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

DAVID EISENBUD, SORIN POPESCU, AND 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 [29]. Walter [36] 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 [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 | dim ker(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, 𝒪_{Z} has a symmetric locally free resolution

0 → L ⟶^{p^{∗}} ℰ ⟶^{ϕ} ℰ^{∗}(L) ⟶^{p} 𝒪_{X} → 𝒪_{Z} → 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.

with p locally the vector of submaximal Pfaffians of ϕ and with Z subcanonical with ω_{Z} ≅ ω_{X}(L^{−1})|_{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 ℙ^{n} is subcanonical if it is (locally) Gorenstein with canonical line bundle ω_{Z} = 𝒪_{Z}(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 ⊂ ℙ_{k}^{N} 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 ℙ^{n} 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 dim_{k(x)}[ℰ(x)∩ℱ(x)] always odd. There is a natural scheme structure on the degeneracy locus

Z := {x ∈ X | dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≥ 3} (0.2)

which is subcanonical if grade(Z) = 3; the structure theorem asserts that every strongly subcanonical codimension 3 subscheme (in the sense of the definition in the introduction of our paper [14]) arises in this way.

Another more complicated but also more explicit description of the structure theorem may be given as follows (see our paper [14] for more details): If one has a subcanonical subvariety with ω_{Z} ≅ L|_{Z}, then the isomorphism 𝒪_{Z} ≅ ω_{Z}(L^{−1})|_{Z} ≅ ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, ω_{X}(L^{−1})) is a sort of symmetry in the derived category. Let L_{1} := ω_{X}(L^{−1}). Then 𝒪_{Z} should have a locally free resolution which is symmetric in the derived category. This is the case if 𝒪_{Z} has a locally free resolution with a symmetric quasi-isomorphism into the twisted shifted dual complex:

[figure: 0; L_{1}; ℰ; ψ; 𝒢; 𝒪_{X}; 𝒪_{Z}; 0; 0; L_{1}; ‖; 𝒢^{∗}(L_{1}); ϕ; −ψ^{∗}; ℰ^{∗}(L_{1}); ϕ^{∗}; 𝒪_{X}; ‖; ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L_{1}); ≅ η; 0] ; (0.3)

If ϕ^{∗} is the identity, then we get the Pfaffian resolution of (0.1). But in general it is not. However, the resolutions in (0.3) mean that Z can be interpreted as the degeneracy locus of (0.2) for the Lagrangian subbundles ℰ, 𝒢^{∗}(L) ⊂ 𝒢 ⊕ 𝒢^{∗}(L).

In this paper we give examples of non-Pfaffian codimension 3 subcanonical subvarieties. Thus we give an answer “No” to Okonek’s question. Because of the numerical conditions of [36], our examples fall into four types:

1. Surfaces S ⊂ ℙ_{k}^{5} with char(k) = 2 and with ω_{S} ≅ 𝒪_{S}(2d) such that h^{1}(𝒪_{S}(d)) is odd.

2. Fourfolds Y ⊂ ℙ^{7} with ω_{Y} ≅ 𝒪_{Y} (2d) such that h^{2}(𝒪_{Y} (d)) is odd.

3. Fourfolds Z ⊂ ℙ_{ℝ}^{7} such that ω_{Z} ≅ 𝒪_{Z}(2d) such that the cup product pairing on H^{2}(𝒪_{Z}(d)) is positive definite (or otherwise not hyperbolic).

4. Examples in ambient varieties other than ℙ^{N}.

Our examples include among others: codimension 3 Schubert subvarieties of orthogonal Grassmannians, non-Pfaffian fourfolds of degree 336 in ℙ^{7}, and nonclassical Enriques surfaces in ℙ^{5} in characteristic 2.

Structure of the paper. In §1 we review well known about quadratic forms, orthogonal Grassmannians and Pfaffian line bundles. In §2 and §3 we review the machinery and results of our previous papers [14] and [36]. The rest of the paper is devoted to the examples of non-Pfaffian subcanonical subschemes. The first example, in §4, is a codimension 3 Schubert variety in 𝕆𝔾_{2n}. In §5 we construct a smooth non-Pfaffian subcanonical fourfold of degree 336 in ℙ^{7} with K = 12H. In §6 we construct some subcanonical fourfolds in ℙ_{ℝ}^{7} which are not Pfaffian over ℝ but become Pfaffian over ℂ. In §7 we analyze Reisner’s example in characteristic 2 of the union of 10 coordinate 2-planes in ℙ^{5}.

In the last two sections §§8,9 we analyze nonclassical Enriques surfaces in characteristic 2 and their Fano models. As a consequence we obtain a (partial) moduli description of Fano-polarized unnodal nonclassical Enriques surfaces as a “quotient” of 𝕆𝔾_{20} modulo SL_{6} (Corollary 8.9). In §9 we identify the closure of the locus of α_{2}-surfaces within 𝕆𝔾_{20} by calculating the action of Frobenius on our resolutions.

The philosophy that (skew)-symmetric sheaves should have locally free resolutions that are (skew)-symmetric up to quasi-isomorphism is also pursued in [15] and [37]. The former deals primarily with methods for constructing explicit locally free resolutions for (skew)-symmetric sheaves on ℙ^{n}. The latter investigates the obstructions (in Balmer’s derived Witt groups [1]) for the existence of a genuinely (skew)-symmetric resolution.

Acknowledgments. We are grateful to Igor Dolgachev, André Hirschowitz, and N. Shepherd-Barron for many useful discussions or suggestions.

The second and third authors would like to thank the Mathematical Sciences Research Institute in Berkeley for its support while part of this paper was being written. The first and second authors also thank the University of Nice–Sophia Antipolis for its hospitality.

Many of the diagrams were set using the diagrams.tex package of Paul Taylor.

1. Orthogonal Grassmannians

In this section we bring together a number of facts about orthogonal Grassmannians which we will need throughout the paper. All the results are well known, so we omit proofs.

Quadratic forms [23] [31]. Let V be vector space of even dimension 2n over a field k. (We impose no restrictions on k; it need not be algebraically closed nor of characteristic ≠ 2.) A quadratic form on V is a homogeneous quadratic polynomial in the linear forms on V, i.e. a member q ∈ S^{2}(V^{∗}). The symmetric bilinear form b : V × V → k associated to q is given by the formula

b(x, y) := q(x + y) − q(x) − q(y). (1.1)

The quadratic form q is nondegenerate if b is a perfect pairing.

A Lagrangian subspace L ⊂ (V, q) is a subspace of dimension n such that q|_{L} ≡ 0. There exist such subspaces if and only if the quadratic form q is hyperbolic. By definition, this means that there exists a coordinate system on V in which one can write q = ∑_{i=1}^{n} x_{i}x_{n+i}. Lagrangian subspaces L satisfy L = L^{⊥}, and the converse is true if char(k) ≠ 2.

If X is an algebraic variety over the field k, then the quadratic form q makes V ⊗ 𝒪_{X} into an orthogonal bundle. A Lagrangian subbundle ℰ ⊂ V ⊗𝒪_{X} is a subbundle of rank n such that all fibers ℰ(x) ⊂ V are Lagrangian with respect to q.

Orthogonal Grassmannians [27] [28] [32]. Let V be a vector space of even dimension 2n with the hyperbolic quadratic form q = ∑_{i=1}^{n} x_{i}x_{n+i}.

The Lagrangian subspaces of (V, q) form two disjoint families such that dim(L ∩ W) ≡ n (mod 2) if L and W are Lagrangian subspaces in the same family, while dim(L ∩ W) ≡ n + 1 (mod 2) otherwise.

Each family of Lagrangian subspaces is parametrized by the k-rational points of an orthogonal Grassmannian (or spinor variety) 𝕆𝔾_{2n} of dimension n(n−1)/2. Let U (resp. W) be a Lagrangian subspace such that dim(L∩U) is even (resp. dim(L ∩ W) is odd) for all L in the first family. Then 𝕆𝔾_{2n} contains the following Schubert varieties:

(1.2)  σ_{2}(U) := {L | dim(L ∩ U) ≥ 2},
(1.3)  σ_{3}(W) := {L | dim(L ∩ W) ≥ 3},
(1.4)  σ_{5}(W) := {L | dim(L ∩ W) ≥ 5}.

We have the following basic results about these subvarieties:

Lemma 1.1. (a) The subvariety σ_{2}(U) ⊂ 𝕆𝔾_{2n} is of codimension 1, and its complement (parametrizing Lagrangian subspaces complementary to U) is an affine space 𝔸_{k}^{n(n−1)/2}.

(b) The subvarieties σ_{3}(W) and σ_{5}(W) are of codimensions 3 and 10 in 𝕆𝔾_{2n}, respectively. Moreover, σ_{3}(W) is nonsingular outside σ_{5}(W).

The reason behind Lemma 1.1(a) is that if one fixes a Lagrangian subspace U^{∗} complementary to U, then the Lagrangian subspaces complementary to U are the graphs of alternating maps U → U^{∗}. This fact also underlies the following lemma.

Lemma 1.2. Suppose L ⊂ (V, q) is a Lagrangian subspace. Then there is a natural identification T_{[L]}(𝕆𝔾_{2n}) ≅ Λ^{2}L^{∗}. Moreover, if dim(L ∩ W) = 3, then there is a natural identification

T_{[L]} (σ_{3}(W)) ≅ {f ∈ Λ^{2}L^{∗} | f|_{L∩W} ≡ 0}.

