LAGRANGIAN SUBBUNDLES AND CODIMENSION 3 SUBCANONICAL SUBSCHEMES

DAVID EISENBUD, SORIN POPESCU, AND CHARLES WALTER

Abstract. We show that a subcanonical subscheme Z ⊂ X of codimension 3 satisfying a mild cohomological condition can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately (and vice versa). The extra condition always holds when X is a projective space (and considerably more generally; all we need is the vanishing of a certain class in a cohomology group of a line bundle). In the local case our structure theorems reduce to that of Buchsbaum-Eisenbud [5] and say that Z is Pfaffian. An important technical point is that pairs of Lagrangian subbundles can be transformed into locally alternating maps of vector bundles, allowing us to define natural scheme structures on Lagrangian degeneracy loci.

In the last section of the paper we prove codimension one symmetric and skew-symmetric analogues of our structure theorems.

Smooth subvarieties of small codimension Z ⊂ X = ℙ^{N} have been extensively studied in recent years, especially in relation to Hartshorne’s conjecture that a smooth subvariety of sufficiently small codimension in ℙ^{N} is necessarily a complete intersection. Although the conjecture remains open, smooth subvarieties of small codimension in ℙ^{N} are known to be subcanonical by the Barth-Larsen-Lefschetz theorem; that is, they are Gorenstein and the canonical line bundle ω_{Z} is the restriction of a line bundle on the ambient variety X.

In this paper we study subcanonical subschemes of codimension 3 in a noetherian scheme X.

There are well known theorems describing the local structure of Gorenstein subschemes of a [regular] noetherian scheme in codimensions ≤ 3. In codimensions 1 and 2 all Gorenstein subschemes are local complete intersections. These results have been globalized: any Z ⊂ X of codimension 1 is the zero locus of a section of a line bundle; while a subcanonical Z ⊂ X of codimension 2 is the zero locus of a section of a rank 2 vector bundle if a certain obstruction in cohomology vanishes (see §6 below). In both cases 𝒪_{Z} has a symmetric resolution by locally free 𝒪_{X}-modules.

In codimension 3 both the local and the global cases become more complicated. Locally, a Gorenstein subscheme of codimension 3 need not be a local complete intersection. Rather, Buchsbaum and Eisenbud [5] showed that such a subscheme is cut out locally by the submaximal Pfaffians of an alternating matrix appearing in a minimal free resolution. Okonek [26] asked whether this local result could be generalized to show that codimension 3 subcanonical schemes are cut out by the Pfaffians of an alternating map of vector bundles. Walter [32] gave a positive answer to Okonek’s question in ℙ^{n} under a mild additional hypothesis, but left open the question of whether this hypothesis is always satisfied.

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.

In our next paper [11] we will show that not every subcanonical subscheme of codimension 3 in ℙ^{n} is Pfaffian, settling Okonek’s question negatively. But in the present paper we show that a different way of looking at the Pfaffian construction does generalize, and gives the desired structure theorem for all subcanonical subschemes of codimension 3. (The question as to which subschemes are Pfaffian can be answered in the derived Witt group of Balmer [1]; see Walter [33].)

The terminology we will use in our structure theorems is as follows. A closed subscheme Z ⊂ X of a noetherian scheme will be said to be subcanonical of codimension d if it satisfies two conditions: A) The subscheme Z is relatively Cohen-Macaulay of codimension d in X,

i.e. ℰxt_{𝒪_{X}}^{i} (𝒪_{Z}, 𝒪_{X}) = 0 for all i ≠ d, and

B) There exists a line bundle L on X such that the relative canonical sheaf

ω_{Z/X} := ℰxt_{𝒪_{X}}^{d} (𝒪_{Z}, 𝒪_{X}) is isomorphic to the restriction of L^{−1} to Z. Unfortunately, these conditions are not enough for our structure theorem in codimension 3 nor for the Serre correspondence in codimension 2. For condition B asserts the existence of an isomorphism

η : 𝒪_{Z} ⟶^{∼} ω_{Z/X}(L) = ℰxt_{𝒪_{X}}^{d} (𝒪_{Z}, L) (1)

which one can think of as an η ∈ H^{0}(ℰxt_{𝒪_{X}}^{d} (𝒪_{Z}, L)) = Ext_{𝒪_{X}}^{d} (𝒪_{Z}, L). In the Yoneda Ext, this η defines a class of “resolutions of 𝒪_{Z} by coherent sheaves”

0 → L → ℱ_{d−1} → ⋯ → ℱ_{1} → ℱ_{0} → 𝒪_{Z} → 0.

In our structural theorem we would like to have ℱ_{0}, …, ℱ_{d−1} locally free. We would also like to have ℱ_{0} = 𝒪_{X}, which means that we would like η ∈ Ext_{𝒪_{X}}^{d} (𝒪_{Z}, L) to lift to Ext_{𝒪_{X}}^{d−1}(ℐ_{Z}, L). We will therefore look at what we call strongly subcanonical subschemes which satisfy conditions A-B plus two additional conditions: C) The 𝒪_{X}-module 𝒪_{Z} is of finite local projective dimension (necessarily

equal to the codimension d), and

D) The isomorphism class η ∈ Ext_{𝒪_{X}}^{d} (𝒪_{Z}, L) of (1) goes to zero under the map

Ext_{𝒪_{X}}^{d} (𝒪_{Z}, L) → Ext_{𝒪_{X}}^{d} (𝒪_{X}, L) = H^{d}(X, L) (2)

induced by the surjection 𝒪_{X} → 𝒪_{Z}.

The Serre construction says that a subscheme of codimension 2 is the zero locus of a rank 2 vector bundle if and only if it is strongly subcanonical (see Griffiths-Harris [16] Proposition 1.33, Vogelaar [31] Theorem 2.1, or Bănică-Putinar [2] §2.1 for references stating variants of condition D explicitly).

Condition C holds automatically if the ambient scheme X is regular. Condition D holds automatically if H^{d}(X, L) = 0. This is the case if X = ℙ^{n} with n ≥ d+1, or if X is an affine scheme. In addition, if the ambient scheme X is a Gorenstein variety over a field k, then condition D can be put into a dual form which looks more natural. For in that case Z ⊂ X is subcanonical (of dimension r) if and only if it is Cohen-Macaulay and there exists a line bundle M on X such that ω_{Z} ≅ M|_{Z}, and condition D holds if and only if the following composite map vanishes

H^{r}(X, M) ⟶^{rest} H^{r}(Z, M|_{Z}) ⟶_{≅}^{η} H^{r}(Z, ω_{Z}) ⟶^{tr} k. (3)

Our main results give a construction for strongly subcanonical subschemes of codimension 3 as a degeneracy locus associated to the following situation. Let 𝒱 be a vector bundle on X of even rank 2n equipped with a nonsingular quadratic form q with values in a line bundle L. Let ℰ and ℱ be a pair of Lagrangian subbundles of (𝒱, q) (i.e. totally isotropic subbundles of rank n). It is then well known that dim[ℰ(x) ∩ ℱ(x)] is locally constant modulo 2.

Now suppose that m is an integer such that dim[ℰ(x)∩ℱ(x)] ≡ m (mod 2) for all x ∈ X. Then there is a degeneracy locus which as a set is given by

Z_{m}(ℰ, ℱ)_{red} := {x ∈ X | dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≥ m} .

In §2 we will define a natural scheme structure on this set in roughly the following manner. Using the data ℰ, ℱ ⊂ (𝒱, q) one defines a composite map

λ : ℰ → 𝒱 ≅ 𝒱^{∗}(L) → ℱ^{∗}(L)

such that ker(λ(x)) = ℰ(x)∩ℱ(x) for all x ∈ X. The map λ is not alternating in the classical sense, but (perhaps after modifying λ slightly as in (7)), it is possible to find local bases in which the matrix of λ is alternating (Proposition 2.1). Although these alternating matrices do not glue together, they are sufficiently compatible that we can define Z_{m}(ℰ, ℱ) as the locus defined by their Pfaffians of order rk(ℰ) − m + 2. This scheme structure is natural in the sense that in a suitably generic setting it is reduced, and it is stable under base change.

For our first main result (Theorem 3.1) we show that if dim_{k(x)}[ℰ(x)∩ℱ(x)] is odd for all x ∈ X, and if Z_{3}(ℰ, ℱ) is of grade 3, the expected value, then Z = Z_{3}(ℰ, ℱ) is strongly subcanonical, i.e. it satisfies conditions A-D above. Moreover, we give explicit locally free resolutions for 𝒪_{Z} (see especially Theorem 4.1).

Conversely, we show (Theorem 6.1) that any strongly subcanonical subscheme Z in a quasi-projective scheme X can be obtained as a locus Z_{3}(ℰ, ℱ) as above. We show that the bundle 𝒱 can even be taken to be hyperbolic— that is, 𝒱 = ℱ ⊕ ℱ^{∗}(L) for some bundle ℱ.

