jan 2, 2012 at 13:33 | comment | added | Jason Starr | My opinion is that this is not a valid question unless you specify at least one definition of the field with one element. Could you please give a reference to the definition you are using? | |
| Jan 2, 2012 at 12:28 | comment | added | Jeffrey Giansiracusa | I think what you mean is that spec $\mathbb{F}_1$ should be a final object, so everything maps to it. This is different from being defined over it. E.g. $\mathbb{Z}$ is initial among rings and so any scheme maps to spec $\mathbb{Z}$, but there are certainly things that are not defined over $\mathbb{Z}$ - i.e., there exists schemes over spec $k$ that are not the pullback (along spec $k$ $\to$ spec $\mathbb{Z}$) of a scheme over $\mathbb{Z}$. | |
| Jan 2, 2012 at 8:03 | comment | added | Spice the Bird | I think this is a crude answer, and so better suited as a comment: $spec\mathbb{F}_1$ is the absolute point and so morally, everything in algebraic geometry should be defined over $spec\mathbb{F}_1$. | |
| Jan 1, 2012 at 22:45 | history | asked | Jeffrey Giansiracusa | CC BY-SA 3.0 |