Timeline for Definition of Hitchin map
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2020 at 2:55 | review | Close votes | |||
| Sep 30, 2020 at 3:05 | |||||
| Sep 24, 2020 at 15:44 | comment | added | Will Sawin | Then the main point of Qfwfq's argument is to check that $\operatorname{Tr} \wedge^i \phi$ makes sense as a map from $S$ to a vector space for a flat family over $S$ of (semistable) pairs $(E,\phi)$. This is fairly straightforward. | |
| Sep 24, 2020 at 9:34 | comment | added | Aoki | @abx Thanks, I'll study it. | |
| Sep 24, 2020 at 5:20 | comment | added | abx | @Aoki: This scheme turns out to be a variety... Before studying the Hitchin map, it might be useful.to study the classical theory of moduli spaces of vector bundles. | |
| Sep 24, 2020 at 3:44 | comment | added | Aoki | It is the scheme M satisfying certain universal property to the functor $\mathscr{M}(r,d): S \mapsto$ $\{$ isom class of flat family over $S$ of semistable pairs $(E,\phi) \}$ | |
| Sep 24, 2020 at 3:07 | comment | added | Will Sawin | What, for you, is the definition of a coarse moduli space? | |
| Sep 24, 2020 at 2:56 | history | edited | Aoki | CC BY-SA 4.0 |
added 30 characters in body
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| Sep 24, 2020 at 2:50 | comment | added | Aoki | @Tabes Bridges I think Hitchin map should be a morphism of the category of scheme. But $(E,\phi)\mapsto \operatorname{Tr}\wedge^{i}\phi$ is only defined at closed points and isn't a morphism between schemes.And, I don't know the usual definition of regular between schemes. | |
| Sep 22, 2020 at 19:34 | comment | added | Tabes Bridges | @Aoki A regular function on $X$ is an element of $\mathcal O_X$, equivalently a map to $\mathbb A^1$. Since you're mapping to a vector space here, you want a tuple of regular functions I think. | |
| Sep 22, 2020 at 16:44 | comment | added | Aoki | Thanks for comments. But what do you mean by regular function? regular map in the category of algebraic variety? | |
| Sep 22, 2020 at 13:03 | comment | added | Qfwfq | Another way of seeing it (perhaps a rewording of abx's comment?) is: the construction $(E,\phi)\mapsto\mathrm{tr}\wedge^i \phi$ works well in families cause it's done fiberwise. This gives a map from the moduli functor (i.e. the functor associating to $S$ the set of appropriate families on $S$) to (the functor of points of) your $SpecSym(V^*)$. By definition of coarse moduli space, this induces a unique map of schemes from the coarse moduli space to $SpecSym(V^*)$ which makes the diagram of the previous maps commute hence is the given map on points/objects. | |
| Sep 22, 2020 at 9:38 | comment | added | abx | Yes, you are on a wrong track. All you have to prove is that $(E,\phi)\mapsto \operatorname{Tr}\wedge^{i}\phi $ is a regular function on $\mathscr{M}(r,d)$. This is clear on a fine moduli space (or stack), then use basic descent. | |
| Sep 22, 2020 at 9:33 | history | asked | Aoki | CC BY-SA 4.0 |