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 = X₁ ⊂ X₂ ⊂⊂ 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 n-th 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 X_n+1 of multiplicity n + 1, and let M_n a fine moduli space of vector bundles on X_n. With suitable hypotheses, we construct a fine moduli space M_n+1 for the vector bundles on X_n+1 whose restriction to X_n belongs to M_n. It is an affine bundle over the subvariety N_n ⊂ M_n of bundles that can be extended to X_n+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 ω_Y is trivial.