Fragmented deformations of primitive multiple curves

TL;DR

This paper studies the deformations of primitive multiple curves, focusing on fragmented deformations where the curve splits into disjoint smooth components, providing a complete description and characterization of such deformations.

Contribution

It offers a comprehensive description of fragmented deformations of primitive multiple curves and characterizes those curves that admit such deformations.

Findings

Fragmented deformations are fully characterized.

Primitive multiple curves with fragmented deformations are identified.

Conditions for the existence of fragmented deformations are established.

Abstract

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that the associated reduced curve Y_red is smooth. The subject of this paper is the study of deformations of Y in curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations in n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.