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 ⊂⊂ Xn = Y , such that Xi 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 Zn = spec(ℂ[t]∕(tn)). The primitive multiple schemes of multiplicity n are obtained by taking an open cover (Ui) of a smooth variety X and by gluing the schemes Ui × Zn using automorphisms of Uij × Zn that leave Uij invariant. This leads to the study of the sheaf of nonabelian groups n of automorphisms of X × Zn 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 Xn 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 = ℙm 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 ⊂⊂ Xn ⊂ Xn+1 ⊂ of non trivial primitive multiple schemes.