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 = C_{1} ⊂⊂ 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 t_{0} ∈ 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.