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 if the line bundle L^* by the zero section.

Let Z_n = spec(ℂ[t]∕(tⁿ)). The primitive multiple schemes of multiplicity n are obtained by taking an open cover (U_i) of a smooth variety X and by gluing the schemes U_i × Z_n using automorphisms of U_ij × Z_n that leave U_ij invariant. This leads to the study of the sheaf of nonabelian groups _n of automorphisms of X × Z_n that leave the X invariant, and to the study of its first cohomology set. If n ≥ 2 there is an obstruction to the extension of X_n to a primitive multiple scheme of multiplicity n + 1, which lies in the second cohomology group H²(X,E) of a suitable vector bundle E on X.

In this paper we study these obstructions and the parametrization of primitive multiple schemes. As an example we show that if X = ℙ_m with m >= 3 all the primitive multiple schemes are trivial. If X = ℙ₂, there are only two non trivial primitive multiple schemes, of multiplicities 2 and 4, which are not quasi-projective. On the other hand, if X is a projective bundle over a curve, we show that there are infinite sequences  X = X₁ ⊂ X₂ ⊂⊂ X_n ⊂ X_n+1 ⊂ of non trivial primitive multiple schemes.