Timeline for Use of Hilbert Schemes in Arithmetic?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 28, 2012 at 15:24 | comment | added | Minhyong Kim | A specific application of moduli schemes (or stacks) might be Faltings's theorem. On the other hand, if you're so inclined I think it's rather a good idea to think about direct applications of Hilbert schemes to arithmetic. They are, after all, more elementary than moduli schemes. There are still many important difficult questions about the arithmetic of moduli schemes, for example, effective versions of Shafarevich's conjecture. It's conceivable that a careful study of the arithmetic of Hilbert schemes could be helpful with them. | |
| Oct 28, 2012 at 8:27 | comment | added | user27056 | As a reference, if you look in Chapter 6 of Mumford's GIT, you'll see him rather directly using Hilbert schemes to build moduli schemes (over localized integer rings) of polarized abelian schemes (which in turn underlie PEL Shimura varieties). This is a specific instance of what Liedtke alludes to early in his answer. In these and other applications, it is crucial that fixing the Hilbert polynomial defines a moduli scheme that is finite type (and not merely locally of finite type) over the base. This relative quasi-compactness is an important output of the construction of Hilbert schemes. | |
| Oct 28, 2012 at 6:58 | answer | added | Christian Liedtke | timeline score: 12 | |
| Oct 28, 2012 at 3:49 | comment | added | Emerton | Dear 36min, Picard schemes (for example) are usually constructed using Hilbert schemes, and it is very common in arithmetic to consider Picard schemes of a curve over an integer ring, or over a DVR. Regards, Matthew | |
| Oct 28, 2012 at 0:32 | history | asked | 36min | CC BY-SA 3.0 |