Paramétrisation des courbes multiples primitives

Abstract

The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Bănică and O. Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z_n = spec(ℂ[t]/(tⁿ)). The curves of multiplicity n are obtained by taking an open cover (U_i) of a smooth curve C and by glueing schemes of type 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 C × Z_n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C_n of multiplicity n to one of multiplicity n + 1, and to extend the quasi locally free sheaf D_n of derivations of C_n to a rank 2 vector bundle on C_n.