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.