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} ⊂ X_{2} ⊂⊂ 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]∕(t^{n})). 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 H^{2}(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 = ℙ_{2}, 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 = X_{1} ⊂ X_{2} ⊂⊂ X_{n} ⊂ X_{n+1} ⊂ of non trivial primitive multiple
schemes.