Pfaffian line bundles [27] [32]. The Picard group of 𝕆𝔾_{2n} is isomorphic to ℤ. Its positive generator 𝒪(1) is the universal Pfaffian line bundle. The divisors σ_{2}(U) of Lemma 1.1(a) are zero loci of particular sections of 𝒪(1). The canonical line bundle of 𝕆𝔾_{2n} is 𝒪(2 − 2n).

On the orthogonal Grassmannian 𝕆𝔾_{2n} there is a universal short exact sequence of vector bundles

0 → 𝒮 → V ⊗ 𝒪_{𝕆𝔾_{2n}} → 𝒮^{∗} → 0 (1.5)

with 𝒮 the universal Lagrangian subbundle of rank n. The determinant line bundle related to the Plücker embedding is det(𝒮^{∗}) = 𝒪(2).

The universal Lagrangian subbundle has the following universal property: If X is an algebraic variety and ℰ ⊂ V ⊗ 𝒪_{X} is a Lagrangian subbundle over X whose fibers lie in the family parametrized by 𝕆𝔾_{2n}, then there exists a unique map f : X → 𝕆𝔾_{2n} such that the natural exact sequence 0 → ℰ → V ⊗ 𝒪_{X} → ℰ^{∗} → 0 is the pullback along f of the universal exact sequence (1.5). The Pfaffian line bundle of ℰ is then Pf(ℰ) := f^{∗}(𝒪(−1)). It is a canonically defined line bundle on X such that Pf(ℰ)^{⊗2} ≅ det(ℰ).

2. The Lagrangian subbundle construction

Let X be a nonsingular algebraic variety over a field k, and let Z ⊂ X be a codimension 3 subvariety. In [14] we called Z ⊂ X subcanonical if the sheaf 𝒪_{Z} has local projective dimension over 𝒪_{X} equal to its grade (in this case 3), and the relative canonical sheaf ω_{Z/X} := ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, 𝒪_{X}) is isomorphic to the restriction to Z of a line bundle L on X. We called Z strongly subcanonical if in addition the map

Ext_{𝒪_{X}}^{3} (𝒪_{Z}, L^{−1}) → Ext_{𝒪_{X}}^{3} (𝒪_{X}, L^{−1}) = H^{3}(X, L^{−1}).

induced by the surjection 𝒪_{X} → 𝒪_{Z} is zero. The last condition is immediate if X = ℙ^{n}, for n ≥ 4, as the right cohomology group is always zero. Thus all subcanonical subvarieties of ℙ^{n} are strongly subcanonical.

The method we use to construct subcanonical subvarieties of codimension 3 was developed in [14]. In fact, all our examples can be constructed using the following result, which is a special case of [14] Theorem 3.1.

Theorem 2.1. Let V be a vector space of even dimension 2n over a field k, and q = ∑_{i=1}^{n} x_{i}x_{n+i} a hyperbolic quadratic form on V. Suppose that X is a nonsingular algebraic variety over k, and that ℰ ⊂ V ⊗ 𝒪_{X} is a Lagrangian subbundle with Pfaffian line bundle L := Pf(ℰ) satisfying L^{⊗2} ≅ det(ℰ). Let W ⊂ (V, q) be a Lagrangian subspace such that dim_{k(x)} [ℰ(x) ∩ W] is odd for all x. If W is sufficiently general, then

Z_{W} = {x ∈ X | dim_{k(x)} (ℰ(x) ∩ W) ≥ 3}, (2.1)

is a (strongly) subcanonical subvariety of codimension 3 in X with

ω_{Z_{W}} ≅ (ω_{X} ⊗ det(ℰ^{∗})) |_{Z_{W}}

and with symmetrically quasi-isomorphic locally free resolutions

[figure: 0; L^{⊗2}; ℰ(L); ψ; W^{∗} ⊗ 𝒪_{X}(L); 𝒪_{X}; 𝒪_{Z_{W}}; 0; L^{⊗2}; ‖; W ⊗ 𝒪_{X}(L); ϕ; −ψ^{∗}; ℰ^{∗}(L); ϕ^{∗}; 𝒪_{X}; ‖; ℰxt _{𝒪_{X}}^{3} (𝒪_{Z_{W}} , L^{⊗2}); ≅ η] ; (2.2)

Moreover, if W is sufficiently general and char(k) = 0, then Sing(Z_{W}) is of codimension 10 in X.

According to Fulton-Pragacz [16] (6.5), the fundamental class of Z_{W} in the Chow ring of X is \frac{1}{4}(c_{1}c_{2} − 2c_{3}), where c_{i} := c_{i}(ℰ^{∗}).

To check the smoothness of Z_{W} in characteristic 0 we proceed as follows. The universal property of the orthogonal Grassmannian gives us a morphism f : X → 𝕆𝔾_{2n}. The group SO_{2n} acts on 𝕆𝔾_{2n}, and it translates the Schubert varieties into other Schubert varieties: g · σ_{i}(Λ) = σ_{i}(gΛ). Since the characteristic is 0, Kleiman’s theorem on the transversality of the general translate [22] applies. So there exists a nonempty Zariski open subset U ⊂ SO_{2n} such that if g ∈ U, then σ_{5}(gΛ) intersects f(X) transversally in codimension 10, and σ_{3}(gΛ) intersects f(X) transversally in codimension 3. Moreover, since SO_{2n} is a rational variety and k is an infinite field, U contains k-rational points. So if W := gΛ with g a k-rational point in U, then W ⊂ (V, q) is a Lagrangian subspace defined over k such that Z_{W} := f^{−1}(σ_{3}(W)) is of codimension 3 and is smooth outside f^{−1}(σ_{5}(W)), which is of codimension 10 in X.

In characteristic p, it will be more complicated to prove that Z_{W} is smooth (see Lemma 5.2 below).

3. Pfaffian subschemes

Theorem 2.1 allows us to construct subcanonical subvarieties Z ⊂ X of codimension 3 with resolutions which are not symmetric. Such a Z might nevertheless possess other locally free resolutions which are symmetric (i.e. Pfaffian). However, the third author gave in [36] a necessary and sufficient condition for a subcanonical subscheme of codimension 3 in projective space to be Pfaffian.

So suppose Z ⊂ ℙ^{n+3} is a subcanonical subscheme of codimension 3 and dimension n such that ω_{Z} ≅ 𝒪_{Z}(ℓ). If n and ℓ are both even, then Serre duality defines a nondegenerate bilinear form

H^{n/2}(𝒪_{Z}(ℓ/2)) × H^{n/2}(𝒪_{Z}(ℓ/2)) ⟶^{∪} H^{n}(𝒪_{Z}(ℓ)) ≅ k. (3.1)

This bilinear form is symmetric if n ≡ 0 (mod 4), and it is skew-symmetric if n ≡ 2 (mod 4). In characteristic 2 it is symmetric in both cases. The result proven was the following.

Theorem 3.1 (Walter [36]). Suppose that Z ⊂ ℙ^{n+3} is a subcanonical subscheme of codimension 3 and dimension n ≥ 1 over the field k such that ω_{Z} ≅ 𝒪_{Z}(ℓ). Then Z is Pfaffian if and only at least one of the following conditions holds: (i) n or ℓ is odd, (ii) n ≡ 2 (mod 4) and char(k) ≠ 2, or (iii) n and ℓ are even, and H^{n/2}(𝒪_{Z}(ℓ/2)) is of even dimension and contains a subspace which is Lagrangian with respect to the cup product pairing of (3.1).

Another way of stating the theorem is that a subcanonical subscheme Z ⊂ ℙ^{n+3} of dimension n with ω_{Z} ≅ 𝒪_{Z}(ℓ) is not Pfaffian if and only either (a) n ≡ 0 (mod 4), or else

(b) n ≡ 2 (mod 4) and char(k) = 2; and at the same time either

(c) ℓ is even and H^{n/2}(𝒪_{X}(ℓ/2)) is odd-dimensional, or else

(d) ℓ is even and H^{n/2}(𝒪_{X}(ℓ/2)) is even-dimensional but has no subspace which is Lagrangian with respect to the cup product pairing of (3.1).

The simplest cases where non-Pfaffian subcanonical subschemes of dimension n in ℙ_{k}^{n+3} could exist are therefore

1. Surfaces S ⊂ ℙ_{k}^{5} with char(k) = 2 and with ω_{S} ≅ 𝒪_{S}(2d) such that h^{1}(𝒪_{S}(d)) is odd.

2. Fourfolds Y ⊂ ℙ^{7} with ω_{Y} ≅ 𝒪_{Y} (2d) such that h^{2}(𝒪_{Y} (d)) is odd.

3. Fourfolds Z ⊂ ℙ_{ℝ}^{7} such that ω_{Z} ≅ 𝒪_{Z}(2d) such that the cup product pairing on H^{2}(𝒪_{Z}(d)) is positive definite (or otherwise not hyperbolic).

4. The codimension 3 Schubert variety in 𝕆𝔾_{2n}

Our first examples of non-Pfaffian subcanonical subschemes of codimension 3 are the codimension 3 Schubert varieties of 𝕆𝔾_{2n}.

Theorem 4.1. Let n ≥ 4, and let σ_{3}(W) ⊂ 𝕆𝔾_{2n} be one of the codimension 3 Schubert varieties of Lemma 1.1(b). Then Z is (strongly) subcanonical with ω_{Z} ≅ 𝒪_{Z}(4 − 2n) but is not Pfaffian.

