Moduli of vector bundles on primitive multiple schemes

A primitive multiple scheme is a Cohen-Macaulay scheme Y such that the associated reduced scheme X = Y-red is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension dim(X) + 1. If n is the multiplicity of Y, there is a canonical filtration X = X-1 subset of X-2 subset of center dot center dot center dot subset of X-n = Y, such that X-i is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle L* by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that Y = X-n is of multiplicity n, and can be extended to Xn+1 of multiplicity n + 1, and let M-n be a fine moduli space of vector bundles on Xn. With suitable hypotheses, we construct a fine moduli space Mn+1 for the vector bundles on Xn+1 whose restriction to Xn belongs to Mn. It is an affine bundle over the subvariety N-n subset of Mn of bundles that can be extended to Xn+1. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes Y such that the dualizing sheaf.Y is trivial.