PRIMITIVE MULTIPLE SCHEMES

Jean-Marc Drézet (1)

1 IMJ - Institut de Mathématiques de Jussieu

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 canon-ical filtration X = X 1 ⊂ X 2 ⊂ · · · ⊂ 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 n-th 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 × Z n using automorphisms of U ij × Z n that leave U ij invariant. This leads to the study of the sheaf of nonabelian groups G n of automorphisms of X × Z 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 non trivial primitive multiple schemes, of mul-tiplicities 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 ⊂ X 2 ⊂ · · · ⊂ X n ⊂ X n+1 ⊂ · · · of non trivial primitive multiple schemes.

Domaines: Géométrie algébrique [math.AG]

Jean-Marc Drézet. PRIMITIVE MULTIPLE SCHEMES. European Journal of Mathematics, In press. ⟨hal-02541179v2⟩