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. 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 be a flat family of sheaves containing E. We find that it is a reduced deformation of E when some canonical family associated to 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 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.