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).
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.