Timeline for Why people usually consider reductive groups in GIT?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 21, 2014 at 4:59 | comment | added | abz | Someone should mention that Mumford was mainly interested in constructing moduli schemes for curves and polarized abelian varieties, for which reductive groups suffice. | |
| Jan 20, 2014 at 23:28 | vote | accept | Li Yutong | ||
| Jan 20, 2014 at 18:40 | answer | added | Jim Humphreys | timeline score: 10 | |
| Jan 20, 2014 at 17:50 | comment | added | Ruadhaí Dervan | As mentioned before reductivity implies the invariant coordinate ring is finitely generated. Frances Kirwan has worked on (and I believe is working on) non-reductive GIT along with her student(s). The focus is on quotients by unipotent groups, by structure theorems for general groups. See for example "Towards non-reductive geometric invariant theory". | |
| Jan 20, 2014 at 17:13 | comment | added | Sasha Anan'in | My guess is that nobody knows how to get (in general) a quotient variety with respect to the other algebraic groups: the lack of general methods. "Mathematicians prefer what they are able to do." | |
| Jan 20, 2014 at 17:05 | comment | added | Daniel Barter | Reductive groups are important in their own right also. Their representation categories are completely reducible (in fact this is why the coordinate rings are finitely generated). Also, every affine group variety you probably care about is reductive, in paticular, all of the classical groups | |
| Jan 20, 2014 at 17:03 | comment | added | Li Yutong | I see, this is a good point. But is reductive group a suitable notion just for such property? | |
| Jan 20, 2014 at 16:59 | comment | added | Daniel Barter | My understanding is this: If $G$ is a reductive group acting on an affine (projective) variety, then the invariants in the coordinate ring (homogeneous coordinate ring) are finitely generated. This is not true for non reductive groups. I will let someone with more knowledge than me fill in the details | |
| Jan 20, 2014 at 16:50 | history | asked | Li Yutong | CC BY-SA 3.0 |