Duke Mathematical Journal

Lagrangian subbundles and codimension 3 subcanonical subschemes

David Eisenbud, Sorin Popescu, and Charles Walter

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We show that a Gorenstein subcanonical codimension 3 subscheme ZX=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.

Article information

Duke Math. J., Volume 107, Number 3 (2001), 427-467.

First available in Project Euclid: 5 August 2004

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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

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