One way to look at these results is as follows. The existence of the symmetric isomorphism

η : 𝒪_{Z} ⟶^{∼} ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L)

means that there should be a symmetric isomorphism in the derived category from the locally free resolution

0 → L ⟶^{q} ℰ ⟶^{ψ} 𝒢 ⟶^{p} 𝒪_{X} → 𝒪_{Z} → 0. (4)

of 𝒪_{Z} into its twisted shifted dual. In general, morphisms in the derived category are complicated objects involving homotopy classes of maps and a calculus of fractions. Nevertheless, in Theorem 4.1 we show that there exist suitable locally free resolutions of 𝒪_{Z}—which depend on the choice of η—for which the symmetric isomorphism in the derived category is induced by a symmetric honest-to-goodness chain map which is a quasi-isomorphism and therefore becomes an isomorphism in the derived category. Okonek’s Pfaffian subschemes correspond to situations where this quasi-isomorphism is an isomorphism.

The philosophy that (skew)-symmetric sheaves should have locally free resolutions that are (skew)-symmetric up to quasi-isomorphism is also pursued in our paper [12] and in Walter [33]. The former deals primarily with methods for constructing explicit locally free resolutions for (skew)- symmetric sheaves on ℙ^{n}. The latter studies the obstructions (in Balmer’s derived Witt groups [1]) for the existence of a genuinely (skew)-symmetric resolution.

The results of this paper give a full characterization of codimension 3 subcanonical subschemes. In [11] we will use this machinery to construct various geometric natural examples of subcanonical subschemes of codimension 3 which are not Pfaffian.

Porteous-type formulas for the fundamental classes of degeneracy loci for skew-symmetric maps ϕ : ℰ → ℰ^{∗}(L) were found by Harris-Tu [18], Józefiak-Lascoux-Pragacz [20], and Pragacz [29]. Harris asked for similar formulas for degeneracy loci related to pairs of Lagrangian subbundles, and they were provided by Fulton [13] [14] and Pragacz-Ratajski [30] (see Fulton-Pragacz [15] for more details). (A natural scheme structure on these degeneracy loci and their generalizations with isotropic flag conditions can be defined in a manner similar to (11).)

Fulton and Pragacz ([15] §9.4) also ask whether one can find “natural” resolutions for the structural sheaves of these kinds of symmetric and skewsymmetric degeneracy loci. From such a resolution one can read off formulas in K_{0}(X). Theorem 3.1 provides an explicit answer in one simple case.

Structure of the paper. In Sections 1 and 2 we review basic facts about Lagrangian subbundles of orthogonal bundles and define the scheme structure on the degeneracy loci Z_{m}(ℰ, ℱ). In Section 3 we review the notion of (strongly) subcanonical subschemes and prove one of our main results Theorem 3.1. In Section 4 we discuss the split case of this construction (Theorem 4.1) which also turns out to be a more practical method for constructing codimension 3 subcanonical subschemes. The computation of local equations for these degeneracy loci is discussed in Section 5.

In Section 6 we prove our other main result, a converse to Theorem 3.1, describing the structure of strongly subcanonical schemes. Sections 7 and 8 discuss at length various examples of codimension 3 subcanonical subschemes, particularly the case of points in ℙ^{3}. Additional and more interesting examples can be found in our paper [11].

Finally, in Section 9 we prove codimension one symmetric and skewsymmetric analogues of all previous results. In particular we state Casnati-Catanese’s structural result ([6] Remark 2.2) with the correct parity condition, and give an example of a self-linked threefold of degree 18 in ℙ^{5} which does not have a symmetric resolution because the parity condition fails.

Acknowledgements. We are grateful for many useful discussions to Igor Dolgachev, Bill Fulton, Joe Harris, and André Hirschowitz.

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. Quadratic forms on vector bundles

In this section we recall the basic definitions of twisted orthogonal bundles and of Lagrangian subbundles. The definitions and results can be found in many standard references such as Fulton-Pragacz [15] Chap. 6, Knus [22], and Mukai [23] §1.

Quadratic forms. Suppose that V is a finite-dimensional vector space 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). (5)

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

Now suppose that 𝒱 is a locally free sheaf of constant finite rank over a scheme X. A quadratic form on 𝒱 with values in a line bundle L is a global section q of S^{2}(𝒱^{∗}) ⊗ L. Such a quadratic form is nonsingular if the induced symmetric bilinear form is a perfect pairing. Equivalently a quadratic form q on 𝒱 is nonsingular if for each point x ∈ X the induced quadratic form q(x) on the fiber vector space 𝒱(x) is nondegenerate. A twisted orthogonal bundle on X is a vector bundle 𝒱 equipped with a nonsingular quadratic form q with values in some line bundle L.

Lagrangian subbundles. If V is a vector space of even dimension 2n equipped with a nondegenerate quadratic form, then a Lagrangian subspace E ⊂ (V, q) is a subspace of V of dimension n such that q|_{E} ≡ 0. If the characteristic is ≠ 2, then E ⊂ (V, q) is Lagrangian if and only if E = E^{⊥} := {x ∈ V | b(x, y) = 0 for all y ∈ E}. But in characteristic 2 this condition is necessary but not sufficient for q to vanish on E, i.e. for E to be Lagrangian.

Similarly, a Lagrangian subbundle ℰ ⊂ (𝒱, q) of a twisted orthogonal bundle of even rank 2n is a subbundle (with locally free quotient sheaf) of rank n such that q|_{ℰ} ≡ 0.

The following result is well known (cf. Bourbaki [4] §6 ex. 18(d), Mumford [24], Mukai [23] Proposition 1.6).

Proposition 1.1. If ℰ and ℱ are Lagrangian subbundles of a twisted orthogonal bundle over a scheme X, then the function on X given by x ↦ dim_{k(x)}[ℰ(x) ∩ ℱ(x)] is locally constant modulo 2.

Hyperbolic bundles. If a Lagrangian subbundle ℰ ⊂ (𝒱, q) has a Lagrangian complement ℱ, then the symmetric bilinear form induces a natural isomorphism ℱ ≅ ℰ^{∗}(L), and (𝒱, q) is isometric to the vector bundle ℰ ⊕ ℰ^{∗}(L) endowed with the hyperbolic quadratic form q(e ⊕ α) := α(e) with values in L. (This q is bilinear on ℰ × ℰ^{∗}(L) but quadratic on ℰ ⊕ ℰ^{∗}(L).) The associated hyperbolic symmetric bilinear form has matrix ( \begin{matrix}0 & I \\ I & 0 \end{matrix} ).

The following result is well known and can be found for instance in [22] Remark I.5.5.4. It applies only to affine schemes.

Lemma 1.2. Suppose that X is an affine scheme and that ℰ ⊂ (𝒱, q) is a Lagrangian subbundle of a twisted orthogonal bundle on X with values in a line bundle L. Then there exists an isomorphism 𝒱 ≅ ℰ ⊕ ℰ^{∗}(L) which is the identity on ℰ and which identifies q with the hyperbolic quadratic form on ℰ ⊕ ℰ^{∗}(L).

The following result describing some of the Lagrangian subbundles of a hyperbolic orthogonal bundle is also well known (see for instance [23] Proposition 1.3). Note that it has the effect of transforming an alternating map into a pair of Lagrangian subbundles.

Lemma 1.3. Suppose that ζ : ℰ → ℰ^{∗}(L) is an alternating map, and let Γ_{ζ} : ℰ ↪ ℰ ⊕ ℰ^{∗}(L) be its graph. Then Γ_{ζ}(ℰ) is a Lagrangian subbundle of the hyperbolic twisted orthogonal bundle ℰ ⊕ ℰ^{∗}(L) complementary to the summand ℰ^{∗}(L). Moreover, all Lagrangian subbundles of ℰ ⊕ ℰ^{∗}(L) complementary to ℰ^{∗}(L) may be obtained in this way.

2. Locally alternating maps and their degeneracy loci

In this section we reverse Lemma 1.3 locally by showing how to convert a pair of “same-family” Lagrangian subbundles of an orthogonal bundle into an alternating map of vector bundles. This allows to define degeneracy loci associated to pairs of Lagrangian subbundles of a twisted orthogonal bundle in a way which generalizes the degeneracy loci for alternating maps of vector bundles ϕ : ℰ → ℰ^{∗}(L).

From Lagrangian subbundles to locally alternating maps. Suppose that f : ℰ ↪ (𝒱, q) and g : ℱ ↪ (𝒱, q) are Lagrangian subbndles of a twisted orthogonal bundle. Consider the composite map

λ : ℰ ⟶^{f} 𝒱 ⟶^{β} 𝒱^{∗}(L) ⟶^{g^{∗}} ℱ^{∗}(L) (6)

where β : 𝒱 ⟶^{≅} 𝒱^{∗}(L) is the isomorphism induced by the quadratic form q. In the situation of Lemma 1.3, λ is the alternating map ζ : ℰ → ℰ^{∗}(L).

