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A primitive multiple curve is a Cohen-Macaulay scheme Y over the field ofcomplex numbers such that the reduced scheme C=Y_red is a smooth curve, andthat Y can be locally embedded in a smooth surface. In general such a curve Ycannot be globally embedded in a smooth surface. If Y is a primitive multiplecurve of multiplicity n, then there is a canonical filtration of Y C=C_1 ...C_n=Y such that C_i is a primitive multiple curve of multiplicity i. The idealsheaf I_C of C in Y is a line bundle on C_{n-1}. Let T be a smooth curve andt_0 a closed point of T. Let D-->T be a flat family of projective smoothirreducible curves, and C=D_{t_0}. Then the n-th infinitesimal neighbourhood ofC in D is a primitive multiple curve C_n of multiplicity n, embedded in thesmooth surface D, and in this case I_C is the trivial line bundle on C_{n-1}.Conversely, we prove that every projective primitive multiple curve Y=C_n suchthat I_C is the trivial line bundle on C_{n-1} can be obtained in this way.

Courbes multiples primitives et d\'eformations de courbes lisses