NON-REDUCED MODULI SPACES OF SHEAVES ON MULTIPLE CURVES

International audienceSome 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 P 2 (cf. [18]). 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. Let E be such a sheaf.-Let 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 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 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