In general λ is not alternating and may have even or odd rank. However, its rank is locally constant modulo 2 because the kernel of λ(x) : ℰ(x) → ℱ^{∗}(L)(x) is ℰ(x) ∩ ℱ(x), allowing us to apply Proposition 1.1. We may assume that the rank is always even since otherwise we may replace the Lagrangian subbundles ℰ, ℱ of 𝒱 by the Lagrangian subbundles ℰ_{1} := ℰ⊕𝒪_{X} and ℱ_{1} := ℱ ⊕ L of the orthogonal bundle 𝒱_{1} := 𝒱 ⊕ 𝒪_{X} ⊕ L. This has the effect of replacing λ by

λ_{1} : ℰ_{1} = ℰ ⊕ 𝒪_{X} \underrightarrow{( \begin{matrix}λ & 0 \\ 0 & 1 \end{matrix} )} ℱ^{∗}(L) ⊕ 𝒪_{X} = ℱ_{1}^{∗}(L), (7)

This makes all the ranks even but does not change the kernel or cokernel of λ. Notice also that ℰ_{1}(x) ∩ ℱ_{1}(x) = ℰ(x) ∩ ℱ(x) for all x ∈ X.

The following proposition is the main result of this section. It shows that when the rank of λ is even, then λ is locally alternating in suitable bases.

Proposition 2.1. Let ℰ, ℱ ⊂ (𝒱, q) be Lagrangian subbundles of a twisted orthogonal bundle. Suppose that the rank of the map λ : ℰ → ℱ^{∗}(L) of (6) is always even, or equivalently that dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≡ rk(ℰ) (mod 2) for all x ∈ X.

(a) Any x ∈ X has a neighborhood U over which ℰ|_{U} and ℱ|_{U} have a common Lagrangian complement ℳ.

(b) The symmetric bilinear form provides natural isomorphisms γ_{1} : ℳ ⟶^{∼} ℰ^{∗}(L)|_{U} and γ_{2} : ℳ ⟶^{∼} ℱ^{∗}(L)|_{U} such that the composite μ_{ℳ} = γ_{2}^{−1} ∘ λ ∘ γ_{1}^{∗} in the commutative diagram (8) is alternating.

[figure: ℰ|_{U}; λ; ℱ^{∗}(L)|_{U}; 𝒱|_{U} ≅ 𝒱^{∗}(L)|_{U}; g^{∗}; f; ℳ^{∗}(L); γ_{1}^{∗} ≅; μ_{ℳ}; δ^{∗}; ℳ; ≅ γ_{2}; δ] ; (8)

(c) If 𝒩 ⊂ 𝒱|_{U} is another common Lagrangian complement to ℰ|_{U} and ℱ|_{U}, then there is an isomorphism ϵ : ℳ ⟶^{∼} 𝒩 such that the μ_{𝒩} = ϵ∘μ_{ℳ} ∘ϵ^{∗}.

Proof. Since dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≡ rk(ℰ) (mod 2) for any x ∈ X, the Lagrangian subspaces ℰ(x), ℱ(x) ⊂ 𝒱(x) have a common Lagrangian complement M by Lemma 2.2 below. Over some neighborhood U of x, this M lifts to a Lagrangian subbundle ℳ ⊂ 𝒱|_{U} which is a common complement of ℰ|_{U} and of ℱ|_{U}. This gives (a). Part (b) follows from a short but straightforward computation using Lemma 1.3, while part (c) follows from the structure of diagram (8).

Proposition 2.1 has a natural analogue converting pairs of Lagrangian subbundles of a twisted symplectic bundle into a locally symmetric map of vector bundles. We leave the exact formulation to the reader.

The existence of common Lagrangian complements, even over small fields, follows from the following elementary but obscure lemma.

Lemma 2.2. Suppose that q is a nondegenerate quadratic form on an evendimensional vector space V, and that U, U′ ⊂ (V, q) are two Lagrangian subspaces such that dim(U ∩ U′) ≡ dim(U) (mod 2). Then there exists a Lagrangian subspace L ⊂ (V, q) complementary to U and to U′.

Proof. Let K = U ∩ U′. Then U = U^{⊥} ⊂ K^{⊥}, and similarly U′ ⊂ K^{⊥}. On dimensional grounds, we must indeed have U + U′ = K^{⊥}. As a result U/K and U′/K are complementary Lagrangian subspaces of K^{⊥}/K. Moreover, by hypothesis they are even-dimensional.

Let f_{1}, …, f_{2m} be a system of vectors in U mapping onto a basis of U/K. Since U/K and U′/K are complementary Lagrangian subspaces of K^{⊥}/K, the symmetric bilinear form b associated to q induces a perfect pairing between them. So there exists a system of vectors g_{1}, …, g_{2m} in U′ such that b(f_{i}, g_{j}) = δ_{ij} for all i, j.

Let N be the subspace spanned by the f_{i} and g_{j}. Then q|_{N} is nondegenerate, so there is an orthogonal direct sum decomposition V = N ⊕N^{⊥} such that q|_{N} and q|_{N^{⊥}} are both nondegenerate. Moreover, K ⊂ (N^{⊥}, q|_{N^{⊥}}) is a Lagrangian subspace, for which there exists a complementary Lagrangian subspace P by our previous remarks. Let p_{1}, …, p_{r} be a basis of P. One may now check that

f_{1} + g_{2}, f_{2} − g_{1}, f_{3} + g_{4}, f_{4} − g_{3}, …, f_{2m−1} + g_{2m}, f_{2m} − g_{2m−1}, p_{1}, …, p_{r}

form a basis for a Lagrangian subspace L ⊂ (V, q) complementary to both U and U′.

It is amusing to note that Lemma 2.2 is optimal in the following sense. Suppose that k = ℤ/2ℤ, that V = k^{4}, and that q = x_{1}x_{3} + x_{2}x_{4}. Let U, U′ and U′′ be the Lagrangian subspaces given by x_{1} = x_{2} = 0, by x_{3} = x_{4} = 0, and by x_{1} +x_{3} = x_{2} +x_{4} = 0, respectively. Each subspace is of dimension 2, and each pair of subspaces has intersection of dimension 0. But there is no Lagrangian subspace of V which is complementary to U, to U′ and to U′′.

Degeneracy Loci. Let ℰ be a vector bundle of odd (resp. even) rank, L a line bundle, and ζ : ℰ → ℰ^{∗}(L) an alternating map. If m is an odd (resp. even) integer, then the degeneracy locus

Z_{m}(ζ) := {x ∈ X | rk(ζ(x)) ≤ rk(ℰ) − m} (9)

has expected codimension m(m − 1)/2. The natural scheme structure on Z_{m}(ζ) is defined locally by the ideal Pf_{rk(ℰ)−m}(ζ) generated by the (rk(ℰ) − m)-Pfaffians of the alternating map ζ. These loci have been studied notably in Harris-Tu [18], and from a different point of view in Okonek [26] and Walter [32].

Harris has remarked (see [15], p. 70) that Z_{m}(ζ) is a special case of a Lagrangian degeneracy locus defined set-theoretically as follows. Let ℰ, ℱ ⊂ (𝒱, q) be Lagrangian subbundles of a twisted orthogonal bundle, and let

Z_{m}(ℰ, ℱ) := {x ∈ X | dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≥ m}. (10)

The natural fundamental classes of these loci are discussed in Fulton-Pragacz [15] Chap. 6, where they are given as polynomials in the Chern classes of ℰ, ℱ, and L.

We now define a natural scheme structure on these Lagrangian degeneracy loci. By (7) we may assume that dim_{k(x)}[ℰ(x) ∩ ℱ(x)] ≡ rk(ℰ) (mod 2) for all x ∈ X. Then by Proposition 2.1 any x ∈ X has a neighborhood U over which ℰ|_{U} and ℱ|_{U} have a common Lagrangian complement ℳ where the map λ|_{U} may be identified with an alternating map μ_{ℳ} : ℳ^{∗}(L) → ℳ. We define the scheme structure on Z_{m}(ℰ, ℱ) by

ℐ_{Z_{m}(ℰ,ℱ)} |_{U} := Pf_{rk(ℰ)−m}(μ_{ℳ} ). (11)

This scheme structure is well-defined because the ideal of Pfaffians is independent of the choice of a local Lagrangian complement by Proposition 2.1(c). Since Z_{m}(ℰ, ℱ) is locally a degeneracy locus of an alternating map, its expected codimension is m(m − 1)/2.

Given an alternating map, we get identical scheme structures:

Lemma 2.3. If ζ : ℰ → ℰ^{∗}(L) is alternating, and m ≡ rk(ℰ) (mod 2), then the degeneracy loci Z_{m}(ζ) and Z_{m}(ℰ, Γ_{ζ}(ℰ)) given in (9) and (10) coincide.

Our interest in this paper is in the locus Z := Z_{3}(ℰ, ℱ) of expected codimension 3.

3. Subcanonical subschemes

In this section we recall what we mean by a subcanonical subscheme. We then prove one of the main results of the paper.

