Lagrangian Subbundles and Codimension 3 Subcanonical Subscheme

David Eisenbud; Sorin Popescu; Charles Walter

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arXiv · math/9906170

Lagrangian Subbundles and Codimension 3 Subcanonical Subscheme

David Eisenbud·Sorin Popescu·Charles Walter

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Abstract

We show that a Gorenstein subcanonical codimension 3 subscheme Z in X = P^N, N > 3, 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 quasiprojective 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 Buchsbaum-Eisenbud and says that Z is Pfaffian. We also prove codimension one symmetric and skew-symmetric analogues of our structure theorems.

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David Eisenbud, Sorin Popescu, Charles Walter. 2000-06-25. Lagrangian Subbundles and Codimension 3 Subcanonical Subscheme. https://arxiv.org/abs/math/9906170

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