A primitive multiple curve is a Cohen-Macaulay scheme Y over ℂ 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 embedded in a smooth surface. If Y is a primitive multiple curve of multiplicity n, then there is a canonical filtration C = C1 ⊂⊂ Cn = Y such that Ci is a primitive multiple curve of multiplicity i. The ideal sheaf C of C in Y is a line bundle on Cn-1.
Let T be a smooth curve and t0 ∈ T a closed point. Let → T be a flat family of projective smooth irreducible curves, and C = t0. Then the n-th infinitesimal neighbourhood of C in is a primitive multiple curve Cn of multiplicity n, embedded in the smooth surface , and in this case C is the trivial line bundle on Cn-1. Conversely, we prove that every projective primitive multiple curve Y = Cn such that C is the trivial line bundle on Cn-1 can be obtained in this way.