Timeline for When are GIT quotients projective?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2009 at 18:29 | comment | added | Anton Geraschenko | Imposing the condition that there are no invariant sections of any power of L does tell you that X^ss(L)/G has trivial affine envelope, but it could still fail to be projective. If X is P^2 minus a point, and G is trivial, any information you could get from regular functions or sections of a line bundle would apply equally well to the case P^2/G, but one of the quotients is projective, and the other is not. | |
| Nov 21, 2009 at 5:31 | comment | added | David E Speyer | Rereading Anton's question, he asked two things: (1) When is X//G projective? (2) When is X^s/G projective? It seems to me that he should also have asked (3) When is the image of X^{ss} in X//G projective? I was thinking about (2) and (3); your answer is a correct answer to (1). | |
| Nov 21, 2009 at 4:43 | comment | added | David E Speyer | That said, I feel like this is addressed somewhere in Mumford's book, and that there is some good criterion which lets you conclude the map is surjective. | |
| Nov 21, 2009 at 4:42 | comment | added | David E Speyer | How do you know that the map from X^{ss}(L) to Proj of the ring of invariant functions is surjective? You are giving a criterion for when that Proj is projective. | |
| Nov 21, 2009 at 4:04 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |