Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that C = Y red is smooth. In this case, L = ICIC2 is a line bundle on C. If Y is of multiplicity 2, i.e. if I C2 = 0, Y is called a ribbon. If Y is a ribbon and h0(L-2) = 0, then Y can be deformed to smooth curves, but in general a coherent sheaf on Y cannot be deformed in coherent sheaves on the smooth curves.

A ribbon with associated line bundle L such that deg(L) = -d < 0 can be deformed to reduced curves having 2 irreducible components if L can be written as

L = OC (- P1 - ⋅⋅⋅ - Pd ),
where P1,⋅⋅⋅,Pd are distinct points of C. In this case we prove that quasi locally free sheaves on Y can be deformed to torsion free sheaves on the reducible curves with two components. This has some consequences on the structure and deformations of the moduli spaces of semi-stable sheaves on Y .