2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78720We 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.AMS-LaTeX, diagrams.sty, 35 pages, minor revisionsAlgebraic GeometryCommutative Algebra14M07, 13D02, 14M12, 14F05text