In the simplest version of the definition, a smooth projective subvariety Y ⊂ ℙ^{N} is subcanonical if its canonical divisor class was a multiple of the hyperplane divisor class, i.e. if K_{Y} = ℓH in the Picard group of Y. More generally, a smooth closed subvariety of a smooth variety Y ⊂ X is subcanonical if there exists a line bundle L on X such that K_{Y} = L|_{Y}.

In the introduction we gave a more complicated definition for a (strongly) subcanonical subscheme of a general scheme which reduces to the above one for smooth varieties. Basically, a closed subscheme Z ⊂ X is subcanonical of codimension 3 if it is relatively Cohen-Macaulay of codimension 3 and there exists a line bundle L on X and an isomorphism η : 𝒪_{Z} ⟶^{∼} ω_{Z/X}(L) making ω_{Z/X} the restriction of a line bundle on X. The subscheme Z ⊂ X is strongly subcanonical if in addition 𝒪_{Z} is of finite local projective dimension over 𝒪_{X}, and if η ∈ Ext_{𝒪_{X}}^{3} (𝒪_{Z}, L) lifts to Ext_{𝒪_{X}}^{2} (ℐ_{Z}, L). These are conditions A–D of the introduction.

One of the two main results of this paper is the following theorem.

Theorem 3.1. Suppose that (𝒱, q) is a twisted orthogonal bundle over a locally noetherian scheme X with values in a line bundle L. Suppose that ℰ, ℱ ⊂ (𝒱, q) are Lagrangian subbundles such that dim_{k(x)}[ℰ(x) ∩ ℱ(x)] is odd for all x ∈ X. Write L_{ℰ,ℱ,𝒱} := det(ℰ)⊗det(ℱ)⊗det(𝒱)^{−1}. Suppose that the submaximal minors of the composite map λ : ℰ ↪ 𝒱 ≅ 𝒱^{∗}(L) ↠ ℱ^{∗}(L) generate an ideal sheaf ℐ of grade 3 (the expected value).

Then the ideal sheaf of the closed subscheme (cf. (11))

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

has grade 3 and satisfies ℐ_{Z}^{2} = ℐ. The sheaf 𝒪_{Z} has locally free resolutions

0 → L_{ℰ,ℱ,𝒱} → ℰ(M) ⟶^{λ} ℱ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0, (12a)

0 → L_{ℰ,ℱ,𝒱} → ℱ(M) ⟶^{−λ^{∗}} ℰ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0. (12b)

with M a line bundle such that M^{⊗2} ≅ L_{ℰ,ℱ,𝒱} ⊗ L^{−1}. Moreover, the natural isomorphism between (12b) and the dual of (12a) defines an isomorphism

η : 𝒪_{Z} ⟶^{≅} ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L_{ℰ,ℱ,𝒱} ) =: ω_{Z/X}(L_{ℰ,ℱ,𝒱} ),

with respect to which Z is (strongly) subcanonical of codimension 3 in X, i.e. satisfies conditions A-D of the introduction.

Corollary 3.2. If, in the situation of the theorem, X is locally Gorenstein, then so is Z, and ω_{Z} ≅ ω_{X}(L_{ℰ,ℱ,𝒱}^{−1} )|_{Z}.

The statement of the theorem remains true even if X is not noetherian provided one defines grade as in Eagon-Northcott [9] and Northcott [25]. The only difference in the proofs is that one uses the non-noetherian generalizations of the Buchsbaum-Eisenbud structure theorems found in these references.

Proof of Theorem 3.1. Let N = ℰ ∩ ℱ be the kernel of the natural map

0 → N \underrightarrow{( \begin{matrix}i \\ j \end{matrix} )} ℰ ⊕ ℱ ⟶^{( f −g)} 𝒱.

If β : 𝒱 ⟶^{∼} 𝒱^{∗}(L) is the isomorphism induced by the quadratic form q, and λ := g^{∗}βf, then we get a commutative diagram

[figure: ℰ; λ; ℱ^{∗}(L); N; ⊂; i; 𝒱; g^{∗}β; ⊂; f; N^{−1} ⊗ L.; j^{∗}; ℱ; λ^{∗}; ⊂; g; ⊂; j; ℰ^{∗}(L); i^{∗}; f^{∗}β] ; (13)

Since the diagonals are short exact sequences, the kernels of λ and of λ^{∗} are both equal to N. In addition, fi = gj.

We claim that N is a line bundle, and that the complexes

0 → N ⟶^{i} ℰ ⟶^{λ} ℱ^{∗}(L) ⟶^{j^{∗}} N^{−1} ⊗ L (14a)

0 → N ⟶^{j} ℱ ⟶^{−λ^{∗}} ℰ^{∗}(L) ⟶^{i^{∗}} N^{−1} ⊗ L (14b)

are exact and are locally free resolutions of 𝒪_{Z}(N^{−1} ⊗ L) for the subscheme Z = Z_{3}(ℰ, ℱ) ⊂ X of grade 3, with ℐ_{Z}^{2} = ℐ. We will prove these claims locally by making λ locally alternating and applying the Buchsbaum-Eisenbud structure theorem [5].

Now the rank of the vector bundles ℰ and ℱ may be even or odd. If the rank is even, we may use (7) to replace λ by

ℰ ⊕ 𝒪_{X} \underrightarrow{( \begin{matrix}λ & 0 \\ 0 & 1 \end{matrix} )} ℱ^{∗}(L) ⊕ 𝒪_{X}

without changing the kernel and cokernel of λ. Thus we may assume that ℰ and ℱ are of odd rank.

By hypothesis, dim_{k(x)}[ℰ(x) ∩ ℱ(x)] is also odd for all x ∈ X. Therefore we may apply Proposition 2.1 to see that our complexes (14a) and (14b) are locally isomorphic to complexes

0 → N|_{U} ⟶^{p} ℳ^{∗}(L) ⟶^{μ_{ℳ}} ℳ ⟶^{p^{∗}} (N^{−1} ⊗ L)|_{U} (15)

with μ_{ℳ} alternating with kernel p = δ^{∗}fi = δ^{∗}gj in the notation of diagrams (8) and (13). Now ℰ and ℳ are of odd rank, μ_{M} is alternating, and the ideal ℐ generated by its submaximal minors is of grade 3. So the Buchsbaum-Eisenbud structure theorem [5] applies. Therefore the kernel N|_{U} is a line bundle, the map p is given by the submaximal Pfaffians of μ_{ℳ} , and the complex (15) is exact and is a resolution of 𝒪_{Z}(N^{−1} ⊗ L)|_{U}. We can also identify the ideal sheaf ℐ generated by the submaximal minors of λ with ℐ_{Z}^{2}. This works because on U the sheaf ℐ_{Z}|_{U} is generated by the submaximal Pfaffians p_{1}, …, p_{n} of the alternating map μ_{M} of (8), while ℐ|_{U} is generated by the submaximal minors. Since the (i, j)-th submaximal minor is ±p_{i}p_{j} ([5] Appendix), we do indeed get ℐ_{Z}^{2}|_{U} = ℐ|_{U}. This verifies our claims.

Now we set M := N ⊗ L^{−1} and twist. We get two dual resolutions

(16a)  0 → M^{⊗2} ⊗ L → ℰ(M) ⟶^{λ} ℱ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0,
(16b)  0 → M^{⊗2} ⊗ L → ℱ(M) ⟶^{−λ^{∗}} ℰ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0.

The alternating product of the determinant line bundles in each resolution is trivial, and therefore M^{⊗2} ⊗ L ≅ L_{ℰ,ℱ,𝒱} .

Let us now verify that Z ⊂ X satisfies conditions A–D of the introduction. The duality between the two resolutions of 𝒪_{Z} shows that Z ⊂ X is relatively Cohen-Macaulay of codimension 3. The duality also induces an isomorphism η : 𝒪_{Z} ⟶^{∼} ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L_{ℰ,ℱ,𝒱} ), making Z ⊂ X subcanonical. Clearly 𝒪_{Z} is of finite local projective dimension. Moreover, η is the Yoneda extension class of (16a) and is thus the image of the class in Ext_{𝒪_{X}}^{2} (ℐ_{Z}, M^{⊗2} ⊗ L) of

0 → M^{⊗2} ⊗ L → ℰ(M) → ℱ^{∗}(L ⊗ M) → ℐ_{Z} → 0.

So Z ⊂ X is strongly subcanonical with respect to η.

4. The split case of Theorem 3.1

In this section we analyze a special case of Theorem 3.1 for which we actually have a slightly more structured result. This version includes many examples of interest, and we will use it in [11] to construct non-Pfaffian subcanonical subschemes of codimension 3 in ℙ^{n}.

Theorem 4.1. Let ℱ be a vector bundle of rank n and L a line bundle on a locally noetherian scheme X. Let ℱ ⊕ ℱ^{∗}(L) be the hyperbolic twisted orthogonal bundle. Suppose that

ℰ ^{⊂}^{( \begin{matrix}ψ \\ ϕ \end{matrix} )} ℱ ⊕ ℱ^{∗}(L) (17)

is a Lagrangian subbundle such that dim_{k(x)} [ℰ(x) ∩ ℱ^{∗}(L)(x)] is odd for all x ∈ X. Let L_{ℰ,ℱ} := det(ℰ) ⊗ det(ℱ)^{−1}.