Proof. We apply Theorem 2.1 with X := 𝕆𝔾_{2n} and ℰ := 𝒮 the universal Lagrangian subbundle. The degeneracy locus is Z := σ_{3}(W) and it has resolutions

[figure: 0; 𝒪(−2); 𝒮(−1); ψ; W^{∗} ⊗ 𝒪(−1); 𝒪; 𝒪_{Z}; 0; 𝒪(−2); ‖; W ⊗ 𝒪(−1); ϕ; −ψ^{∗}; 𝒮^{∗}(−1); ϕ^{∗}; 𝒪; ‖; ℰxt^{3}(𝒪_{Z}, 𝒪(−2)); ≅ η] ; (4.1)

Moreover, ω_{Z} ≅ ω_{𝕆𝔾_{2n}}(2)|_{Z} ≅ 𝒪_{Z}(4 − 2n) since ω_{𝕆𝔾_{2n}} ≅ 𝒪(2 − 2n).

It remains to show that Z is not Pfaffian. Let U be a general totally isotropic subspace of dimension n − 3, and let i : 𝕆𝔾_{6}(U^{⊥}/U) ↪ 𝕆𝔾_{2n} be the natural embedding. If Z were Pfaffian, its symmetric resolution would be of the form

0 → 𝒪(−2) → ℰ(−1) → ℰ^{∗}(−1) → 𝒪 → 𝒪_{Z} → 0 (4.2)

(cf. (0.1) in the introduction). It would thus pull back to a symmetric resolution of i^{−1}(Z) on 𝕆𝔾_{6}(U^{⊥}/U) of the same form.

Now 𝕆𝔾_{6} parametrizes one family of ℙ^{2}’s contained in a smooth hyperquadric in ℙ^{5}. Thus 𝕆𝔾_{6} ≅ ℙ^{3} (see Jessop [21] or Griffiths-Harris [18]), and i^{−1}(Z) is a codimension 3 Schubert subvariety of 𝕆𝔾_{6} ≅ ℙ^{3} and indeed a point Q. If we pull the resolution (4.2) back to ℙ^{3} and twist, we get a resolution

0 → 𝒪_{ℙ^{3}}(−3) → i^{∗}ℰ(−2) → i^{∗}ℰ^{∗}(−2) → 𝒪_{ℙ^{3}}(−1) → 𝒪_{Q}(−1) → 0.

This gives 1 = χ(𝒪_{Q}(−1)) = 2χ(i^{∗}ℰ(−2)), which is a contradiction. This completes the proof.

The contradiction obtained at the end of the proof above is of the same nature as the obstructions in Theorem 3.1 (see [36]). The contradiction can also be derived from [14] Theorem 7.2.

5. A non-Pfaffian subcanonical 4-fold in ℙ^{7}

In this section we use the construction of Theorem 2.1 to give an example of a smooth non-Pfaffian subcanonical 4-fold in ℙ^{7} over an arbitrary infinite field k.

Let V = H^{0}(𝒪_{ℙ^{7}}(1))^{∗}, so that ℙ^{7} is the projective space of lines in V. We fix an identification Λ^{8}V ≅ k. The 70-dimensional vector space Λ^{4}V has the quadratic form q(u) = u^{(2)}, the divided square in the exterior algebra. (If char(k) ≠ 2, then u^{(2)} = \frac{1}{2}u ∧ u.) The associated symmetric bilinear form is b(u, v) = u ∧ v. This quadratic form is hyperbolic because it has Lagrangian subspaces, as we shall see in a moment.

We now look for a Lagrangian subbundle of Λ^{4}V ⊗ 𝒪_{ℙ^{7}}. On ℙ^{7} there is a canonical map 𝒪_{ℙ^{7}} → V ⊗ 𝒪_{ℙ^{7}}(1). Using this, one constructs a Koszul complex of which one part is:

⋯ → Λ^{3}V ⊗ 𝒪_{ℙ^{7}}(−1) ⟶^{d} Λ^{4}V ⊗ 𝒪_{ℙ^{7}} → Λ^{5}V ⊗ 𝒪_{ℙ^{7}}(1) → ⋯ (5.1)

The image and cokernel of d fit into an exact sequence

0 → Ω_{ℙ^{7}}^{4}(4) → Λ^{4}V ⊗ 𝒪_{ℙ^{7}} → Ω_{ℙ^{7}}^{3}(4) → 0. (5.2)

The fiber of Ω_{ℙ^{7}}^{4}(4) over a point \overline{ξ} ∈ ℙ^{7}, viewed as a subspace of Λ^{4}V, is the image over \overline{ξ} of the map d of (5.1), namely W_{ξ} := ξ ∧ Λ^{3}V. These spaces are all totally isotropic of dimension 35, half the dimension of Λ^{4}V. Hence Ω_{ℙ^{7}}^{4}(4) is a Lagrangian subbundle of Λ^{4}V ⊗ 𝒪_{ℙ^{7}}.

The fibers W_{ξ} of Ω_{ℙ^{7}}^{4}(4) are all members of the same family of Lagrangian subspaces of the hyperbolic quadratic space Λ^{4}V. Let W be a general member of that family, and (to abuse notation) W^{∗} a general member of the opposite family of Lagrangian subspaces of Λ^{4}V. Then Λ^{4}V = W ⊕ W^{∗}, and the symmetric bilinear form on Λ^{4}V induces a natural isomorphism between W^{∗} and the dual of W. Finally, let

Z_{W} := \{\overline{ξ} ∈ ℙ^{7} | dim(W_{ξ} ∩ W) ≥ 3\}.

We now apply Theorem 2.1 and obtain the following result.

Theorem 5.1. If the base field k is infinite, and if W ⊂ Λ^{4}V is a Lagrangian subspace in the same family as the W_{ξ} which is sufficiently general, then Z_{W} is a nonsingular subvariety of ℙ^{7} of dimension 4 which is subcanonical but not Pfaffian. It is of degree 336 and has ω_{Z_{W}} ≅ 𝒪_{Z_{W}} (12), and its structural sheaf has locally free resolutions (where we write 𝒪 := 𝒪_{ℙ^{7}})

[figure: 0; 𝒪(−20); Ω_{ℙ^{7}}^{4}(−6); ψ; W^{∗} ⊗ 𝒪(−10); 𝒪; 𝒪_{Z_{W}}; 0; 𝒪(−20); ‖; W ⊗ 𝒪(−10); ϕ; −ψ^{∗}; Ω_{ℙ^{7}}^{3}(−6); ϕ^{∗}; 𝒪; ‖; ω_{Z_{W}} (−12); ≅ η] ; (5.3)

Proof. The existence of Z_{W} and the resolutions follow from Theorem 2.1, using the fact that det(Ω_{ℙ^{7}}^{4}(4)) = 𝒪_{ℙ^{7}}(−20). The subcanonical subvariety Z_{W} is not Pfaffian because ω_{Z} = 𝒪_{Z}(12) and h^{2}(𝒪_{Z}(6)) = 1 (as one sees from (5.3)). Using the resolutions and a computer algebra package, one computes the Hilbert polynomial of Z_{W} as:

336\binom{n + 3}{4} − 2520\binom{n + 2}{3} + 9814\binom{n + 1}{2} − 25571n + 49549

So the degree of Z_{W} is 336. In characteristic 0 the smoothness of the general Z_{W} follows from Kleiman’s transversality theorem as explained after Theorem 2.1. In characteristic p the smoothness follows from Lemma 5.2 below.

The free resolution of R/I_{Z_{W}} is of the form

(5.4) 0 → R(−14) → R(−13)^{8} → R(−12)^{28} ⊕ R(−20)

→ R(−11)^{56} → R(−10)^{35} → R → R/I_{Z_{W}} → 0.

Lemma 5.2. In Theorem 5.1, if k is an infinite field and W is general, then Z_{W} is smooth.

Proof. Let

Υ := {W ∈ 𝕆𝔾_{70} | ℙ^{7} ∩ σ_{5}(W) = ∅,

and ℙ^{7} ∩ σ_{3}(W) = Z_{W} is nonsingular of dimension 4}

We will show that Υ is nonempty without invoking Kleiman’s theorem on the transversality of a general translate. We will do this by showing that the complement has dimension at most dim 𝕆𝔾_{70} −1.

Let n := 35, and let Gr(3, 2n) be the Grassmannian parametrizing threedimensional subspaces of Λ^{4}V. Let

Y := {(ξ, F, W) ∈ ℙ^{7} × Gr(3, 2n) × 𝕆𝔾_{2n} | F ⊂ W_{ξ} ∩ W }.

To choose a point of Y, one first chooses ξ ∈ ℙ^{7}, then F in a fiber of a Grassmannian bundle with fiber Gr(3, n) , then W in a fiber of an orthogonal Grassmannian bundle with fiber 𝕆𝔾_{2(n−3)}. We see that Y is nonsingular with dim(Y) = dim 𝕆𝔾_{2n} +4. A similar computation shows that

Y_{1} := {(ξ, F, W) ∈ Y | dim(W_{ξ} ∩ W) ≥ 5}

is of dimension equal to dim 𝕆𝔾_{2n} −3. So the image of Y_{1} under the projection π : Y → 𝕆𝔾_{2n} is not dominant. Over a point W ∉ π(Y_{1}), one has π^{−1}(W) = Z_{W}. Hence it is enough to show that the dimension of

Y_{2} := {(ξ, F, W) ∈ Y | dπ : T_{(ξ,F,W)}Y → T_{W} 𝕆𝔾_{2n} is not surjective}

is at most dim 𝕆𝔾_{2n} −1. The essential question is: if (ξ, F, W) ∈ Y − Y_{1}, so that π^{−1}(W) = Z_{W}, when is the dimension of T_{ξ}Z_{W} = ker(dπ)_{(ξ,F,W)} equal to 4, and when is it more?

