Primitive multiple schemes

Abstract

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme $X=Y_{\mathrm{red}}$ 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 $X=X_1\subset X_2\subset \cdots \subset X_n=Y$, 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 nth infinitesimal neighborhood of X, embedded in the line bundle $L^*$ by the zero section. Let . 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 of automorphisms of $X\,{\times }\, \mathbf{Z}_n$ that leave the X invariant, and to the study of its first cohomology set. If $n\geqslant 2$ 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 $X={{\mathbb {P}}}_m$ with $m\geqslant 3$, all the primitive multiple schemes are trivial. If $X={{\mathbb {P}}}_2$, 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 $X=X_1\subset X_2\subset \cdots \subset X_n\subset X_{n+1}\subset \cdots$ of nontrivial primitive multiple schemes.