sheaves on thickened nodal cubics

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.

flag	asked May 20 2010 at 19:07 paul Monsky 2,796●1●7●18

Hi Paul, welcome to MO! – Hailong Dao May 20 2010 at 20:05

@paul Monsky: I think I can classify 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 2010 at 5:20

Apologies--I didn't see your comment until now. I'm hoping, doubtless in vain, for something in arbitrary rank. (I recently used the Drozd-Greuel classification in calculating the dimension of the quotient of F[x,y,z], char F=p, by the ideal generated by h and the q th powers of some fixed homogeneous elements, where q=1,p,p^2,p^3,... There's a very simple formula for all these dimensions, and the computer suggests that something similar is happening when h is replaced by a power of h). But any info you can give is welcome. You can send me mail at Brandeis University. – paul Monsky Jul 9 2010 at 21:34