Courbes multiples primitives et d\'eformations de courbes lisses

TL;DR

This paper studies primitive multiple curves, their local and global embeddings, and the structure of their ideal sheaves, establishing a link between such curves and infinitesimal neighborhoods in smooth surfaces.

Contribution

It characterizes primitive multiple curves with trivial ideal sheaves as infinitesimal neighborhoods in smooth surface families, providing a classification framework.

Findings

Primitive multiple curves have a canonical filtration by multiplicity.

The ideal sheaf I_C is a line bundle on C_{n-1}.

Primitive multiple curves with trivial I_C are exactly those arising from infinitesimal neighborhoods.

Abstract

A primitive multiple curve is a Cohen-Macaulay scheme Y over the field of complex numbers such that the reduced scheme C=Y_red is a smooth curve, and that Y can be locally embedded in a smooth surface. In general such a curve Y cannot be globally embedded in a smooth surface. If Y is a primitive multiple curve of multiplicity n, then there is a canonical filtration of Y C=C_1 ... C_n=Y such that C_i is a primitive multiple curve of multiplicity i. The ideal sheaf I_C of C in Y is a line bundle on C_{n-1}. Let T be a smooth curve and t_0 a closed point of T. Let D-->T be a flat family of projective smooth irreducible curves, and C=D_{t_0}. Then the n-th infinitesimal neighbourhood of C in D is a primitive multiple curve C_n of multiplicity n, embedded in the smooth surface D, and in this case I_C is the trivial line bundle on C_{n-1}. Conversely, we prove that every projective primitive multiple curve Y=C_n such that I_C is the trivial line bundle on C_{n-1} can be obtained in this way.