Timeline for Does the orbit in geometric invariant theory have natural scheme structure
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| Oct 20, 2023 at 13:47 | comment | added | R. van Dobben de Bruyn | A good reference for algebraic groups is Milne's book (a preliminary version is available on the author's website). See Def. 9.5 for the scheme theoretic orbit, and Thm. 7.35 for the existence of quotients of quasi-projective algebraic groups. It turns out the affine case is only easier if the subgroup is normal (see §5d); otherwise the quotient need not be affine (e.g. $\mathbf P^n$ is a quotient of $\operatorname{GL}_{n+1}$), so cannot be constructed using Hopf algebra methods (or representation theory, depending on your point of view). | |
| Oct 20, 2023 at 13:36 | comment | added | R. van Dobben de Bruyn | It is not true in general that the image of a morphism of $k$-varieties is locally closed; for instance the image of $\mathbf A^2 \to \mathbf A^2$ given by $(x,y) \mapsto (x,xy)$ is $D(x) \cup \{(0,0)\}$. Chevalley's theorem says that the image is constructible, but in this example it doesn't carry a scheme structure. | |
| Oct 19, 2023 at 20:38 | history | edited | user267839 | CC BY-SA 4.0 |
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| Oct 19, 2023 at 20:36 | comment | added | user267839 | ah, and yes, it would be comforting to assume $G$ to be affine :) | |
| Oct 19, 2023 at 20:30 | comment | added | user267839 | domain of the map is an algebraic group/ group scheme. At all it seems to reduce basically to the question when the set theoretic image of map is open in it's scheme theoretic image, which in turn by definition is endowed naturally with scheme structure,right? | |
| Oct 19, 2023 at 20:26 | comment | added | user267839 | But I have still some questions about the statement that in case $G$ is algebraic group over field the orbit (as set) is open in scheme theoretic image. Firstly, do you know a reference for this statement? Secondly, is this really a statement which deals with group schemes or is it a statement of type: Let $f:X \to Y$ morphism of "nice enough" $k$-schemes, then the image (as set!) is open in it's scheme theoretic image? Why I'm asking this it because the orbit as defined 2nd paragraph above is given as nothing but an image, so this rase the question how "neccessary" is it to assume that the | |
| Oct 19, 2023 at 20:17 | comment | added | user267839 | @R.vanDobbendeBruyn: That's seems to be exactly (...modulo generalization of base scheme $S$) I was looking for! Indeed, if we know that the orbit is (a priori only topologically open) open subscheme in scheme theoretic image, then the way it is going to be endowed with scheme structure is canonical, since the scheme structure of scheme theoretic image is declared canonically. | |
| Oct 19, 2023 at 20:16 | comment | added | user267839 | Is this exactly the scheme structure Remy van Dobben de Bruyn mentioned? Namely, if $G$ is algebraic group over field, then the orbit is open subscheme of it's scheme theoretic image, and since the latter has "canonical" associated scheme structure (namely that one I descirbed in 4th paragraph in the question), and it induces in turn "canonically" a scheme structure on the orbit regarded as it's open subscheme? If yes, then the question becomes in which situations the orbit is open in it's scheme theoretic closure? | |
| Oct 19, 2023 at 19:59 | comment | added | user267839 | scheme structure the orbit is endowed with? You remarked in your comment, that a orbit can be non reduced. Sure, but that's the point! When you say that it " is nonreduced", then you already know it's scheme structure, right? And the question is from "where" does one know the scheme structure of the orbit carries? Is there a canonical way to give the orbit a scheme structure from knowledge of the action and the involved spaces? | |
| Oct 19, 2023 at 19:52 | comment | added | user267839 | @JasonStarr: not exactly, I posed a much simpler question, namely just if the orbit in context of GIT carries a "natural" scheme structure. I saw very often in literature (just to pick one example, eg this one , see in Def 1.4, page 107. There one tacitly regards an orbit as a scheme (... since one talks about it's smoothness), and my question is simply what is the | |
| Oct 19, 2023 at 12:31 | comment | added | R. van Dobben de Bruyn | The orbit can also be thought of as $G/H$ where $H$ is the stabiliser, making it a question about representability of quotients. There is quite a bit to be said about this; a starting point is SGA 3$_\text{I}$, Exp. VI$_\text{B}$, §9. I am not really an expert in this, so I don't know when it can and when it cannot be done. | |
| Oct 19, 2023 at 12:29 | comment | added | R. van Dobben de Bruyn | Note that for algebraic groups over a field, the orbit is always locally closed, so it makes sense to view it as an open subscheme of the scheme-theoretic image. I'm not sure if local closedness is true over a general base. I do expect that the situation for affine group schemes is much better (which is often the case you're interested in when doing geometric invariant theory). | |
| Oct 19, 2023 at 12:28 | comment | added | Jason Starr | The orbit can be nonreduced, e.g., if $G$ itself is nonreduced. Is that what you are asking? | |
| Oct 18, 2023 at 22:31 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 18, 2023 at 21:40 | history | edited | user267839 | CC BY-SA 4.0 |
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| Oct 18, 2023 at 19:59 | history | edited | user267839 | CC BY-SA 4.0 |
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| Oct 18, 2023 at 18:22 | history | edited | user267839 | CC BY-SA 4.0 |
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| Oct 18, 2023 at 15:08 | history | edited | user267839 | CC BY-SA 4.0 |
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| Oct 18, 2023 at 15:02 | history | asked | user267839 | CC BY-SA 4.0 |