Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z] is an irreducible cubic form defining a plane curve C with a node. A lot is known about sheaves on C; for example Drozd and Greuel classified indecomposable torsion free sheaves in terms of combinatorial data. I'd like to know if there's anything comparable when h is replaced by a power of h.

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rank 1, torsion-free sheaves on the curve defined by h^2=0. Is this the sort you are after? If so, I can try to write something up. – jlk Jun 11 '10 at 5:20