Talk:GIT quotient

GIT Notes

http://www-fourier.univ-grenoble-alpes.fr/~mbrion/lin_rev.pdf Superschool https://www.springer.com/gp/book/9783319916255 — Preceding unsigned comment added by Wundzer (talk • contribs) 21:29, 12 February 2020 (UTC)

Definition of stable point

I don't know enough about GIT to be comfortable editing the page I'm pretty sure the definition of stable points here, which makes no reference to the choice of linearization, is wrong. Based on the notes by Richard Thomas which are linked on the page, it seems the correct thing is that not that x itself has closed orbit and finite stabilizer, but rather that this is true of some lift to the affine cone. Cycle space (talk) 22:38, 12 September 2020 (UTC)

This is correct and well-spotted. And different choices of linearisation of your G-action can change the set of stable points, so it isn't enough to say "one lift" of the affine cone either, you absolutely do need to make reference to the specific choice of lift. I had planned to make a sweep of the GIT/Kempf–Ness theorem/Hilbert–Mumford criterion pages at some point and fix all these inconsistencies and bring it inline with Richard Thomas's notes (which, frankly, could almost be put directly onto wikipedia without change).

This fact is kind of explained on the Geometric invariant theory page (insofar as everything there is stated for a vector space, and in GIT that vector space is always the (completion of the) affine cone of your projective space, but even there it is not really made clear. In fact I'm not sure why there are two pages, one for GIT quotient and one for Geometric invariant theory. If you don't feel comfortable boldly fixing these errors, I will return to these pages sometime in the near future and set the record straight.Tazerenix (talk) 00:38, 13 September 2020 (UTC)

To respond to the original comment, if you follow the definition in the article, the notion of stable points does depend on a choice of an ample line bundle as well as a linearlization (since the notion of semistable points does have that dependency). So, the definition here is correct (in fact, it comes from the original source).

As for why this separate page exists; it’s mainly because the construction of a GIT quotient seems a bit distracting to be put in the GIT page. But also, in Wikipedia, we have separate pages for the notions in GIT; e.g., we have equivariant sheaf (of which a linearlized line bundle is a special case) as a separate article; so it seems it makes sense to have a separate page for a GIT quotient as well. —- Taku (talk) 01:01, 13 September 2020 (UTC)

I see. This is explained at the bottom of page 12 of Richard Thomas's notes. It could still be worth having a remark about the different characterisations of semistable/stable points (although perhaps that should be better explained and left to the GIT page instead of this one.Tazerenix (talk) 01:11, 13 September 2020 (UTC)

One additional comment: basically there are two (equivalent) definitions of stable/semistable points, one given here and the other pointed by the original poster here (namely, one in terms of a lift in the affine cone). *I think* the equivalence of the two definitions is shown in Mumford’s original GIT book. But this should be clearly explained somewhere somehow (as you said). Incidentally, there is also a characterization of a stable points in terms of the movement map (if remember); this is basically a key component of the Kempf–Ness theorem and this too should be mentioned in somehow. —- Taku (talk) 01:21, 13 September 2020 (UTC)