Reducible deformations and smoothing 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 C = Y red is smooth. In this case, L = ICIC2 is a line bundle on C.

This paper continues the study of deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y ). We prove that a primitive double curve can be deformed to reduced curves with smooth components intersecting transversally if and only if  h0(L-1) = 0. We give also some properties of reducible deformations in the case of multiplicity n > 2.