Timeline for Field of definition for general type surfaces
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 10, 2019 at 10:36 | comment | added | Daniel Loughran | I think providing your moduli stack is sufficiently nice (e.g Deligne-Mumford and quasi-compact and quasi-separated) then the answer is yes. I'm not sure about in general as there are some very bad stacks out there. | |
| Dec 8, 2019 at 22:14 | comment | added | Jonny Evans | For example, are you saying that if I have an isolated point in the moduli space of complex surfaces then it is biholomorphic to an algebraic surface defined over $\bar{\mathbb{Q}}$? This seems to follow from what you're saying (unless, as is very possible, there's something I'm missing). | |
| Dec 8, 2019 at 21:17 | comment | added | Jonny Evans | It's the moduli space whose points correspond to isomorphism classes of complex surfaces with ample canonical bundle. Each of these is biholomorphic to some $Proj(R)$ for a finitely generated graded ring $R$ over $\mathbb{C}$, but I don't see why, for some dense set of surfaces in the moduli space, $R$ should be of the form $R'\otimes_{\bar{\mathbb{Q}}}\mathbb{C}$ for $R'$ defined over $\bar{\mathbb{Q}}$. Sorry, the Q in the question at the end of my comment should have been \bar{Q} not just Q. | |
| Dec 8, 2019 at 20:54 | comment | added | Daniel Loughran | Sorry I don't know what the Gieseker coarse moduli space is. The coarse moduli space of a stack satisfies a universal property so is unique. This is the right space to work with; I would guess that it agrees with your space on some dense open subset. If I understand you question correctly: yes a variety over $\mathbb{Q}$ need not have any $\mathbb{Q}$-rational points. | |
| Dec 7, 2019 at 9:29 | comment | added | Jonny Evans | Thanks for this answer. What's still bothering me is that I'm really interested in the Gieseker coarse moduli space (of canonically polarised complex surfaces). Presumably your moduli stack has an associated coarse moduli space. Is it clear that these coarse moduli spaces are the same, or is yours just a subset? Could there be components of the Gieseker moduli space of complex surfaces that just don't contain any surfaces which are biholomorphic to surfaces which are defined over Q? | |
| Dec 6, 2019 at 22:04 | history | edited | Daniel Loughran | CC BY-SA 4.0 |
added 49 characters in body
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| Dec 6, 2019 at 21:56 | history | answered | Daniel Loughran | CC BY-SA 4.0 |