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 ⊂⊂ C_n = Y  such that C_i is a primitive multiple curve of multiplicity i. The ideal sheaf _C of C in Y is a line bundle on C_n-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 C_n of multiplicity n, embedded in the smooth surface , and in this case _C is the trivial line bundle on C_n-1. Conversely, we prove that every projective primitive multiple curve Y = C_n such that _C is the trivial line bundle on C_n-1 can be obtained in this way.