Timeline for When are GIT quotients projective?
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2010 at 7:49 | vote | accept | Anton Geraschenko | ||
| Nov 27, 2009 at 19:59 | comment | added | Alicia Garcia-Raboso | @Anton: Thanks for the explanation. | |
| Nov 21, 2009 at 7:59 | comment | added | Georges Elencwajg | @Joel: thanks for your link to Thomas's notes. They are, as you say, written in a style very friendly to non-specialists. | |
| Nov 21, 2009 at 4:50 | answer | added | Avan Thiyagarajan | timeline score: 1 | |
| Nov 21, 2009 at 4:04 | answer | added | Ben Webster♦ | timeline score: 2 | |
| Nov 21, 2009 at 0:31 | answer | added | Greg Stevenson | timeline score: 6 | |
| Nov 21, 2009 at 0:13 | comment | added | Anton Geraschenko | @Alberto: A "modular interpretation" of a space is a description of it as the moduli space of something. For example, consider the moduli space of smooth curves of a given genus. It turns out that the space is not compact. When you compactify, you've thrown in some extra points. The points of the original space can be interpreted as smooth curves, but what about the new points? A map from X to the original space is just a family of smooth curves over X, but what is a map to the compactification? It turns out, it's a family of curves which might have nodes. That's a modular interpretation. | |
| Nov 21, 2009 at 0:09 | comment | added | Joel Fine | Since Anton went to the trouble of giving some background on GIT, perhaps here is a good place to mention some notes on GIT (arxiv.org/abs/math/0512411) which are excellent but maybe less well-known than the standard references. I should also stress that they make a real effort to make a technical subject accessible to people without a strong algebro-geometric background. They may also be extra-useful for you Buzzard, but for a different reason: if you want to ask the author a question, you can knock on his door easily enough! | |
| Nov 20, 2009 at 23:57 | comment | added | Alicia Garcia-Raboso | I'm not sure if asking here is the correct MO etiquette, but here goes: what is a "modular interpretation of the compactification"? | |
| Nov 20, 2009 at 22:49 | answer | added | David Steinberg | timeline score: 3 | |
| Nov 20, 2009 at 21:57 | comment | added | Charles Siegel | Buzzard, the reason that there's a natural compactification is that the quotient comes with an ample line bundle, so we can always take the smallest very ample power of it, use it to embed, and take the closure. | |
| Nov 20, 2009 at 21:55 | comment | added | Anton Geraschenko | As for viewing source, see tip number 5 at mathoverflow.net/tips. The source for this question is at mathoverflow.net/revisions/2ee0d583-d3f1-4eb3-b9a6-fd44d512eb02/… | |
| Nov 20, 2009 at 21:54 | comment | added | Anton Geraschenko | The quotient comes with an ample line bundle (descended from L). Different very ample powers should give you projective embeddings that differ by d-uple embeddings, so I thought they should all give you isomorphic projective closures, but maybe this is wrong. The compactness results I've seen are along the same lines, but that means you have to guess a modular interpretation of the compactification (decide what "something worse" is), which I want to avoid for some reason. | |
| Nov 20, 2009 at 21:21 | comment | added | Kevin Buzzard | Third unrelated comment: Anton---you do a very good job of laying out questions. Lots of fonts, links, boxes etc. Very clear. Can I "view source" somehow? | |
| Nov 20, 2009 at 21:19 | comment | added | Kevin Buzzard | For what it's worth, with the moduli spaces I've considered in my mathemical lifetime (abelian varieties plus extra structure), compactness (properness) is the same as "given a family of ab vars + structure, can it degenerate into something worse?" and the technique is always "think about the Neron model". In other words, use the moduli problem itself, and not the GIT construction of it. I guess what I'm saying is that in the cases I know, properness, when it's true, comes from a study of the functor, not the construction of the representing object. | |
| Nov 20, 2009 at 21:16 | comment | added | Kevin Buzzard | Dumb comment: can you explain a bit more how "the quotient is automatically quasi-proj, so there is a natural choice of compactification" bit goes? I don't think quasi-proj means "I come with a given map to a projective space", I think it is something weaker. But maybe I'm missing something in this context. | |
| Nov 20, 2009 at 20:59 | history | asked | Anton Geraschenko | CC BY-SA 2.5 |