Now T_{ξ}Z_{W} = T_{W_{ξ}}(ℙ^{7} ∩ σ_{3}(W)). In order to identify this space we must identify T_{W_{ξ}}(𝕆𝔾_{2n}) and the two subspaces T_{W_{ξ}}ℙ^{7} and T_{W_{ξ}}(σ_{3}(W)).

There is a well known isomorphism T_{W_{ξ}}(𝕆𝔾_{2n}) ≅ Λ^{2}W_{ξ}^{∗}. Essentially, given a first-order deformation of W_{ξ} as a Lagrangian subspace of Λ^{4}V, any vector w ∈ W_{ξ} deforms within the deforming subspace as a w + εu(w) with u(w) well-defined modulo W_{ξ}. This gives a map W_{ξ} → Λ^{4}V/W_{ξ} ≅ W_{ξ}^{∗}. The vector w + εu(w) remains isotropic if and only if u is alternating. To describe u as an alternating bilinear form on W_{ξ}, one looks at

⟨w_{1}, w_{2} + εu(w_{2})⟩ = ε⟨w_{1}, u(w_{2})⟩.

If W ∩ W_{ξ} is a space F of dimension 3, then T_{W_{ξ}}(σ_{3}(W)) = {u ∈ Λ^{2}W_{ξ}^{∗} | u|_{F ×F} ≡ 0} according to Lemma 1.2. I.e. T_{W_{ξ}}(σ_{3}(W)) is the kernel of the natural map Λ^{2}W_{ξ}^{∗} → Λ^{2}F^{∗}.

Now let V_{ξ} := V/⟨ξ⟩. Then there is a natural isomorphism T_{ξ}ℙ^{7} ≅ V_{ξ}. There is also a natural isomorphism W_{ξ} = ξ ∧ Λ^{3}V ≅ Λ^{3}V_{ξ}. A vector θ ∈ T_{ξ}ℙ^{7} = V_{ξ} corresponds to a first-order infinitesimal deformation ξ + εθ of ξ, and hence to the first-order infinitesimal deformation (ξ + εθ) ∧ Λ^{3}V of W_{ξ}. The corresponding alternating bilinear form on W_{ξ} = ξ ∧ Λ^{3}V is computed by

⟨ξ ∧ α, (ξ + εθ) ∧ β⟩ = ε(ξ ∧ α ∧ θ ∧ β) ∈ ε(Λ^{8}V).

If we use the isomorphism W_{ξ} ≅ Λ^{3}V_{ξ}, then the corresponding alternating bilinear form on Λ^{3}V_{ξ} is therefore

g_{θ}(α, β) = −α ∧ β ∧ θ ∈ Λ^{7}V_{ξ}.

Hence if we think of F := W ∩ W_{ξ} as a three-dimensional subspace of Λ^{3}V_{ξ} ≅ W_{ξ}, then T_{W_{ξ}}(ℙ^{7} ∩ σ_{3}(W)) ⊂ V_{ξ} is the orthogonal complement of the image of Λ^{2}F in Λ^{6}V_{ξ} ≅ V_{ξ}^{∗}. As a result (ξ, F, W) ∈ Y_{2} if and only if the map Λ^{2}F → Λ^{6}V_{ξ} is not injective.

Every vector in Λ^{2}F is of the form α ∧ β. We therefore look at the set

T_{1}(ξ) := {(α, β) ∈ (Λ^{3}V_{ξ} − {0})^{2} | α ∧ β = 0}.

We stratify this locus according to the rank of the map m_{α} : V_{ξ} → Λ^{4}V_{ξ} defined by multiplication by α. Generically this map is injective, and the dual multiplication map m_{α}^{∗} : Λ^{3}V_{ξ} → Λ^{6}V_{ξ} is surjective. For (α, β) to be in T_{1}(ξ), β must be in the codimension-7 subspace ker(m_{α}^{∗}) ⊂ Λ^{3}V_{ξ}. Hence T_{1}(ξ) contains a stratum of dimension 2n − 7 where m_{α} is injective.

The locus T_{1}(ξ) contains two other strata, but they are of smaller dimension. In one of them the α have the form α′ ∧α′′ with α′ ∈ V_{ξ} and α′′ ∈ Λ^{2}V_{ξ} indecomposable. Since α′ is really only well-defined up to a scalar multiple, and α′′ is really well-defined only in Λ^{2}(V_{ξ}/⟨α′⟩), these form a locus of dimension 21 = n − 14 in Λ^{3}V. For these α the rank of m_{α} is 6, so this stratum in T_{1}(ξ) has dimension 2n − 20.

In the final stratum the α are of the form α_{1}∧α_{2}∧α_{3}, a locus of dimension 13 = n − 22. For these α the rank of m_{α} is 4, so this stratum of T_{1}(ξ) has dimension 2n − 26.

Thus dim T_{1}(ξ) = 2n − 7. As a result the locus

T_{2}(ξ) := {F_{0} ∈ Gr(2, W_{ξ}) | F_{0} = Span(α, β) with α ∧ β = 0}

is of dimension 2n − 11. Consequently the locus

T_{3}(ξ) := {F ∈ Gr(3, W_{ξ}) | F ⊃ F_{0} for some F_{0} in T_{2}(ξ)}

is of dimension at most dim T_{2}(ξ) + n − 3 = 3n − 14.

For each F ∈ T_{3}(ξ), to give a W ∈ 𝕆𝔾_{2n} such that F ⊂ W ∩ W_{ξ} is equivalent to giving a member of one of the families of Lagrangian subspaces of F^{⊥}/F. Consequently, such W form a family of dimension (n−3)(n−4)/2. As a result, the locus

T_{4}(ξ) := {(F, W) ∈ Gr(3, W_{ξ}) × 𝕆𝔾_{2n} | F ⊂ W ∩ W_{ξ} and F ∈ T_{3}(ξ)}

is of dimension at most (3n−14)+(n−3)(n−4)/2 = n(n−1)/2−8. Finally

T_{5} := {(ξ, F, W) | (F, W) ∈ T_{4}(ξ)}

is of dimension at most n(n − 1)/2 − 1. But the part of T_{5} where F = W ∩ W_{ξ} is exactly the part of Y_{2} lying in Y − Y_{1}. So dim Y_{2} < n(n − 1)/2 = dim(𝕆𝔾_{2n}).

As a result Y_{2} does not dominate 𝕆𝔾_{2n}, and Υ ⊂ 𝕆𝔾_{2n} is a nonempty open subset. As before, Υ must then contain a rational point because the base field is infinite, and we conclude.

6. Extension of scalars and non-Pfaffian examples

In this section we prove the following extension of Theorem 5.1. It will allow us to construct subcanonical subschemes of codimension 3 in ℙ_{k}^{N} which are not Pfaffian over k but are Pfaffian over a finite extension of k.

Theorem 6.1. Let k be an infinite field, let E be a k-vector space of dimension d ≥ 1, and let b : E × E → k be a nondegenerate symmetric bilinear form. There exists a nonsingular subcanonical fourfold X ⊂ ℙ_{k}^{7} of degree (1000d^{3} + 8d)/3 with K_{X} = (20d − 8)H such that the cup product pairing (3.1) on the middle cohomology

H^{2}(𝒪_{X}(10d − 4)) × H^{2}(𝒪_{X}(10d − 4)) → H^{4}(𝒪_{X}(20d − 8)) ≅ k

may be identified with b : E × E → k.

Proof. If e_{1}, …, e_{d} is a basis of E and b_{ij} := b(e_{i}, e_{j}), then the quadratic form on E ⊗ Λ^{4}V given by

Q (∑_{i} e_{i} ⊗ u_{i}) := ∑_{i} b_{ii} u_{i}^{(2)} + ∑_{i<j} b_{ij} u_{i} ∧ u_{j}

is nondegenerate and has the property that if L is a Lagrangian subspace of (Λ^{4}V, q), then E⊗L is a Lagrangian subspace of (E⊗Λ^{4}V, Q). Consequently E ⊗ Ω_{ℙ^{7}}^{4}(4) is a Lagrangian subbundle of (E ⊗ Λ^{4}V ⊗ 𝒪_{ℙ^{7}}, Q). Also, if we make a standard identification Λ^{4}V^{∗} ≅ Λ^{4}V, then the symmetric bilinear form associated to Q is b ⊗ 1 : E ⊗ Λ^{4}V → E^{∗} ⊗ Λ^{4}V.

We now choose a general Lagrangian subspace W ⊂ (E ⊗ Λ^{4}V, Q) and apply the construction of Theorem 2.1. This produces a subcanonical fourfold X ⊂ ℙ_{k}^{7} with K_{X} = (20d − 8)H. The degree of X can be computed from the formula after Theorem 2.1. If char(k) = 0, then X is nonsingular by Theorem 2.1. If k is infinite of positive characteristic, X is nonsingular by an argument similar to Lemma 5.2 whose details we leave to the reader.

Let N := 10d − 4. The isomorphism η : 𝒪_{X}(N) → ω_{X}(−N) induces an isomorphism

H^{2}(𝒪_{X}(N)) ⟶^{∼} H^{2}(ω_{X}(−N)) ≅ H^{2}(𝒪_{X}(N))^{∗} (6.1)

which corresponds to the cup product pairing of Serre duality (3.1). Our problem is to identify this map.

