Non-reduced moduli spaces of sheaves on multiple curves

Abstract

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 ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also 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 M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.