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Lagrangian subbundles and codimension 3 subcanonical subschemes

By David Eisenbud, Sorin Popescu and Charles Walter

Abstract

We show that a Gorenstein subcanonical codimension 3 subscheme Z ⊂ X = P N, N ≥ 4, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately and conversely. We extend this result to singular Z and all quasi-projective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of D. Buchsbaum and D. Eisenbud [6] and says that Z is Pfaffian. We also prove codimension 1 symmetric and skew-symmetric analogues of our structure theorems. Smooth subvarieties of small codimension Z ⊂ X = P N have been extensively studied in recent years, especially in relation to R. Hartshorne’s conjecture that a smooth subvariety of sufficiently small codimension in P N is a complete intersection

Publisher: F. Tonoli
Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.190.9120
Provided by: CiteSeerX
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