Now twisting the diagram (2.2) of Theorem 2.1 produces a commutative diagram with exact rows (where we write 𝒪 := 𝒪_{ℙ^{7}} to keep things within the margins):

[figure: 0; 𝒪(−N − 8); f; E ⊗ Ω_{ℙ^{7}}^{4}; ψ; W^{∗} ⊗ 𝒪(−4); g; 𝒪(N); 𝒪_{X}(N); 0; 𝒪(−N − 8); ‖; g^{∗}; W ⊗ 𝒪(−4); ϕ; −ψ^{∗}; E^{∗} ⊗ Ω_{ℙ^{7}}^{3}; ϕ^{∗}; f^{∗}; 𝒪(N); ‖; ω_{X}(−N); ‖] ; (6.2)

The rows of the diagram produce natural isomorphisms H^{2}(𝒪_{X}(N)) ≅ E and H^{2}(ω_{X}(−N)) ≅ E^{∗}. Moreover, the Serre duality pairing

H^{2}(𝒪_{X}(N)) × H^{2}(ω_{X}(−N)) → H^{4}(ω_{X}) ⟶^{tr} k

corresponds to the canonical pairing E×E^{∗} → k since the bottom resolution is dual to the top resolution, so the hypercohomology of the bottom resolution is naturally Serre dual to the hypercohomology of the top resolution.

It remains to identify the map H^{2}(𝒪_{X}(N)) → H^{2}(ω_{X}(−N)) with the map b : E → E^{∗}.

We do this by using the following commutative diagram with exact rows.

[figure: 0; 𝒪(−N − 8); f; E ⊗ Ω_{ℙ^{7}}^{4}; ψ; W^{∗} ⊗ 𝒪(−4); g; 𝒪(N); 𝒪_{X}(N); 0; 𝒪(−N − 8); ‖; W ⊗ 𝒪(−4); ϕ; −β; E ⊗ Ω_{ℙ^{7}}^{3}; α; 𝒪(N); ‖; 𝒪_{X}(N); ‖; 0; 𝒪(−N − 8); ‖; g^{∗}; W ⊗ 𝒪(−4); ‖; −ψ^{∗}; E^{∗} ⊗ Ω_{ℙ^{7}}^{3}; b⊗1; f^{∗}; 𝒪(N); ‖; ω_{X}(−N); ≅ η] ; (6.3)

The maps α and β in this diagram come from the identification E ⊗ Λ^{4}V ≅ W^{∗} ⊕ W, which yields a short exact sequence

0 → E ⊗ Ω_{ℙ^{7}}^{4} \underrightarrow{( \begin{matrix}ψ \\ ϕ \end{matrix} )} (W^{∗} ⊗ 𝒪(−4)) ⊕ (W ⊗ 𝒪(−4)) ⟶^{( α β)} E ⊗ Ω_{ℙ^{7}}^{3} → 0. (6.4)

The exactness of (6.4) implies that the top middle square of diagram (6.3) commutes, and that ϕ induces an isomorphism ker(ψ) ⟶^{∼} ker(β), while α induces an isomorphism coker(ψ) ⟶^{∼} coker(β). As a result the second row of the diagram is exact, and the squares in the top row are all commutative.

To see that the squares in the bottom row are all commutative, recall how the maps α, β, ϕ^{∗} and ψ^{∗} are defined. The inclusions of W^{∗} and W in W^{∗} ⊕ W ≅ E ⊗ Λ^{4}V, together with the bilinear form and the Koszul complex, give rise to a diagram with a commutative square

[figure: W ⊗ 𝒪(−4); ⊂; i; E ⊗ Λ^{4}V ⊗ 𝒪(−4); b⊗1; E^{∗} ⊗ Λ^{4}V ⊗ 𝒪(−4); W^{∗} ⊗ 𝒪(−4); ⊂; i′; E ⊗ Ω_{ℙ^{7}}^{3}; π; b⊗1_{Ω}; E^{∗} ⊗ Ω_{ℙ^{7}}^{3}; π′] ; (6.5)

such that

α = πi′,  β = πi,  ψ^{∗} = π′(b ⊗ 1)i,  ϕ^{∗} = π′(b ⊗ 1)i′.

Since the square in (6.5) is commutative, we get ψ^{∗} = (b ⊗ 1_{Ω})β and ϕ^{∗} = (b⊗1_{Ω})α. The first equality means that the bottom set of squares in diagram (6.3) commutes. The second equality means that the composite of the two chain maps of diagram (6.3) is the chain map of diagram (6.2).

Now the first and second rows of diagram (6.3) produce compatible identifications H^{2}(𝒪_{X}) ≅ E. And the second and third rows of diagram (6.3) show that the map H^{2}(𝒪_{X}(N)) → H^{2}(ω_{X}(−N)) induced by η may be identified with b : E → E^{∗}. This completes the proof of the theorem.

As an example, we may apply Theorem 6.1 with b a positive definite symmetric bilinear form on ℝ^{2}. There are no Lagrangian subspaces of (ℝ^{2}, b), but there are Lagrangian subspaces of (ℂ^{2}, b_{ℂ}). So by the Pfaffianness criterion of Theorem 3.1, we get the following corollary.

Corollary 6.2. There exists a nonsingular subcanonical fourfold Y_{ℝ} ⊂ ℙ_{ℝ}^{7} of degree 2672 with K_{Y} = 32H which is not Pfaffian over ℝ but whose complexification Y_{ℂ} ⊂ ℙ_{ℂ}^{7} is Pfaffian over ℂ.

7. Reisner’s example in ℙ^{5}

Another example of a codimension 3 subcanonical subscheme Z ⊂ ℙ^{5} is provided by the Stanley-Reisner ring of the minimal triangulation of the real projective plane ℝℙ^{2} in characteristic 2. This is Reisner’s original example of a monomial ideal whose minimal free resolution depends on the characteristic of the base field. See Hochster [20], Reisner [33], Stanley [34].

Recall that if Δ is a simplicial complex on the vertex set E = {e_{0}, … e_{n}}, the corresponding face variety X(Δ) ⊂ ℙ_{k}^{n} in the sense of Stanley, Hochster and Reisner (see [34] for more details) is the locus defined by the ideal I_{Δ} generated by monomials corresponding to the simplexes (faces) not contained in Δ. Geometrically, X(Δ) is a “projective linear realization” of Δ. It is the union of linear subspaces of ℙ_{k}^{n} corresponding to the simplices in Δ, where for each m-simplex {e_{i_{0}}, …, e_{i_{m}}} one includes the m-plane spanned by the vertices P_{i_{0}}, …, P_{i_{m}} of the standard coordinate system on ℙ_{k}^{n}.

If the topological realization |Δ| is a manifold, then X(Δ) is a locally Gorenstein scheme with ω_{X(Δ)}^{⊗2} ≅ 𝒪_{X(Δ)}. Moreover, ω_{X(Δ)} is trivial if and only if |Δ| is orientable over the field k. The cohomology of X(Δ) is given by the formula H^{i}(𝒪_{X(Δ)}) = H^{i}(Δ, k). The Hilbert polynomial of X(Δ), which coincides with the Hilbert function for strictly positive values, is completely determined by the combinatorial data (see [34] for more details).

Figure 1. The Minimal Triangulation of ℝℙ^{2}

We apply this theory with Δ the triangulation of ℝℙ^{2} given in Figure 1, and with k a field of characteristic 2. Then X := X(Δ) is a union of ten 2-planes in ℙ_{k}^{5} corresponding to the ten triangles. Moreover, ω_{X} ≅ 𝒪_{X} since ℝℙ^{2} is orientable in characteristic 2. The cohomology is given by

h^{0}(𝒪_{X}) = 1,  h^{1}(𝒪_{X}) = 1,  h^{2}(𝒪_{X}) = 1,

since this is the cohomology of ℝℙ^{2} over a field of characteristic 2. We may now prove

Proposition 7.1 (char 2). Let X ⊂ ℙ^{5} be the face variety corresponding to the triangulation of ℝℙ^{2} given in Figure 1. Then X is subcanonical of codimension 3 but not Pfaffian.

Proof. The subvariety X is a locally Gorenstein surface with ω_{X} = 𝒪_{X} but h^{1}(𝒪_{X}) = 1. Thus it is subcanonical but not Pfaffian by Theorem 3.1.

In addition, X ⊂ ℙ^{5} is linearly normal (all X(Δ) are) and is not contained in a hyperquadric (since all pairs of vertices are joined by an edge in Δ). This will allow us to conclude below that X is a degenerate member of a family of nonclassical Enriques surfaces (see Proposition 8.7 and (8.5)). Over the complex numbers this is a Type III degeneration in Morrison [26] Corollary 6.2 (iii a). See also Symms [35] and Altmann-Christophersen (in preparation) for the deformation theory of this scheme.

8. Enriques surfaces in ℙ^{5}

In their extension to characteristic p of the Enriques-Castelnuovo-Kodaira classification of surfaces not of general type, Bombieri and Mumford [4] characterize Enriques surfaces as nonsingular projective surfaces X with numerically trivial canonical class K_{X} ≡ 0 and with χ(𝒪_{X}) = 1. In characteristic 2, Enriques surfaces are divided into three types:

h^{1}(𝒪_{X}) h^{2}(𝒪_{X}) canonical class F : H^{1}(𝒪_{X}) → H^{1}(𝒪_{X})
classical  0  0  2K_{X} = 0, K_{X} ≠ 0  —
μ_{2}  1  1  K_{X} = 0  injective
α_{2}  1  1  K_{X} = 0  0

The map F : H^{1}(𝒪_{X}) → H^{1}(𝒪_{X}) is the action of Frobenius on the cohomology. There is a voluminous literature on Enriques surfaces which we do not cite here. Our Pfaffianness criterion Theorem 3.1 yields immediately

Proposition 8.1 (char 2). If X ⊂ ℙ^{5} is a nonclassical Enriques surface, then X is subcanonical but not Pfaffian.