If the sheaf of ideals ℐ generated by the submaximal minors of ψ is of grade 3 (the expected value), then ideal sheaf of the closed subscheme

Z = Z_{3}(ℰ, ℱ^{∗}(L)) = {x ∈ X | dim_{k(x)} [ℰ(x) ∩ ℱ^{∗}(L)(x)] ≥ 3},

has grade 3 and satisfies ℐ_{Z}^{2} = ℐ. There is a commutative diagram with exact rows

[figure: 0; L_{ℰ,ℱ}; ℰ(M); ψ; ℱ(M); 𝒪_{X}; 𝒪_{Z}; 0; L_{ℰ,ℱ}; ‖; ℱ^{∗}(L ⊗ M); ϕ; −ψ^{∗}; ℰ^{∗}(L ⊗ M); ϕ^{∗}; 𝒪_{X}; ‖; ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L_{ℰ,𝒢} ); ≅ η] ; (18)

with M a line bundle on X such that M^{⊗2} ≅ L^{−1} ⊗ L_{ℰ,ℱ} and with ϕ^{∗}ψ alternating. Moreover, ω_{Z/X} ≅ L_{ℰ,ℱ}^{−1} |_{Z}, and Z is strongly subcanonical of codimension 3 in X with respect to η.

Corollary 4.2. If, in the situation of the theorem, X is locally Gorenstein, then so is Z, and ω_{Z} ≅ ω_{X}(L_{ℰ,ℱ}^{−1} )|_{Z}.

Proof of Theorem 4.1. The only things we need to show which do not already follow from Theorem 3.1 are that (18) commutes and that ϕ^{∗}ψ is alternating. But ℰ is a Lagrangian subbundle, so any local section e ∈ Γ(U, ℰ) satisfies

0 = q(ψ(e) ⊕ ϕ(e)) = ⟨ψ(e), ϕ(e)⟩ = ⟨ϕ^{∗}ψ(e), e⟩. (19)

Thus ϕ^{∗}ψ is alternating, and the central part of diagram (18) commutes. The rest is easy and left to the reader.

Pfaffian subschemes. Okonek’s Pfaffian subschemes [26] are the special case of the construction of Theorem 4.1 with ℰ = ℱ^{∗}(L) and ϕ = 1. For if

ℰ ^{⊂}^{( \begin{matrix}ψ \\ 1 \end{matrix} )} ℰ^{∗}(L) ⊕ ℰ

is a Lagrangian subbundle, then ψ is alternating by Lemma 1.3 or by (19). So in this case the two resolutions of (18) reduce to

0 → L ⊗ M^{⊗2} → ℰ(M) ⟶^{ψ} ℰ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0,

with ψ alternating. Thus Z ⊂ X is one of Okonek’s Pfaffian subschemes.

From non-split to split bundles. When the orthogonal bundle (𝒱, q) of Theorem 3.1 does not split as ℱ ⊕ ℱ^{∗}(L), we cannot fill in the diagram (18) with direct arrows

[figure: 0; L_{ℰ,ℱ,𝒱}; i; ℰ(M); λ; ℱ^{∗}(L ⊗ M); j^{∗}; 𝒪_{X}; 0; 0; L_{ℰ,ℱ,𝒱}; ‖; j; ℱ(M); −λ^{∗}; ℰ^{∗}(L ⊗ M); i^{∗}; 𝒪_{X}; ‖; 0.] ; (20)

Nevertheless, by modifying the orthogonal bundle and its Lagrangian subbundles, we can usually realize the same degeneracy locus as a Lagrangian degeneracy locus of a split orthogonal bundle.

We know two strategies to accomplish this under different hypotheses. One is to apply the converse structure theorem (Theorem 6.1). The other strategy works whenever the quadratic form q ∈ Γ(X, (S^{2}𝒱^{∗})(L)) is the image of an α ∈ Γ(X, (𝒱^{∗} ⊗ 𝒱^{∗})(L)), (for instance if 2 ∈ Γ(X, 𝒪_{X})^{×}), then (𝒱, q) ⊥ (𝒱, −q) is hyperbolic because of the inverse isometries

(𝒱, q) ⊥ (𝒱, −q) [figure: ( \begin{matrix}1 & 1 \\ α & −α^{∗} \end{matrix} ); \underline{( \begin{matrix}β^{−1}α^{∗} & β^{−1} \\ β^{−1}α & −β^{−1} \end{matrix} )}] 𝒱 ⊕ 𝒱^{∗}(L)

with β := α + α^{∗} the nonsingular symmetric bilinear form associated to q.

The composite map ℰ ⊕ ℱ ↪ (𝒱, q) ⊥ (𝒱, −q) ≅ 𝒱 ⊕ 𝒱^{∗}(L), or more explicitly

ℰ ⊕ ℱ \underrightarrow{( \begin{matrix}f & g \\ αf & −α^{∗}g \end{matrix} )} 𝒱 ⊕ 𝒱^{∗}(L)

embeds ℰ⊕ℱ as a Lagrangian subbundle of the hyperbolic bundle 𝒱⊕𝒱^{∗}(L). We may then fill in the diagram (20) with a sequence of quasi-isomorphisms going in both directions:

[figure: 0; L_{ℰ,ℱ,𝒱}; ℰ(M); g^{∗}βf; ℱ^{∗}(L ⊗ M); 𝒪_{X}]

[figure: 0; L_{ℰ,ℱ,𝒱}; ‖; (ℰ ⊕ ℱ)(M); ( 1 0 ); ( f g); 𝒱(M); g^{∗}β; 𝒪_{X}; ‖]

[figure: 0; L_{ℰ,ℱ,𝒱}; ‖; 𝒱^{∗}(L ⊗ M); ( αf −α^{∗}g); ( \begin{matrix}−f^{∗} \\ −g^{∗} \end{matrix} ); (ℰ^{∗} ⊕ ℱ^{∗})(L ⊗ M); ( \begin{matrix}f^{∗}α^{∗} \\ −g^{∗}α^{∗} \end{matrix} ); 𝒪_{X}; ‖]

[figure: 0; L_{ℰ,ℱ,𝒱}; ‖; ℱ(M); βg_{∪}; −f^{∗}βg; ℰ^{∗}(L ⊗ M); (\begin{matrix}1 \\ 0 \end{matrix})_{∪}; 𝒪_{X}; ‖]

Another way of looking at this is to say: let 𝒫_{▪} and 𝒬_{▪} denote the first two lines of the last diagram. If one has 𝒱 ≅ ℱ ⊕ ℱ^{∗}(L) as in Theorem 4.1, then the chain map of that theorem is induced by a twisted shifted nonsingular quadratic form on the chain complex 𝒫_{▪} given by a chain map D_{2}(𝒫_{▪}) → L_{ℰ,ℱ,𝒱} [3]. In general, no such chain map exists, but if we can lift q to α as above, then there is a pair of chain maps

D_{2}(𝒫_{▪}) ⟵^{∼} D_{2}(𝒬_{▪}) → L_{ℰ,ℱ,𝒱} [3] (21)

with the first arrow a quasi-isomorphism. This means that in Theorem 3.1, we are also dealing with a sort of twisted shifted nonsingular quadratic form on 𝒫_{▪}, but only in the derived category.

5. Local equations for the degeneracy locus

Let Z ⊂ X be a subcanonical subscheme of codimension 3 which is the degeneracy locus as in diagram (18). We give two strategies for computing equations which define this degeneracy locus locally. The first is based on the idea of using a common Lagrangian complement to skew-symmetrize λ as in Proposition 2.1 and in the proof of Theorem 3.1. The second is based on finding standard local forms for a pair of Lagrangian submodules.

Strategy 1: Alternating homotopies. We start with a pair of Lagrangian subbundles ℰ and ℱ^{∗}(L) of a twisted orthogonal bundle. As in the proof of Theorem 3.1, we may reduce to the case where the rank of ℰ and ℱ are odd using (7). Also since we are working locally, we may assume that the orthogonal bundle splits as ℱ ⊕ ℱ^{∗}(L) because of Lemma 1.2. Then locally ℰ and ℱ^{∗}(L) have a common Lagrangian complement ℳ according to Proposition 2.1(a). This ℳ is necessarily the graph of an alternating map h : ℱ → ℱ^{∗}(L) by Lemma 1.3. We use this h as an alternating local homotopy to transform the commutative diagram on the left below into the one on the right

[figure: ℰ; ψ; ℱ; ℱ^{∗}(L); ϕ; −ψ^{∗}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; h; ℰ^{∗}(L); ϕ^{∗}; ℰ; ψ; ℱ; ℱ^{∗}(L); ϕ−hψ; −ψ^{∗}; ℰ^{∗}(L); ϕ^{∗}+ψ^{∗}h] (22)

Then ϕ − hψ is an isomorphism because it is the projection of ℰ onto ℱ^{∗}(L) along their common complement ℳ. The dual map ϕ^{∗} + ψ^{∗}h is also an isomorphism. Thus diagram (18), with a symmetric quasi-isomorphism between the resolutions of 𝒪_{Z}, can be modified locally by an alternating homotopy to get a diagram (valid locally) with a symmetric isomorphism from the resolution into its dual

[figure: 0; L_{ℰ,ℱ}; ℰ(M); ψ; ℱ(M); 𝒪_{X}; 𝒪_{Z}; 0; L_{ℰ,ℱ}; ‖; ℱ^{∗}(L ⊗ M); ϕ−hψ; −ψ^{∗}; ℰ^{∗}(L ⊗ M); ϕ^{∗}+ψ^{∗}h; 𝒪_{X}; ‖; ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L_{ℰ,𝒢} ); ≅ η]

Thus locally 𝒪_{Z} has a symmetric resolution

0 → L_{ℰ,ℱ} → ℰ(M) ⟶^{μ} ℰ^{∗}(L ⊗ M) → 𝒪_{X} → 𝒪_{Z} → 0

where μ = ϕ^{∗}ψ + ψ^{∗}hψ is alternating (and is essentially the map μ_{ℳ} of (8) and Proposition 2.1). The submaximal Pfaffians of μ give local equations for the degeneracy locus Z.

