I'm currently working through Frenkel's beautiful paper: 
<http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf>. 
I'm looking for a good example of a projective curve to get my hands dirty, and go through the general constructions that Frenkel shows there and try to do them manually for this example of a curve. Are there any good instructive examples for doing this? (Or does it always get out of hand very quickly?)



A good example of a curve for geometric Langlands
Unfortunately I don't think geometric Langlands is very easy on any curve.
The only curve where the objects are readily accessible is $P^1$, but even there the general statement is kind of tricky (see Lafforgue's note [here](http://people.math.jussieu.fr/~vlafforg/geom.pdf)). I would look at Frenkel's writings on the Gaudin model, which is a concrete illustration of the Beilinson-Drinfeld-Feigin-Frenkel approach to geometric Langlands for $P^1$ with several punctures. Also Arinkin and Lysenko worked out explicitly a case of geometric Langlands (in a stronger sense) on $P^1$ minus 4 points -- see the first four papers on a mathscinet search for Arinkin. So the answer is try $P^1$ with some punctures, but don't be surprised if things are rather tricky already there.


(I also think geometric Langlands on an elliptic curve should be accessible, but as far as I
know it hasn't been worked out very explicitly.)