A Fano polarization on an Enriques surface X is a numerically effective divisor H such that H^{2} = 10 and such that H · F ≥ 3 for every numerically effective divisor F with F^{2} = 0. It is well known that every Enriques surface admits a Fano polarization, and that for any Fano polarization H the linear system |H| is base-point-free and defines a map φ_{H} : X → ℙ^{5} which is birational onto its image. The image φ_{H}(X) ⊂ ℙ^{5} is called a Fano model of X. A surface Y ⊂ ℙ^{5} is a Fano model of an Enriques surface if and only if it is of degree 10 with at worst rational double points as singularities and has χ(𝒪_{Y}) = 1 and numerically trivial canonical class K_{Y} ≡ 0.

An Enriques surface is called nodal if it contains a smooth rational curve C with C^{2} = −2. Fano models of unnodal Enriques surfaces are necessarily smooth. The following theorem was proven in characteristic ≠ 2 by Cossec [8]. Igor Dolgachev has informed us that the theorem also holds in characteristic 2 using arguments similar to Dolgachev-Reider [13].

Theorem 8.2. Let H be a Fano polarization of an Enriques surface X. Then the following are equivalent: (a) X is nodal, and (b) one of the Fano models φ_{H}(X) or φ_{H+K_{X}} (X) is contained in a hyperquadric.

We will now give an algebraic method for realizing Fano models of unnodal nonclassical Enriques surfaces as degeneracy loci associated to Lagrangian subbundles of an orthogonal bundle on ℙ^{5}. We will need the following technical lemmas about divided squares. Both lemmas hold for even m in all characteristics.

Lemma 8.3. Suppose that R is a commutative algebra over a field of characteristic 2, that m ≥ 3 is an odd integer, and that E a free R-module of rank 2m. Then there exists a well-defined divided square operation Λ^{m}E → Λ^{2m}E ≅ R whose formula, with respect to any basis e_{1}, …, e_{m}, is given by

(∑ x_{i_{1}...i_{m}}e_{i_{1}...i_{m}})^{(2)} := ∑_{\begin{matrix}i_{1}<···<i_{m} \\ j_{1}<···<j_{m} \\ i_{1}<j_{1} \\ \{i_{k}\}∩\{j_{ℓ}\}=∅ \end{matrix}} x_{i_{1}···i_{m}}x_{j_{1}···j_{m}}e. (8.1)

Proof. It is enough to prove the lemma for R a field, since one can reduce to the case where the coefficients x_{i_{1}...i_{m}} are independent indeterminates, and R is the field of rational functions in these indeterminates.

We have to show that (8.1) is invariant under changes of basis in GL(E) = GL_{n}(R). But since R is now assumed to be a field, GL_{n}(R) is generated by three special kinds of changes of basis: permutations of the basis elements, operations of the form e_{i} ⇝ λe_{i}, and operations of the form e_{j} ⇝ e_{j} + αe_{i}. The formula (8.1) is almost trivially invariant under the first two kinds of operations, and it is invariant under the third kind of operation because the portions

x_{ijk_{1}...k_{m−2}}x_{ℓ_{1}...ℓ_{m}}  and  x_{ik_{1}...k_{m−1}}x_{jℓ_{1}...ℓ_{m−1}} + x_{iℓ_{1}...ℓ_{m−1}}x_{jk_{1}...k_{m−1}}

do not change.

Lemma 8.4. Suppose that m ≥ 3 is an odd integer, that V is a vector space of dimension 2m over a field of characteristic 2, and that Q is a nonzero quadratic form on Λ^{m}V. Then Q(x ∧ w) = 0 for all x ∈ V and w ∈ Λ^{m−1}V if and only if Q is a constant multiple of the divided-square quadratic form of Lemma 8.3.

Proof. Suppose first that Q is a multiple of the divided-square quadratic form, and that 0 ≠ x ∈ V and w ∈ Λ^{m−1}V. Choosing a basis of V of the form e_{1} = x, e_{2}, …, e_{m} and applying formula (8.1), one sees easily that Q(x ∧ w) = 0.

Conversely, suppose that Q(x ∧ w) = 0 for all x ∈ V and w ∈ Λ^{m−1}V. Write Q(∑ y_{I}e_{I}) := ∑_{I⪯J} α_{IJ}y_{I}y_{J}. The hypothesis implies that Q(e_{I}) = α_{II} = 0 for all I. It also implies that each Q(e_{iA} + e_{iB}) = α_{iA,iB} = 0. Hence α_{IJ} = 0 if the multi-indices I and J are not disjoint. Finally Q((e_{i} + e_{j}) ∧ (e_{A} + e_{B})) = α_{iA,jB} + α_{iB,jA} = 0. In other words α_{IJ} = α_{KL} if I and J are disjoint, and K and L are disjoint, and I and K contain exactly m − 1 common indices. One then easily deduces that all the α_{IJ} with I and J disjoint must be equal. But then Q is the same as the quadratic form of (8.1) up to a constant.

Now let V = H^{0}(𝒪_{ℙ^{5}}(1))^{∗}, and consider the exact sequence of vector bundles on ℙ^{5}.

0 → Ω_{ℙ^{5}}^{3}(3) → Λ^{3}V ⊗ 𝒪_{ℙ^{5}} → Ω_{ℙ^{5}}^{2}(3) → 0. (8.2)

As in (5.2), if ξ ∈ V, then the fiber of Ω_{ℙ^{5}}^{3}(3) over \overline{ξ} ∈ ℙ^{5} = ℙ(V) may be identified with the subspace ξ ∧ Λ^{2}V ⊂ Λ^{3}V. Therefore Lemmas 8.3 and 8.4 have the following corollary.

Corollary 8.5 (char 2). There exists a nondegenerate quadratic form Q on Λ^{3}V, unique up to a constant multiple, such that Ω_{ℙ^{5}}^{3}(3) is a Lagrangian subbundle of the orthogonal bundle (Λ^{3}V ⊗ 𝒪_{ℙ^{5}}, Q).

We now apply the construction of Theorem 2.1 to the Lagrangian subbundle Ω_{ℙ^{5}}^{3}(3) ⊂ Λ^{3}V ⊗ 𝒪_{ℙ^{5}}. We get the following result.

Theorem 8.6 (char 2). If W ⊂ (Λ^{3}V, Q) is a general Lagrangian subspace in the family opposite to the one containing the fibers of Ω_{ℙ^{5}}^{3}(3), then the degeneracy locus

Z_{W} := {x ∈ ℙ^{5} | dim [Ω_{ℙ^{5}}^{3}(3)(x) ∩ W ] ≥ 3} (8.3)

is a smooth Fano model of a nonclassical unnodal Enriques surface with symmetrically quasi-isomorphic resolutions

[figure: 0; 𝒪_{ℙ^{5}}(−6); Ω_{ℙ^{5}}^{3}; ψ; W^{∗} ⊗ 𝒪_{ℙ^{5}}(−3); 𝒪_{ℙ^{5}}; 𝒪_{Z_{W}}; 0; 𝒪_{ℙ^{5}}(−6); ‖; W ⊗ 𝒪_{ℙ^{5}}(−3); ϕ; −ψ^{∗}; Ω_{ℙ^{5}}^{2}; ϕ^{∗}; 𝒪_{ℙ^{5}}; ‖; ω_{Z_{W}}; ≅ η] ; (8.4)

Proof. The general Z_{W} is a subcanonical surface of degree 10 with ω_{Z_{W}} ≅ 𝒪_{Z_{W}} and with the resolutions (8.4) according to Theorem 2.1. It is smooth by an argument similar to Lemma 5.2. Since K_{Z_{W}} = 0 and h^{i}(𝒪_{Z_{W}}) = 1 for i = 0, 1, 2, it is a nonclassical Enriques surface by the Bombieri-Mumford classification. Since the degree is 10, it is a Fano model. Moreover, according to the resolution, it is not contained in a hyperquadric. Hence it is unnodal by Theorem 8.2.

The theorem has the following converse.

Proposition 8.7 (char 2). Let X ⊂ ℙ^{5} be a locally Gorenstein subscheme of dimension 2 and degree 10, with ω_{X} ≅ 𝒪_{X}, with h^{i}(𝒪_{X}) = 1 for i = 0, 1, 2, and which is linearly normal and not contained in a hyperquadric. Let V = H^{0}(𝒪_{ℙ^{5}}(1))^{∗}, and let Q be the divided square on Λ^{3}V. Then there exists a unique Lagrangian subspace W ⊂ (Λ^{3}V, Q) such that X = Z_{W} (cf. Theorem 8.6) with locally free resolutions as in (8.4).

For example, the face variety X = X(Δ) of Proposition 7.1 satisfies all the hypotheses of the proposition and is defined as X = Z_{W} for the Lagrangian subspace W ⊂ Λ^{3}V spanned by the exterior monomials

e_{013}, e_{014}, e_{023}, e_{025}, e_{045}, e_{124}, e_{125}, e_{135}, e_{234}, e_{345}. (8.5)

Any Z_{W} in (8.3) which is of dimension 2 satisfies all the hypotheses of Proposition 8.7.

The following refinement of the Castelnuovo-Mumford Lemma is needed for the proof of Proposition 8.7. Its proof is left to the reader.

Lemma 8.8. Let ℳ be a coherent sheaf on ℙ^{n} with no nonzero skyscraper subsheaves, and let 0 ≤ q ≤ n − 1 and m be integers. Suppose that ℳ is (m + 1)-regular, that H^{i}(ℳ(m − i)) = 0 for i ≥ n − q, and that the comultiplication maps

H^{i}(ℳ(m − 1 − i)) → V ⊗ H^{i}(ℳ(m − i))

are surjective for 1 ≤ i < n − q. Then for any 0 ≤ j ≤ q, the module of j-th syzygies of H_{∗}^{0}(ℳ) is generated in degrees ≤ m + j.

