Abstract
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 IC of C in Y is a line bundle on Cn-1. Let T be a smooth curve and t0 ∈ T a closed point. Let D→ T be a flat family of projective smooth irreducible curves, and C = Dt0. Then the n-th infinitesimal neighbourhood of C in D is a primitive multiple curve Cn of multiplicity n, embedded in the smooth surface D, and in this case IC is the trivial line bundle on Cn-1. Conversely, we prove that every projective primitive multiple curve Y = Cn such that IC is the trivial line bundle on Cn-1 can be obtained in this way.
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