The choice of a another common Lagrangian complement ℳ gives a different alternating homotopy h, and vice-versa.

Strategy 2: Standard local forms. Suppose that R is a commutative local ring with maximal ideal 𝔪 and residue field k := R/𝔪. Let F be a free R-module of finite rank, and equip F ⊕ F^{∗} with the hyperbolic quadratic form. Suppose that E ⊂ F ⊕ F^{∗} is a Lagrangian submodule, i.e. a totally isotropic direct summand of rank equal to that of F. Let ψ : E → F and ϕ : E → F^{∗} be the two components of the inclusion.

Lemma 5.1. In the above situation, there exist bases of E and F and a dual basis of F^{∗} in which the matrices of ψ and ϕ are of the form

ψ = (\begin{matrix}β & 0 \\ 0 & I \end{matrix})  ϕ = (\begin{matrix}I & 0 \\ 0 & γ \end{matrix})

with the blocks in the two matrices of the same size, and with β and γ alternating.

Proof. We begin by choosing bases for E and F and the dual basis of F^{∗}, so that we can treat ψ and ϕ as matrices. Since E is a direct summand of F ⊕ F^{∗}, the columns of the total matrix ( \begin{matrix}ψ \\ ϕ \end{matrix} ) are linearly independent even modulo 𝔪. Moreover, by (19) above ϕ^{∗}ψ is an alternating matrix because E ⊂ F ⊕ F^{∗} is a totally isotropic submodule.

We now begin a series of row and column operations on ψ and ϕ which will put them into the required form. The column operations (resp. row operations) correspond to changes of basis of E (resp. of F and F^{∗}) and to the action of invertible matrices P (resp. Q) on ψ and ϕ via ψ ⇝ Q^{−1}ψP and ϕ ⇝ Q^{∗}ϕP.

Choose a maximal invertible minor of ϕ. After row and column operations, we can assume that the corresponding submatrix is an identity block lying in the upper left corner of ϕ and that the blocks below and to the right of it are 0. Thus we can assume that

ϕ = (\begin{matrix}I & 0 \\ 0 & δ \end{matrix})  ψ = (\begin{matrix}ψ_{11} & ψ_{12} \\ ψ_{21} & ψ_{22} \end{matrix})

where the blocks of the two matrices are of the same size, the on-diagonal blocks are square, and the coefficients of δ lie in 𝔪. Since

ϕ^{∗}ψ = (\begin{matrix}ψ_{11} & ψ_{12} \\ δ^{∗}ψ_{21} & δ^{∗}ψ_{22} \end{matrix})

is alternating, we see that all the coefficients of ψ_{12} also lie in 𝔪. Hence all the coefficients in the last block of columns of ( \begin{matrix}ψ \\ ϕ \end{matrix} ) lie in 𝔪 except those in ψ_{22}. Since these columns must be linearly independent modulo 𝔪, it follows that ψ_{22} must be invertible. Applying a new set of column operations to ϕ and ψ, we may assume that ψ = ( \begin{matrix}ψ_{11} & ϵ \\ ψ_{21} & I \end{matrix} ) and that ϕ = ( \begin{matrix}I & 0 \\ 0 & γ \end{matrix} ). Moreover, ϕ^{∗}ψ remains alternating, which actually means that ψ_{11} and γ are alternating, and ϵ = −ψ_{21}^{∗}γ. A final set of row and column operations using the matrices Q = ( \begin{matrix}I & ψ_{21}^{∗}γ \\ 0 & I \end{matrix} ) and P = ( \begin{matrix}I & 0 \\ −ψ_{21} & I \end{matrix} ) puts ψ and ϕ into the form required by the lemma.

Corollary 5.2. Let R be a commutative local ring with residue field k, let F be a free R-module of odd rank, and let E ⊂ F ⊕ F^{∗} be a Lagrangian submodule such that dim_{k} [(E ⊗ k) ∩ (F ⊗ k)] is odd. Let ψ : E → F and ϕ : E → F be the two components of the inclusion.

(a) The determinant of ϕ is of the form det ϕ = af^{2} with a invertible.

(b) If det(ϕ) is not a zero-divisor, and if ψ degenerates along an ideal I of height and grade 3 (as expected), then this ideal is the conductor I = Pf(ϕ^{∗}ψ) : f where Pf(ϕ^{∗}ψ) is the ideal generated by the submaximal Pfaffians of ϕ^{∗}ψ, and where f is as in part (a).

Proof. (a) We put the matrices of ϕ and of ψ in the special form of Lemma 5.1, and we set f := Pf(γ). The determinant of the matrix of ϕ is then f^{2}. Consequently the determinant of the matrix of ϕ with respect to any bases of E and F is of the form af^{2}, with a an invertible element of R coming from the determinants of the change-of-basis matrices.

(b) Using the special forms for ϕ and ψ given in Lemma 5.1, we find that I is generated by the submaximal Pfaffians p_{1}, …, p_{2s+1} of β, while the ideal Pf(ϕ^{∗}ψ) is generated by fp_{1}, …, fp_{2s+1}. Since we suppose det(ϕ) and therefore f are not zero-divisors, this gives (b).

6. The converse structure theorem

In this section we prove a structure theorem for strongly subcanonical subschemes of codimension 3 which is a converse to Theorems 3.1 and 4.1. These strongly subcanonical subschemes Z ⊂ X satisfy conditions A–D which were explained in the introduction, namely, 𝒪_{Z} is a Cohen-Macaulay 𝒪_{X}-module of grade and local projective dimension 3, and there is a line bundle L on X and an isomorphism

η : 𝒪_{Z} ⟶^{∼} ω_{Z/X}(L) = ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L)

such that η ∈ Ext_{𝒪_{X}}^{3} (𝒪_{Z}, L) lifts to Ext_{𝒪_{X}}^{2} (ℐ_{Z}, L). The obstruction to this lifting lives in H^{3}(X, L) because of the long exact sequence

⋯ → Ext_{𝒪_{X}}^{2} (ℐ_{Z}, L) → Ext_{𝒪_{Z}}^{3} (𝒪_{Z}, L) → H^{3}(X, L) → · · · . (23)

This lifting condition holds automatically if H^{3}(X, L) = 0 for instance if X is a projective space. The analogous lifting condition is also necessary for the codimension 2 Serre construction (see e.g. Bănică-Putinar [2] §2.1).

We show that such a Z can be obtained as a Lagrangian degeneracy locus as in Theorem 4.1 if the ambient scheme is quasi-projective over a noetherian ring.

Theorem 6.1. Let A be a noetherian ring, and X ⊂ ℙ_{A}^{N} a locally closed subscheme. Suppose that Z ⊂ X is a strongly subcanonical subscheme (as just described). Then Z has symmetrically quasi-isomorphic locally free resolutions

[figure: 0; L; ℰ; ψ; 𝒢; 𝒪_{X}; 𝒪_{Z}; 0; 0; L; ‖; 𝒢^{∗}(L); ϕ; −ψ^{∗}; ℰ^{∗}(L); ϕ^{∗}; 𝒪_{X}; ‖; ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L); ≅ η; 0] ; (24)

with ϕ^{∗}ψ : ℰ → ℰ^{∗}(L) an alternating map.

We will need the following two lemmas in the proof of the theorem.

Lemma 6.2. Let A be a noetherian ring, let X ⊂ ℙ_{A}^{N} be a locally closed subscheme, let ℱ, 𝒢 be coherent sheaves on X, let ℳ be a vector bundle on X, and let p > 0.

(a) If ξ ∈ Ext_{𝒪_{X}}^{p} (ℱ, 𝒢), then there exists a vector bundle ℰ on X and a surjection f : ℰ ↠ ℱ such that the pullback class f^{∗}ξ ∈ Ext_{𝒪_{X}}^{p} (ℰ, 𝒢) vanishes.

