Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

Abstract

A primitive multiple curve is a Cohen–Macaulay irreducible projective curve [Formula: see text] that can be locally embedded in a smooth surface, and such that [Formula: see text] is smooth. In this case, [Formula: see text] is a line bundle on [Formula: see text]. If [Formula: see text] is of multiplicity 2, i.e. if [Formula: see text], [Formula: see text] is called a ribbon. If [Formula: see text] is a ribbon and [Formula: see text], then [Formula: see text] can be deformed to smooth curves, but in general a coherent sheaf on [Formula: see text] 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 [Formula: see text] such that [Formula: see text] can be deformed to reduced curves having two irreducible components if [Formula: see text] can be written as [Formula: see text] where [Formula: see text] are distinct points of [Formula: see text]. In this case we prove that quasi-locally free sheaves on [Formula: see text] 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 [Formula: see text].