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 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 subset of X-2 subset of ... subset of X-n = Y, such that X-i is a primitive multiple scheme ofmultiplicity 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. 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 xZ(n) using automorphisms of U-ij xZ(n) that leave U-i j invariant. This leads to the study of the sheaf of nonabelian groups G(n) of automorphisms of X xZ(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 nontrivial 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 subset of X-2 subset of ... X-n subset of Xn+1 subset of ... of nontrivial primitive multiple schemes.