Timeline for Generalize the Proj construction?
Current License: CC BY-SA 2.5
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| Dec 3, 2010 at 15:25 | comment | added | Anton Geraschenko | @Ben: I don't understand your objection. The recipe produces something over Spec of the zero-graded (invariant) subring. In your example, you get the non-separated line over the line Spec(k[xy]). It's weird, but what's so unreasonable about it? You're right that cutting out the fixed locus is probably the wrong thing to do, since you could artificially change this locus--the right thing is to remove points with large stabilizers. But once you do that, this recipe is basically the quotient of the pre-stable locus. It seems like I don't have to pick line bundle to construct things like ℙ^n. | |
| Dec 3, 2010 at 8:42 | comment | added | Ben Webster♦ | Anton- You should know better than this. By your recipe, what is the Proj of k[x,y] with x of weight 1 and y of weight -1? This is one of the standard examples for why geometric invariant theory is a good reason and it involves (surprise, surprise) picking a positive direction. The monoid you use for your proj should be a submonoid of the ample cone on the result; it doesn't sound like you really want to have lots of units. Such line bundles can only see the affinization of your variety. | |
| Dec 3, 2010 at 8:15 | history | answered | Anton Geraschenko | CC BY-SA 2.5 |