(b) If ζ ∈ Ext_{𝒪_{X}}^{p} (Λ^{2}ℳ, 𝒢), then there exists a surjection of vector bundles 𝒫 ↠ ℳ such that the pullback of ζ to Ext_{𝒪_{X}}^{p} (Λ^{2}𝒫, 𝒢) vanishes.

Proof. (a) Extending ℱ to a coherent sheaf on the closure \overline{X} ⊂ ℙ_{A}^{N} and then applying Serre’s Theorem A, we see that there exists a surjection of the form g : 𝒪_{X}(−n)^{r} ↠ ℱ. Pulling back gives us a class g^{∗}ξ ∈ H^{p}(X, 𝒢(n))^{r}.

Let R be the homogeneous coordinate ring of \overline{X}, and let I ⊂ R be the homogeneous ideal of strictly positive degree elements vanishing on the closed subset \overline{X} \ X. Extend 𝒢 to a coherent sheaf on \overline{X}, and let G be a finitely generated graded R-module whose associated sheaf is this extension of 𝒢. Then H_{∗}^{p}(X, 𝒢)^{r} ≅ H_{I}^{p+1}(G)^{r}. Consequently, g^{∗}ξ, as a member of a local cohomology module, is annihilated by some power I^{m} of I. A finite set of homogeneous generators of I^{m} gives surjections ⨁_{i} R(−a_{i}) ↠ I^{m} and ⨁_{i} 𝒪_{X}(−a_{i}) ↠ 𝒪_{X}, such that the pullback of g^{∗}ξ along the induced map ⨁_{i} 𝒪_{X}(−a_{i} − n)^{r} ↠ 𝒪_{X}(−n)^{r} vanishes.

(b) For the same reasons as in part (a), ζ is killed by some power of I. For convenience we assume that the same I^{m} as in part (a) kills ζ. Let ⨁_{i} 𝒪_{X}(−a_{i}) ↠ 𝒪_{X} be the surjection used in part (a). Then the surjection ⨁_{i} ℳ(−a_{i}) ↠ ℳ kills ζ because the exterior square factors as Λ^{2}(⨁_{i} ℳ(−a_{i})) ↠ ⨁_{i≤j}(Λ^{2}ℳ)(−a_{i} − a_{j}) ↠ Λ^{2}ℳ.

Lemma 6.3 (Serre; [27] Lemma 5.1.2). Let A be a noetherian local ring and M a finitely generated A-module of projective dimension at most 1. Suppose that ζ ∈ Ext_{A}^{1}(M, A) corresponds to the extension 0 → A → N → M → 0. Then N is a free A-module if and only if ζ generates the A-module Ext_{A}^{1}(M, A).

Proof of Theorem 6.1. By hypothesis, η lifts to a class in Ext_{𝒪_{X}}^{2} (ℐ_{Z}, L). By Lemma 6.2(a) there exists a vector bundle ℳ and a surjection and kernel 0 → 𝒦 → ℳ → ℐ_{Z} → 0 such that η lifts further to a class ζ ∈ Ext_{𝒪_{X}}^{1} (𝒦, L). This defines an extension 0 → L → ℰ → 𝒦 → 0. Attaching these extensions gives an acyclic complex

0 → L → ℰ → ℳ → 𝒪_{X} → 𝒪_{Z} → 0. (25)

We claim that ℰ is locally free. Our reasoning is as follows. Since the local projective dimension of 𝒪_{Z} is at most 3, the local projective dimension of 𝒦 is at most 1. By Lemma 6.3, ℰ will be locally free if ζ generates the sheaf ℰxt_{𝒪_{X}}^{1} (𝒦, L). Moreover the sheaves 𝒪_{Z}, ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L)), and ℰxt_{𝒪_{X}}^{1} (𝒦, L) are all isomorphic, and their respective global sections 1, η, and ζ correspond under these isomorphisms.

ζ ∈ Ext_{𝒪_{X}}^{1} (𝒦, L) \overline{}H^{0}(ℰxt_{𝒪_{X}}^{1} (𝒦, L))

[figure: η ∈ Ext_{𝒪_{X}}^{3} (𝒪_{Z}, L); ≅; H^{0}(ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L)); ≅; ≅; H^{0}(𝒪_{Z}) ∋ 1]

Since 1 generates 𝒪_{Z}, the section ζ generates ℰxt_{𝒪_{X}}^{1} (𝒦, L). Thus ℰ is locally free.

The complex

𝒜_{▪} : 0 → L → ℰ → ℳ → 𝒪_{X} → 0 (26)

is now a locally free resolution of 𝒪_{Z}. As in Buchsbaum-Eisenbud [5] and Walter [32], we try to make this into a commutative associative differential graded algebra resolution of 𝒪_{Z} by constructing a map D_{2}(𝒜_{▪}) → 𝒜_{▪} from the divided square covering the identity in degree 0:

[figure: ⋯; ℳ(L) ⊕ D_{2}ℰ; L ⊕ (ℰ ⊗ ℳ); ℰ ⊕ Λ^{2}ℳ; ℳ; 𝒪_{X}; ⋯; 0; . . . . . . . . .; L; . . . . . . . . .; ℰ; . . . . . . . . .; ℳ; ‖; 𝒪_{X}; ‖] ; (27)

Now Λ^{2}ℳ maps into the kernel 𝒦 of ℳ → 𝒪_{X}. Hence the first problem in filling in the dotted arrows above is to carry out a lifting

\begin{aligned}& & & & & Λ^{2}ℳ \\ 0 & \overline{} & L & \overline{} & ℰ & \overline{} & ................... & 𝒦 & & \overline{} & 0 \end{aligned}

The obstruction to carrying out the lifting is a class ζ ∈ Ext_{𝒪_{X}}^{1} (Λ^{2}ℳ, L). There is no reason for this class to vanish. So the liftings sought in (27) need not exist. But there is a way around this.

By Lemma 6.2(b) there is a surjection from another vector bundle 𝒢 ↠ ℳ such that the pullback of ζ to Ext_{𝒪_{X}}^{1} (Λ^{2}𝒢, L) vanishes. We now redo the construction of the complex and get commutative diagrams with exact rows and columns

[figure: ℛ = = = ℛ; 0; L; ℱ; ∩; 𝒦′; ∩; 0; □; 0; L; ‖; ℰ; 𝒦; 0; ℛ = = = ℛ; 0; 𝒦′; ∩; 𝒢; ∩; ℐ_{Z}; 0; 0; 𝒦; ℳ; ℐ_{Z}; ‖; 0]

This allows us to construct a new complex

ℬ_{▪} : 0 → L → ℱ ⟶^{ψ} 𝒢 → 𝒪_{X} → 0.

One sees easily that ℛ and therefore ℱ are also vector bundles. But this time, the composite map Λ^{2}𝒢 → 𝒦′ → 𝒦, lifts to ℰ since the obstruction is the class in Ext_{𝒪_{X}}^{1} (Λ^{2}𝒢, L) which we got to vanish using Lemma 6.2(b). Since the square marked with the □ is cartesian, we get a lifting Λ^{2}𝒢 → ℱ.

The other liftings

[figure: ⋯; 𝒢(L) ⊕ D_{2}ℱ; L ⊕ (ℱ ⊗ 𝒢); ℱ ⊕ Λ^{2}𝒢; 𝒢; 𝒪_{X}; ⋯; 0; . . . . . . . . .; L; . . . . . . . . .; ℱ; . . . . . . . . .; 𝒢; ‖; 𝒪_{X}; ‖] ; (28)

now occur automatically. We therefore get a chain map D_{2}ℬ_{▪} → ℬ_{▪} which makes ℬ_{▪} into a commutative associative differential graded algebra with divided powers.

We now claim that having this differential graded algebra structure gives us all the properties we want and puts us into the situation of Theorem 4.1. Indeed, as in Buchsbaum-Eisenbud [5], the multiplication gives pairings ℬ_{i} ⊗ ℬ_{3−i} → ℬ_{3} = L, and therefore maps ℬ_{i} → ℬ_{3−i}^{∗}(L). These maps are compatible with the differential, and as a result, the following diagram commutes:

[figure: 0; L; ℱ; ψ; 𝒢; 𝒪_{X}; 𝒪_{Z}; 0; 0; L; ‖; 𝒢^{∗}(L); ϕ; −ψ^{∗}; ℱ^{∗}(L); ϕ^{∗}; 𝒪_{X}; ‖; ℰxt_{𝒪_{X}}^{3} (𝒪_{Z}, L); ≃ η; 0]

The top row is exact by construction, and the bottom row is exact because it is the dual of the top row which is a resolution of a sheaf of grade 3. Since η is an isomorphism, one sees that

0 → ℱ \underrightarrow{( \begin{matrix}ψ \\ ϕ \end{matrix} )} 𝒢 ⊕ 𝒢^{∗}(L) ⟶^{( ϕ^{∗} ψ^{∗})} ℱ^{∗}(L) → 0

is exact. Thus ℱ embeds in 𝒢⊕𝒢^{∗}(L) as a subbundle which is totally isotropic for the hyperbolic symmetric bilinear form on 𝒢⊕𝒢^{∗}(L). The subbundle ℱ is even totally isotropic for the hyperbolic quadratic form, since the restriction of this form to local sections of ℱ is the function e ↦ ⟨ϕ(e), ψ(e)⟩, and this function vanishes because the composite map from diagram (28)

[figure: D_{2}ℱ; ℱ ⊗ 𝒢; L; f ⊗ f; f ⊗ ψ(f); ⟨ϕ(f), ψ(f)⟩]

factors through 0 and hence vanishes identically. Thus ℱ is a Lagrangian subbundle of 𝒢 ⊕ 𝒢^{∗}(L). We have thus constructed all the structure of Theorem 4.1 and completed the proof of the theorem.

