Non-reduced moduli spaces of sheaves on multiple curves

Abstract: Some coherent sheaves on projective varieties have a non reduced versal deformation space. For example, this is the case for most unstable rank 2 vector bundles on ${\mathbb P}_2$. In particular, it may happen that some moduli spaces of stable sheaves are non reduced. We consider the case of some sheaves on ribbons (double structures on smooth projective curves): the quasi locally free sheaves of rigid type. Le $E$ be such a sheaf. -- Let ${\mathcal E}$ be a flat family of sheaves containing $E$. We find that it is a reduced deformation of $E$ when some canonical family associated to ${\mathcal E}$ is also flat. -- We consider a deformation of the ribbon to reduced projective curves with two components, and find that $E$ can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components $\bf M$ of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and $\bf M$ appears as the "limit" of varieties with two components, whence the non reduced structure of $\bf M$.