Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3
David Eisenbud, Sorin Popescu, and Charles Walter
We give examples of subcanonical subvarieties of codimension $3$ in projective $n$-space which are not Pfaffian, i.e., defined by the ideal sheaf of submaximal Pfaffians of an alternating map of vector bundles. This gives a negative answer to a question asked by Okonek
Walter had previously shown that a very large majority of subcanonical subschemes of codimension $3$ in $\mathbb P^n$ are Pfaffian, but he left open the question whether the exceptional non-Pfaffian cases actually occur. We give non-Pfaffian examples of the principal types allowed by his theorem, including (Enriques) surfaces in $\mathbb P^5$ in characteristic 2 and a smooth 4-fold in $\mathbb P^7_{\mathbb C}$.
These examples are based on our previous work showing that any strongly subcanonical subscheme of codimension $3$ of a Noetherian scheme can be realized as a locus of degenerate intersection of a pair of Lagrangian (maximal isotropic) subbundles of a twisted orthogonal bundle.