What is known about unstable bundles on curves ?
What is "maximally unstable bundle" and what means it corresponds to Schwarzian differential equation ?


Consider the P^1 with pairs of points glued p1=q1, p2=q2, ...,
what can be said for such curves ?


Let me mention that for such curves bundles can be described just by the set of matrices M1, M2, ....
which correspond to "gluing the fibers". 
This can be made precise algebraically as follows: functions on "glued curve"
are function on P^1 which satisfy conditions f(p1)=f(q1), f(p2)=f(q2),...
then modules can be described as vector functions on P1, such that s(p1)=M1 s(q1) , s(p2)=M2 s(q2) ,... clearly this is a module over the algebra above.
It has been further studied in Serre's "Algebraic groups and class fields".
Let me also mention our papers: <http://arxiv.org/abs/hep-th/0303069> Hitchin system on singular curves I, <http://arxiv.org/abs/hep-th/0309059> Hitchin systems on singular curves II. Gluing subschemes 


PS
Questions are inspired by discussion with Misha Kapovich after his answer on the following MO question [What is the number field analogue of the Narasimhan-Seshadri theorem ?](https://mathoverflow.net/questions/96187/what-is-the-number-field-analogue-of-the-narasimhan-seshadri-theorem/96401#96401)



Unstable bundles on curves. "The maximally unstable ones correspond to Schwarzian differential equation on Riemann surfaces" ? 
Vector bundles on curves come equipped with a canonical filtration, the Harder Narasimhan filtration, defined (iteratively) maximal destabilizing subbundles. Maximally unstable just means that the HN filtration is a filtration by line bundles, ie is as refined as possible (so bundle is as far from stable as possible). There's a canonical example of such a bundle (up to the choice of spin structure/theta characteristic in the even rank case, if we're fixing the determinant to be trivial) which is an iterated extension of half-integral powers of the canonical bundle (with step of size one: there's a unique nontrivial extension of $\Omega^{k-1/2}$ by $\Omega^{k+1/2}$ up to isomorphism, represented by the fundamental class of the surface) -- these are called "indigenous bundles" in Gunning's books eg (or the work of S.Mochizuki). Connections on such a bundle are the same as so-called $sl\_n$-opers - in the case of $n=2$, such connections are in bijection with projective structures on the Riemann surface, whose relation to Schwarzian derivatives (or the Virasoro central extension) is well documented in many places, eg Gunning again or the paper of Beilinson-Schechtman. This connection is elementary - you take the corresponding projective bundle, it carries a canonical section from the filtration and a connection which is everywhere transverse to this section, so integrating it defines a projective atlas on the curve.


