Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

Abstract

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 = Yred is smooth. In this case, L = JC/J2C is a line bundle on C. If Y is of multiplicity 2, i.e. if JC2 = 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. It has been proved in [Reducible deformations and smoothing of primitive multiple curves, Manuscripta Math. 148 (2015) 447-469] that a ribbon with associated line bundle L such that deg(L) = -d < 0 can be deformed to reduced curves having two irreducible components if L can be written as L = C(-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.