Fragmented deformations of primitive multiple curves
  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. 
J.-M. Drezet
math.AG