Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3

Details Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3 Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3

1999-06-25

arxiv.org/abs/math/9906171v1

Mathematics

Algebraic Geometry

AMS-Latex 1.2 with diagrams.tex and epsf.tex; one postscript figure; 24 pages

Scientific paper

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^7. These examples are based on our previous work math.AG/9906170 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.

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