MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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).

share|cite|improve this question
princiPAL bundle – Urs Schreiber Oct 29 '09 at 17:45
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.