ENRIQUES SURFACES AND OTHER NON-PFAFFIAN SUBCANONICAL SUBSCHEMES OF CODIMENSION 3

DAVID EISENBUD SORIN POPESCU CHARLES WALTER

Abstract. 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 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 ℙ^{5} in characteristic 2 and a smooth 4-fold in ℙ_{ℂ}^{7}.

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.