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I have a very specific question: does anyone know of a (non-trivial) example of a projective curve which is also a homogenous space (or just a principal bundle)? The trivial example being CP^1 = SU(2)/U(1).

edited Oct 29 2009 at 18:15

asked Oct 29 2009 at 17:40

John McCarthy
686●3●10

David Speyer · Accepted Answer · 2009-10-29 18:17:32Z UTC

CP^2 is not a curve. So you may have misstated your question. Nonetheless, here is my answer:

Every curve of genus 1 is a principal homogenous space for its Jacobian. Over an algebraically closed field, a principal homogenous space is just the group itself, and that is what happens in this case.

For genus g >= 2, no algebraic curve has more than 84(g-1) algebraic automorphisms. In particular, no curve can be a homogenous space.

EDIT The comment about CP^2 refers to an earlier version of the question.

Urs Schreiber · Answer 2 · 2009-10-29 17:45:36Z UTC

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princiPAL bundle

answered Oct 29 2009 at 17:45

Urs Schreiber
6,468●13●21

-1. This belongs as a comment, not as an answer. – Anton Geraschenko♦ Oct 29 2009 at 18:06

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sorry, didn't realize that I can post a comment on a main question instead of an answer. But this kind of conversation with a number in front of a sentence is interesting. I imagine a boy and girl dating. She says: -2 You are late. He says: +5 you look beautiful. She: -3, you are just trying to distract from the fact that you are late. He: +2, no really not, trust me. She -1, you are lying. He: -3 you get on my nerves. ;-) – Urs Schreiber Oct 29 2009 at 20:49

+1 now you know. It's a convention taken from Stack Overflow. Votes are normally anonymous, but it's considered good form to leave an explanation of why you downvoted something. The "-1" is shorthand for " I'm the one who downvoted this and here's why." The idea is that if you get feedback about what was wrong with the post, it's easier to make better posts in the future. On the other hand, you can only vote once on a question, and sometimes you want to vote it up a lot, so you leave a comment saying "+1 this is a great answer because x, y, and z" so that other people vote it up too. – Anton Geraschenko♦ Nov 3 2009 at 5:17

Okay, sure. Thanks. I am mainly sorry that I seem to have offended the original poster. Maybe my capital letters came across as shouting. All I meant to do is point out a typo (I think that day I was writing referee reports and was marking typos like a machine). I understand that what was meant to be just kindly helpful came across badly for some reasons. I think I know what to do next time. – Urs Schreiber Nov 6 2009 at 8:06

Projective Curves which are Principal Bundles

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