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Lagrangian subbundles and codimension 3 subcanonical subschemes. (English) Zbl 1069.14053
Summary: We show that a Gorenstein subcanonical codimension 3 subscheme $$Z \subset X=\mathbb{P}^N$$, $$N\geq 4$$, 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. A. Buchsbaum and D. Eisenbud [Am. J. Math. 99, 447–485 (1977; Zbl 0373.13006)] and says that $$Z$$ is Pfaffian.
We also prove codimension 1 symmetric and skew-symmetric analogues of our structure theorems.

##### MSC:
 14M07 Low codimension problems in algebraic geometry 13D02 Syzygies, resolutions, complexes and commutative rings 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14M12 Determinantal varieties
Zbl 0373.13006
Full Text:
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