Zulip Chat Archive

Stream: maths

Topic: Projective spaces


Adam Topaz (Mar 03 2022 at 21:38):

Hi all,

I made a new PR with the definition of the projectivization of a vector space in #12438

It is tagged as RFC, because I would really like to hear your comments! This is only an initial PR, with the definition of projectivization K V, and some other very trivial basics (like constructing elements of the projectivization, maps between projective spaces associated to injective (semi)linear maps, etc.).

If you see anything glaring that's missing and should be added to this PR, please do let me know!

Eventually, I will like to work toward a proof of the fundamental theorem of projective geometry (and I may supervise an undergrad project this summer with this goal). Some other interesting future projects could be to construct the manifold structure in the case of projective spaces over R\mathbb{R} or C\mathbb{C}, and/or to identify the points of the scheme-theoretic projective space (once it's defined) with this "classical" one.

Riccardo Brasca (Mar 03 2022 at 21:44):

Is it worth to do at least the basic definitions over a ring? Considering multiplication by a unit.

Adam Topaz (Mar 03 2022 at 21:45):

I thought about that, but I'm not convinced! Note that Pn(R)\mathbb{P}^n(R) is not the naive thing when RR is not a field. (Here by Pn\mathbb{P}^n I mean ProjZ[T0,,Tn]Proj \mathbb{Z}[T_0,\ldots,T_n], of course!)

Riccardo Brasca (Mar 03 2022 at 22:00):

Yes, in general they are different, but sometimes the two are related, right? For example if any projective module is free, a point of the true projective space should gives a point of the naive one if I am not confused

Adam Topaz (Mar 03 2022 at 22:01):

Yeah that's right.

Adam Topaz (Mar 03 2022 at 22:01):

Okay, maybe it is worthwhile to try to generalize a bit at this point.

Riccardo Brasca (Mar 03 2022 at 22:04):

If the generalization is essentially painless I would say "why not", but I totally agree to assume field if working over a ring causes problems.

Riccardo Brasca (Mar 03 2022 at 22:06):

Mmh, the natural condition would be to take n elements that generate the whole ring, and this already looks complicated

Eric Wieser (Mar 03 2022 at 22:50):

Riccardo Brasca said:

Considering multiplication by a unit.

I said the same thing in the PR - it makes some of the field cases easier to prove too


Last updated: Dec 20 2023 at 11:08 UTC