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Pré-Publication, Document De Travail Année : 2011

Courbes multiples primitives et déformations de courbes lisses

Jean-Marc Drézet
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Résumé

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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Dates et versions

hal-00742543 , version 1 (16-10-2012)
hal-00742543 , version 2 (12-02-2013)

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  • HAL Id : hal-00742543 , version 1

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Jean-Marc Drézet. Courbes multiples primitives et déformations de courbes lisses. 2011. ⟨hal-00742543v1⟩
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