Proof of Proposition 8.7. The Hilbert polynomial of X is χ(𝒪_{X}(t)) = 5t^{2}+1 by Riemann-Roch. The reader may easily verify that the sheaves 𝒪_{X}(t) have cohomology as given in the following table:

h^{0}(𝒪_{X}(t)) h^{1}(𝒪_{X}(t)) h^{2}(𝒪_{X}(t))
t > 0 5t^{2} + 1  0  0
t = 0  1  1  1
t < 0  0  0  5t^{2} + 1

Now according to the table 𝒪_{X} is 3-regular; the only group obstructing its 2-regularity is H^{2}(𝒪_{X}); and the comultiplication H^{2}(𝒪_{X}(−1)) → V ⊗ H^{2}(𝒪_{X}) is dual to the multiplication V^{∗}⊗H^{0}(𝒪_{X}) → H^{0}(𝒪_{X}(1)) and thus is an isomorphism. So we may apply Lemma 8.8 to see that all the generators of H_{∗}^{0}(𝒪_{X}) are in degrees ≤ 2, and all the relations are in degrees ≤ 3. But for all t ≤ 2 the restriction maps H^{0}(𝒪_{ℙ^{5}}(t)) → H^{0}(𝒪_{X}(t)) are bijections. So the generators and relations of H_{∗}^{0}(𝒪_{ℙ^{5}}) and of H_{∗}^{0}(𝒪_{X}) in degrees ≤ 2 are the same—one generator in degree 0 and no relations. Hence H_{∗}^{0}(𝒪_{X}) has only a single generator in degree 0, and its relations are all in degree 3. In other words H_{∗}^{1}(ℐ_{X}) = 0, and the homogeneous ideal I(X) is generated by a 10-dimensional vector space W^{∗} of cubic equations.

We now go through the argument in [14] Theorem 6.2 which constructs a pair of symmetrically quasi-isomorphic locally free resolutions of a subcanonical subscheme of codimension 3. We have an exact sequence

0 → 𝒦 → W^{∗} ⊗ 𝒪_{ℙ^{5}}(−3) → 𝒪_{ℙ^{5}} → 𝒪_{X} → 0.

There is a natural isomorphism

Ext_{𝒪_{ℙ^{5}}}^{1} (𝒦, 𝒪_{ℙ^{5}}(−6)) ≅ H^{4}(𝒦)^{∗} ≅ H^{2}(𝒪_{X})^{∗} ≅ k,

and a nonzero member of this group induces an extension which we can attach to give a locally free resolution

0 → 𝒪_{ℙ^{5}}(−6) → ℱ → W^{∗} ⊗ 𝒪_{ℙ^{5}}(−3) → 𝒪_{ℙ^{5}} → 𝒪_{X} → 0.

The intermediate cohomology of ℱ comes from the intermediate cohomology of X and is given by H_{∗}^{i}(ℱ) = 0 for i = 1, 2, 4 and H_{∗}^{3}(ℱ) ≅ H_{∗}^{1}(𝒪_{X}) = k. According to Horrocks’ Theorem [30], this means that ℱ ≅ Ω_{ℙ^{5}}^{3}⊕⨁ 𝒪_{ℙ^{5}}(a_{i}). Since ℱ and Ω_{ℙ^{5}}^{3} are both of rank 10, they are isomorphic. This gives the top locally free resolution of (8.4). The rest of diagram (8.4) is now constructed as in the proof of [14] Theorem 6.2.

It remains to identify the quadratic spaces W^{∗} ⊕ W and Λ^{3}V, and to show that W is uniquely determined by X. However, (8.4) gives us an exact sequence

0 → Ω_{ℙ^{5}}^{3}(3) \underrightarrow{( \begin{matrix}ψ \\ ϕ \end{matrix} )} (W^{∗} ⊕ W) ⊗ 𝒪_{ℙ^{5}} ⟶^{( ϕ^{∗} ψ^{∗})} Ω_{ℙ^{5}}^{2}(3) → 0.

This is the unique nontrivial extension of Ω_{ℙ^{5}}^{2}(3) by Ω_{ℙ^{5}}^{3}(3), and thus coincides with the truncated Koszul complex (8.2). There is therefore a natural identification of W^{∗} ⊕ W with Λ^{3}V which identifies the hyperbolic quadratic form on W^{∗} ⊕ W with a quadratic form Q on Λ^{3}V for which Ω_{ℙ^{5}}^{3}(3) ⊂ Λ^{3}V ⊗ 𝒪_{ℙ^{5}} is a Lagrangian subbundle. By Corollary 8.5, Q is necessarily the divided square quadratic form.

The identification of the quadratic spaces W^{∗} ⊕ W and Λ^{3}V is unique up to homothety, once the morphisms ψ, ϕ are chosen. However, the diagram (8.4) is unique up to an alternating homotopy α : W^{∗}⊗𝒪_{ℙ^{5}} → W ⊗𝒪_{ℙ^{5}}. One may check that this alternating homotopy does not change the Lagrangian subspace W ⊂ Λ^{3}V but only the choice of the Lagrangian complement W^{∗} (cf. [14] §5). Thus W is uniquely determined by X. This completes the proof.

Corollary 8.9 (char 2). There is a universal family of Fano models in ℙ^{5} = ℙ(V) of unnodal nonclassical Enriques surfaces parametrized by an SL(V)-invariant Zariski open subset U ⊂ 𝕆𝔾_{20}(Λ^{3}V). The SL(V)-orbits correspond to isomorphism classes of Fano-polarized unnodal nonclassical Enriques surfaces (X, H).

In particular, unnodal nonclassical Enriques surfaces form an irreducible family. We will see below that its general member is a μ_{2}-surface (Proposition 9.1).

The universal family of unnodal nonclassical Enriques surfaces has the following universal locally free resolution over 𝔾 := 𝕆𝔾_{20}(Λ^{3}V). Let 𝒮 ⊂ Λ^{3}V ⊗ 𝒪_{𝔾} by the universal Lagrangian subbundle on 𝔾. According the machinery of [14] Theorem 2.1, the two subbundles p^{∗}Ω_{ℙ^{5}}^{3}(3) and q^{∗}𝒮 on ℙ^{5} × 𝔾 (where p and q are the two projections) define a degeneracy locus Z, which one can verify has the expected codimension 3 using an incidence variety argument. Therefore Z ⊂ ℙ^{5} × 𝔾 has a locally free resolution

0 → 𝒪(−6, −2) → (p^{∗}Ω_{ℙ^{5}}^{3})(0, −1) → (q^{∗}𝒮^{∗})(−3, −1) → 𝒪 → 𝒪_{Z} → 0. (8.6)

Remark 8.10. (a) Barth and Peters have shown that a generic Enriques surface over ℂ has exactly 2^{14} · 3 · 17 · 31 distinct Fano polarizations ([2] Theorem 3.11).

(b) The fact that the Fano model of a unnodal Enriques surface is defined by 10 cubic equations was first observed by F. Cossec (cf. [8] [9]).

(c) (char 2) The following geometrical construction of a general Fanopolarized unnodal μ_{2}-surface is inspired by Mori-Mukai ([25] Proposition 3.1). Let E be an elliptic curve with Hasse invariant one, and let i : E → E be the involution given by translating by the unique nontrivial 2-torsion point. Let π : E → E′ := E/⟨i⟩ be the quotient map. Let D′ be a divisor of degree 3 on E′ and let D := π^{∗}(D′). The linear system |D| gives an embedding E ↪ ℙ^{5} with image an elliptic normal sextic curve. The involution i on E extends to an involution of ℙ^{5}. As in Mori-Mukai [25], (3.1.1) we may choose three general i-invariant hyperquadrics Q_{1}, Q_{2}, Q_{3} containing E, such that X := Q_{1} ∩ Q_{2} ∩ Q_{3} ⊂ ℙ^{5} is a smooth K3 surface, such that i|_{X} has no fixed points. Let L be the hyperplane divisor class on X. Then S := X/⟨i⟩ is an Enriques surface of type μ_{2} (since it has an étale double cover by a K3 surface, cf. [4]). The divisor class L + E descends to a Fano polarization H on S, and the elliptic curve E ⊂ X descends to E′ ⊂ S. By [7] or [13] Theorem 1, (which Igor Dolgachev informs us also holds in characteristic 2), S is unnodal if and only if |H − 2E′| = ∅. But this follows since |L−E| = ∅ because E ⊂ ℙ^{5} is nondegenerate. An easy dimension count shows that this construction gives rises to 10 moduli of Fano polarized unnodal μ_{2} Enriques surfaces, and thus to a general one.

9. Calculating the action of Frobenius

We would like to give a method for identifying when a Fano model of a nonclassical Enriques surface constructed in Theorem 8.6 is of type μ_{2} or type α_{2}. This involves computing explicitly the action of Frobenius on the resolutions. We follow the method used for cubic plane curves in [19] Proposition IV.4.21.

Let 𝒪 := 𝒪_{ℙ^{5}}. Then 𝒪 has a Frobenius endomorphism given on local sections by x ↦ x^{p}. If the ideal sheaf ℐ_{X} is locally generated by sections f_{1}, …, f_{r}, then F (ℐ_{X})𝒪 is locally generated by f_{1}^{p}, … f_{r}^{p}. Let F (X) denote the subscheme of ℙ^{5} defined by the ideal sheaf F (ℐ_{X})𝒪. Then ℐ_{F (X)} ⊂ ℐ_{X}, and the action of F on 𝒪_{X} factors as

𝒪_{X} ^{F} 𝒪_{F (X)} ^{π} [figure] 𝒪_{X} (9.1)

where π is the canonical surjection.

