Timeline for When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group?
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5 events
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| Jul 26, 2012 at 15:37 | comment | added | Phillip Williams | OK, thanks! I had been looking at Mumford but this directs my reading a lot more. | |
| Jul 25, 2012 at 19:55 | comment | added | Olivier Benoist | The reference I know is Mumford's GIT. More precisely : apply the theory of Chap.1, par.4 to $X$ and to the trivially $G$-linearized line bundle $\mathcal{O}_X$. First, considering the section $1\in H^0(X,\mathcal{O}_X)^G$, you see that all points of $X$ are semistable. The hypothesis about the closed orbit of maximal dimension shows that the stable locus is nonempty, by Amplification 1.11. Then you look at the categorical quotient provided by Thm. 1.10. Restricted to the stable locus, the quotient is geometric. For such a quotient, the dimensions add up (see the discussion before Prop 0.2). | |
| Jul 25, 2012 at 16:41 | comment | added | Phillip Williams | This criterion will certainly be useful; thank you. Can you give me a reference for the statements in your first paragraph? | |
| Jul 25, 2012 at 16:28 | vote | accept | Phillip Williams | ||
| Jul 24, 2012 at 20:40 | history | answered | Olivier Benoist | CC BY-SA 3.0 |