Reducible deformations and smoothing of primitive multiple curves

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 C = Y red is smooth. In this case, \({L=\mathcal{I}_{C}/{\mathcal{I}}_{C}^{2}}\) 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). If a primitive double curve Y can be deformed to reduced curves with smooth components intersecting transversally, then \({h^{0}(L^{-1}){\neq}0}\). We prove that conversely, if L is the ideal sheaf of a divisor with no multiple points, then Y can be deformed to reduced curves with smooth components intersecting transversally. We give also some properties of reducible deformations in the case of multiplicity n > 2.