Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

TL;DR

This paper studies the deformation of sheaves on ribbons, a type of primitive multiple curve, showing how certain sheaves can be deformed to sheaves on reducible curves, impacting the understanding of moduli spaces.

Contribution

It demonstrates that quasi locally free sheaves on ribbons can be deformed to torsion free sheaves on reducible curves, revealing new insights into moduli space deformations.

Findings

Sheaves on ribbons can be deformed to sheaves on reducible curves.

Deformation conditions depend on the line bundle L and its divisor.

Implications for the structure of moduli spaces of semi-stable 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=Y_red is smooth. In this case, L=I_C/I_C^2 is a line bundle on C. If Y is of multiplicity 2, i.e. if I_C^2=0, Y is called a ribbon. If Y is a ribbon and h^0(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=O_C(-P1-...-Pd)$, where P1,...,P_d 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.