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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Annales de la Faculté des sciences de Toulouse : Mathématiques
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.