Abstract
We show that a Gorenstein subcanonical codimension 3 subscheme Z⊂X=ℙN, 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.
Citation
David Eisenbud. Sorin Popescu. Charles Walter. "Lagrangian subbundles and codimension 3 subcanonical subschemes." Duke Math. J. 107 (3) 427 - 467, 15 April 2001. https://doi.org/10.1215/S0012-7094-01-10731-X
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