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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 [29]. Walter [36] had previously shown that a very large majority of subcanonical subschemes of codimension 3 in P 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 in characteristic 2 and a smooth 4-fold in PC. These examples are based on our previous work [14] 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. There are many relations between the vector bundles E on a nonsingular algebraic variety X and the subvarieties Z ⊂ X. For example, a globalized form of the Hilbert-Burch theorem allows one to realize any codimension 2 locally Cohen-Macaulay subvariety as a degeneracy locus of a map of vector bundles. In addition the Serre construction gives a realization of any subcanonical codimension 2 subvariety Z ⊂ X as the zero locus of a section of a rank 2 vector bundle on X. The situation in codimension 3 is more complicated. In the local setting, Buchsbaum and Eisenbud [5] described the structure of the minimal free resolution of a Gorenstein (i.e. subcanonical) codimension 3 quotient ring of a regular local ring. Their construction can be globalized: If φ : E → E∗(L) is an alternating map from a vector bundle E of odd rank 2n+1 to a twist of its dual by a line bundle L, then the 2n × 2n Pfaffians of φ define a degeneracy locus Z = {x ∈ X | dimker(φ(x)) ≥ 3}. If X is nonsingular and codim(Z) = 3, the largest possible value, (or more generally if X is locally Noetherian and grade(Z) = 3), then, after twisting E and L appropriately, OZ has a symmetric locally free resolution 0 → L p −→ E φ −→ E(L) p −→ OX → OZ → 0 (0.1) Partial support for the authors during the preparation of this work was provided by the NSF. The authors are also grateful to MSRI Berkeley and the University of Nice Sophia-Antipolis for their hospitality. 1 2 DAVID EISENBUD, SORIN POPESCU, AND CHARLES WALTER with p locally the vector of submaximal Pfaffians of φ and with Z subcanonical with ωZ ∼= ωX(L )|Z . (The reader will find a general discussion of subcanonical subschemes in the introduction of our paper [14] and in Section 2 below; for the purpose of this introduction it may suffice to know that a codimension 3 subvariety Z of Pn is subcanonical if it is (locally) Gorenstein with canonical line bundle ωZ = OZ(d).) Okonek [29] called such a Z a Pfaffian subvariety, and he asked: Question (Okonek). Is every smooth subcanonical subvariety of codimension 3 in a smooth projective variety a Pfaffian subvariety? This paper is one of a series which we have written in response to Okonek’s question. In the first paper (Walter [36]) one of us found necessary and sufficient numerical conditions for a codimension 3 subcanonical subscheme X ⊂ Pk to be Pfaffian, but was unable to give subcanonical schemes failing the numerical conditions. In the second paper [14] we found a construction for codimension 3 subcanonical subschemes which is more general than that studied by Okonek, and proved that this construction gives all subcanonical subschemes of codimension 3 satisfying a certain necessary lifting condition. (This lifting condition always holds if the ambient space is Pn or a Grassmannian of dimension at least 4.) The construction is as follows: Let E,F be a pair of Lagrangian subbundles of a twisted orthogonal bundle with dimk(x) [