Duke Mathematical Journal
- Duke Math. J.
- Volume 107, Number 3 (2001), 427-467.
Lagrangian subbundles and codimension 3 subcanonical subschemes
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  and says that Z is Pfaffian.
We also prove codimension 1 symmetric and skew-symmetric analogues of our structure theorems.
Duke Math. J., Volume 107, Number 3 (2001), 427-467.
First available in Project Euclid: 5 August 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14M07: Low codimension problems
Secondary: 13D02: Syzygies, resolutions, complexes 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14M12: Determinantal varieties [See also 13C40]
Eisenbud, David; Popescu, Sorin; Walter, Charles. Lagrangian subbundles and codimension 3 subcanonical subschemes. Duke Math. J. 107 (2001), no. 3, 427--467. doi:10.1215/S0012-7094-01-10731-X. https://projecteuclid.org/euclid.dmj/1091737019