Primitive multiple schemes

A primitive multiple scheme is a complex 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 $X=X_1\subset X_2\subset\cdots\subset X_n=Y$,suchthat$X_i$ 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 $L^{\ast }$ by the zero section. Let ${\mathbf{Z}}_n=spec(C[t]\mathrm{/}(t^n))$. 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 $U_i\times {\mathbf{Z}}_n$ using automorphisms of $U_{ij}\times {\mathbf{Z}}_n$ that leave $U_{ij}$ invariant. This leads to the study of the sheaf of nonabelian groups $G_n$ of automorphisms of $X\times {\bf Z}n$thatleavethe$X$ invariant, and to the study of its first cohomology set. If $n\ge 2$ 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 $X=X_1\subset X_2\subset\cdots\subset X_n\subset X{n+1}\subset\cdots$ of non trivial primitive multiple schemes.