Since ℙ^{5} is a regular scheme, 𝒪 is a flat F (𝒪)-algebra (Kunz [24]). So if one applies F to a locally free resolution of 𝒪_{X} one obtains a locally free resolution of 𝒪_{F (X)}. Now the nonclassical (and sometimes degenerate) Enriques surfaces X ⊂ ℙ^{5} of §§7,8 have minimal free resolutions similar to that of (5.4). The factorization (9.1) lifts to the locally free resolutions and gives a commutative diagram with exact rows:

[figure: 0; 𝒪(−6); d; 𝒪(−5)^{6} ⊕ 𝒪(−6); 𝒪(−4)^{15}; 𝒪(−3)^{10}; d; 𝒪; 𝒪_{X}]

[figure: 0; 𝒪(−12); F; F (d); 𝒪(−10)^{6} ⊕ 𝒪(−12); F; 𝒪(−8)^{15}; F; 𝒪(−6)^{10}; F; F (d); 𝒪; F; 𝒪_{F (X)}; F]

[figure: 0; 𝒪(−6); g; d; 𝒪(−5)^{6} ⊕ 𝒪(−6); 𝒪(−4)^{15}; 𝒪(−3)^{10}; d; 𝒪; ‖; 𝒪_{X}; π]

where the upper set of vertical maps are given by the action of Frobenius, and the lower set of vertical maps are 𝒪-linear maps lifting π. The action of Frobenius on the cohomology of 𝒪_{X} can be computed using the hypercohomology of the resolutions in the above diagram. Only the terms farthest to the left contribute to H^{1}(𝒪_{X}). So the action of Frobenius on H^{1}(𝒪_{X}) is the composite map

H^{5}(ℙ^{5}, 𝒪(−6)) ⟶^{F} H^{5}(ℙ^{5}, 𝒪(−12)) ⟶^{g} H^{5}(ℙ^{5}, 𝒪(−6)).

The action of Frobenius on the generator of the Čech Ȟ^{5}(ℙ^{5}, 𝒪(−6)), computed with respect to the standard cover, is given by

\frac{1}{∏ x_{i}}  \frac{1}{(∏ x_{i})^{2}}  coeff (\frac{g}{(∏ x_{i})^{2}} , \frac{1}{∏ x_{i}} ) = coeff(g, ∏ x_{i}) · \frac{1}{∏ x_{i}}

We conclude that the Hasse invariant of X is the coefficient of ∏ x_{i} in g. If this coefficient is nonzero, then X is a μ_{2}-surface. Otherwise it is an α_{2}-surface.

In the case of Reisner’s monomial example X(Δ) in §7 the minimal free resolution is ℤ^{6}-multigraded, and the minimal generators of each syzygy module have different squarefree weights. In such a situation, one can lift the projection π : R/F (I)R → R/I to a chain map between minimal free resolutions in which all the matrices are diagonal with nonzero squarefree monomial entries. The map g : R(−12) → R(−6) must therefore be ∏ x_{i}. In particular the Hasse invariant is one. This can also be verified with direct computations using Macaulay/Macaulay2 [3] [17].

Macaulay/Macaulay2 can also be used in connection with the methods of this chapter to produce examples of smooth Fano models of unnodal nonclassical Enriques surface of both types μ_{2} and α_{2} defined over the field with two elements.

The previous diagram may be globalized by taking the global resolution (8.6), replacing p^{∗}Ω_{ℙ^{5}}^{3}(0, −1) by its Koszul free resolution, and applying Frobenius. One obtains a diagram of sheaves on ℙ^{5} × 𝕆𝔾_{20}.

[figure: 0; 𝒪(−6, −1); 𝒪(−5, −1)^{6} ⊕ 𝒪(−6, −2); ⋯; 𝒪; 𝒪_{Z}]

[figure: 0; 𝒪(−12, −2); F; 𝒪(−10, −2)^{6} ⊕ 𝒪(−12, −4); F; ⋯; 𝒪; F; 𝒪_{F (Z)}; F]

[figure: 0; 𝒪(−6, −1); g; 𝒪(−5, −1)^{6} ⊕ 𝒪(−6, −2); ⋯; 𝒪; ‖; 𝒪_{Z}; π]

The coefficient of ∏ x_{i} in g is now a section of 𝕆𝔾_{20}(1). We conclude:

Proposition 9.1. The closure in 𝕆𝔾_{20} of the locus parametrizing Fano models of unnodal α_{2}-surfaces is an SL_{6}-invariant hyperplane section of 𝕆𝔾_{20}.

References

[1] P. Balmer, Derived Witt Groups of a Scheme, J. Pure Appl. Algebra, to appear. See

http://www.math.uiuc.edu/K-theory/.

[2] W. Barth, C. Peters, Automorphisms of Enriques surfaces, Invent. Math. 73 (1983), no. 3, 383–411.

[3] D. Bayer, M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download via anonymous ftp from

ftp://math.harvard.edu/macaulay.

[4] E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p. III, Invent. Math. 35 (1976), 197–232.

[5] D. Buchsbaum and D. Eisenbud, Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3, Amer. J. Math. 99 (1977), 447–485.

[6] F. Catanese, Homological algebra and algebraic surfaces, in: Algebraic geometry— Santa Cruz 1995, pp. 3–56, Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc., 1997.

[7] A. Conte, A. Verra, Reye constructions for nodal Enriques surfaces, Trans. Amer. Math. Soc. 336 (1993), no. 1, 79–100.

[8] F. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751.

[9] , I. Dolgachev, Smooth rational curves on Enriques surfaces, Math. Ann. 272 (1985), no. 3, 369–384.

[10] and , Enriques Surfaces I, Progress Math. 76, Birkhäuser, 1989.

[11] I. Dolgachev, Enriques Surfaces II, in preparation.

[12] , Enriques Surfaces: What is Left?, in: Problems in the theory of surfaces and their classification (Cortona, 1988), 81–97, Sympos. Math., XXXII, Academic Press, 1991.

[13] , I. Reider, On rank 2 vector bundles with c_{1}^{2} = 10 and c_{2} = 3 on Enriques surfaces, Algebraic geometry (Chicago, IL, 1989), 39–49, Lecture Notes in Math. 1479, Springer, 1991.

[14] D. Eisenbud, S. Popescu, and C. Walter, Lagrangian Subbundles and Codimension 3 Subcanonical Subschemes, preprint.

[15] , and , Symmetric locally free resolutions of coherent sheaves, in preparation.

[16] W. Fulton and P. Pragacz, Schubert Varieties and Degeneracy Loci, Springer Lect. Notes Math. 1689 (1998).

[17] D. Grayson, M. Stillman, Macaulay 2: a software system devoted to supporting research in algebraic geometry and commutative algebra. Contact the authors, or download source and binaries from ftp://ftp.math.uiuc.edu/Macaulay2 via anonymous ftp.

[18] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.

[19] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

[20] M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in: Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223. Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, 1977.

[21] C. M. Jessop, A treatise on the line complex, Chelsea Publishing Co., New York 1969.

[22] S. Kleiman, The Transversality of a General Translate, Compositio Math. 28 (1974), 287–297.

[23] M-A. Knus, Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften, 294 Springer-Verlag, 1991.

[24] E. Kunz, Characterizations of regular local rings for characteristic p, Amer. J. Math. 91 (1969), 772–784.

[25] S. Mori, S. Mukai, The uniruledness of the moduli space of curves of genus 11, in: Algebraic geometry (Tokyo/Kyoto, 1982), 334–353, Lecture Notes in Math. 1016, Springer, 1983.

[26] D. Morrison, Semistable degenerations of Enriques’ and hyperelliptic surfaces, Duke Math. J. 48 (1981), no. 1, 197–249.

[27] S. Mukai, Curves and symmetric spaces. I. Amer. J. Math. 117 (1995), no. 6, 1627– 1644.

[28] D. Mumford, Theta Characteristics of an Algebraic Curve, Ann scient. Ec. Norm. Sup. (4) 4 (1971), 181–192.

[29] C. Okonek, Notes on varieties of codimension 3 in ℙ^{N}, Manuscripta Math. 84 (1994), no. 3-4, 421–442.

[30] , M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progress in Math. 3, Birkhäuser, 1980.

[31] A. Pfister, Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, 217, Cambridge University Press, Cambridge, 1995.

[32] A. Pressley and G. Segal, Loop Groups, Oxford Univ. Press, 1986.

[33] G. Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49.

[34] R. Stanley, Combinatorics and commutative algebra, Second edition, Progress in Mathematics 41, Birkhäuser, 1996.

[35] J. Symms, On deformations of unions of planes in projective space, Math. Z. 224 (1997), no. 3, 363–384.

[36] C. Walter, Pfaffian Subschemes, J. Alg. Geometry 5 (1996), 671–704.

[37] , Complexes which are Symmetric up to Quasi-Isomorphism, in preparation.

Department of Mathematics, University of California, Berkeley, Berkeley CA 94720, USA

E-mail address: de@msri.org

URL: http://www.msri.org/people/staff/de/

Department of Mathematics, Columbia University, New York, NY 10027, USA

E-mail address: psorin@math.columbia.edu
URL: http://www.math.columbia.edu/~psorin/

Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS), Université de Nice – Sophia Antipolis, 06108 Nice Cedex 02, France

E-mail address: walter@math.unice.fr
URL: http://math1.unice.fr/~walter/