7. Points in ℙ^{3}

In this and the following section we discuss several classes of examples which satisfy some or all of the conditions A-D in the introduction and thus where Theorem 6.1 may apply. More geometric applications and examples can be found in our paper [11].

Okonek [26], p. 429, has shown that any reduced set of points in ℙ^{3} is Pfaffian. By carefully analyzing the conditions of Theorem 6.1, we will describe Pfaffian resolutions of locally Gorenstein zero-dimensional subschemes in ℙ^{3} (see Remark 7.4).

For a locally Gorenstein zero-dimensional subscheme Z ⊂ ℙ_{k}^{3} over a field k, there are many isomorphisms η : 𝒪_{Z} ⟶^{∼} ω_{Z}(t). Which triples (Z, ω_{ℙ^{3}}(t), η) satisfy all the conditions of Theorem 6.1, and which do not? In particular (and this is the only condition which causes trouble), when does the image of η in H^{3}(ℙ^{3}, ω_{ℙ^{3}}(t)) vanish?

We will use the following notation. Let I ⊂ R := k[x_{0}, x_{1}, x_{2}, x_{3}] be the homogeneous ideal of Z, let A := R/I be its homogeneous coordinate ring, and let ω_{A} := Ext_{R}^{3}(A, R(−4)) be its canonical module. Note that η ∈ H_{∗}^{0}(ω_{Z}) ⊃ ω_{A}. Also if M is a graded R-module, then let M′ be its dual as a graded k-vector space, endowed with the natural dual R-module structure.

Proposition 7.1. Let Z ⊂ ℙ^{3} be a locally Gorenstein subscheme of dimension zero, and η : 𝒪_{Z} ⟶^{∼} ω_{Z}(t) an isomorphism. Then the triple (Z, ω_{ℙ^{3}}(t), η) is subcanonical and satisfies conditions A-C, and it satisfies condition D if and only if η ∈ ω_{A}.

Proof. The map η : 𝒪_{Z} → ω_{Z}(t) may be identified with an element of Ext_{𝒪_{ℙ^{3}}}^{3} (𝒪_{Z}, ω_{ℙ^{3}}(t)) ≅ H^{0}(𝒪_{Z}(−t))′. Because of the long exact sequence (23) the subscheme Z ⊂ ℙ^{3} satisfies condition D for η if and only if η is in the image of Ext_{𝒪_{ℙ^{3}}}^{2} (ℐ_{Z}, ω_{ℙ^{3}}(t)) ≅ H^{1}(ℐ_{Z}(−t))′.

Local duality and Serre duality give identifications

ω_{A} := Ext_{R}^{3}(A, R(−4)) ≅ H_{𝔪}^{1}(A)′ ≅ H_{∗}^{1}(ℐ_{Z})′

and H_{∗}^{0}(ω_{Z}) ≅ H_{∗}^{0}(𝒪_{Z})′ which are compatible with the inclusions. So η satisfies condition D in the introduction if and only if η ∈ ω_{A}.

Theorem 7.2. Let Z ⊂ ℙ^{3} be a locally Gorenstein subscheme of dimension 0, and let η ∈ H^{0}(ω_{Z}(t)). Suppose that (a) η generates the sheaf ω_{Z}, (b) η ∈ ω_{A}, and (c) if t = −2ℓ is even, then the following nondegenerate symmetric bilinear form on H^{0}(𝒪_{Z}(ℓ)) is metabolic (i.e. contains a Lagrangian subspace):

H^{0}(𝒪_{Z}(ℓ)) × H^{0}(𝒪_{Z}(ℓ)) → H^{0}(𝒪_{Z}(2ℓ)) ⟶^{η} H^{0}(ω_{Z}) ⟶^{tr} k. (29)

Then there exists a locally free resolution

0 → 𝒪_{ℙ^{3}}(t − 4) → ℱ^{∗}(t − 4) ⟶^{ψ} ℱ → 𝒪_{ℙ^{3}} → 𝒪_{Z} → 0 (30)

with ψ alternating and ℐ_{Z} generated by the submaximal Pfaffians of ψ and such that the Yoneda extension class of (30) is η ∈ Ext_{𝒪_{ℙ^{3}}}^{3} (𝒪_{Z}, 𝒪_{ℙ^{3}}(t−4)) ≅ H^{0}(ω_{Z}(t)).

Conversely, if there exists a locally free resolution of 𝒪_{Z} as in (30) with ψ alternating, then its Yoneda extension class η satisfies conditions (a), (b) and (c) above.

In order for the symmetric bilinear form (29) to be metabolic, it is necessary for deg(Z) to be even. If the base field k is closed under square roots, this is also sufficient.

In any case, the conditions of the theorem always hold if t is large and odd and η is general. This proves the following result, which was proven for reduced sets of points by Okonek [26], p. 429.

Corollary 7.3. A zero-dimensional subscheme of ℙ^{3} is Pfaffian if and only if it is locally Gorenstein.

Proof of Theorem 7.2. We show how to start the proof off. But we will stop when we reach the point where it becomes identical to the proof of the main result of [32].

Suppose that Z, t, η satisfy conditions (a), (b), and (c) of the theorem. Condition (a) implies that the map η : 𝒪_{Z} → ω_{Z}(t) is an isomorphism. So η and Serre duality induce a symmetric perfect pairing

H_{∗}^{0}(𝒪_{Z}) × H_{∗}^{0}(𝒪_{Z}) ⟶^{mult} H_{∗}^{0}(𝒪_{Z}) ⟶^{η} H_{∗}^{0}(ω_{Z}(t)) ⟶^{tr} k(t) (31)

which pairs H^{0}(𝒪_{Z}(n)) with H^{0}(𝒪_{Z}(−n − t)) for all n.

Condition (c) implies that H_{∗}^{0}(𝒪_{Z}) contains a Lagrangian submodule M for this symmetric perfect pairing. Indeed if t is odd, one can pick M := ⨁_{n>−t/2} H^{0}(𝒪_{Z}(n)). If t is even, then there exists a Lagrangian subspace W ⊂ H^{0}(𝒪_{Z}(−t/2)), and one can pick M := W ⊕ ⨁_{n>−t/2} H^{0}(𝒪_{Z}(n)).

The two submodules A ⊂ H_{∗}^{0}(𝒪_{Z}) and ω_{A} ⊂ H_{∗}^{0}(ω_{Z}) are orthogonal complements of each other under the Serre duality pairing; see for example [10]. Hence condition (b), that η ∈ ω_{A}, implies that ηA ⊂ ω_{A} and therefore that A = ω_{A}^{⊥} ⊂ (ηA)^{⊥}. Now the orthogonal complement of ηA ⊂ H_{∗}^{0}(ω_{Z}) under the Serre duality pairing corresponds to the orthogonal complement of A ⊂ H_{∗}^{0}(𝒪_{Z}) under our pairing (31). So condition (b) implies that A ⊂ A^{⊥}. In other words A ⊂ H_{∗}^{0}(𝒪_{Z}) is sub-Lagrangian.

It now follows that there exists a Lagrangian submodule L such that 0 ⊂ A ⊂ L = L^{⊥} ⊂ A^{⊥} ⊂ H_{∗}^{0}(𝒪_{Z}). For instance, pick L := A + (M ∩ A^{⊥}) (cf. Knus [22] Lemma I.6.1.2).

One easily checks that A_{n} = (A^{⊥})_{n} = H^{0}(𝒪_{Z}(n)) for n ≫ 0, and that A_{n} = (A^{⊥})_{n} = 0 for n ≪ 0. Consequently A^{⊥}/A is of finite length. It has an induced nondegenerate symmetric bilinear form, and it has a Lagrangian submodule L/A.

We now claim that we can construct a locally free resolution

0 → 𝒪_{ℙ^{3}}(t − 4) ⟶^{α} ℱ^{∗}(t − 4) ⟶^{ψ} ℱ ⟶^{β} 𝒪_{ℙ^{3}} → 𝒪_{Z} → 0

with ψ alternating and such that H_{∗}^{1}(ℱ) ≅ L/A, and H_{∗}^{2}(ℱ) = 0. Moreover, β induces a surjection H_{∗}^{0}(ℱ) ↠ H_{∗}^{0}(ℐ_{Z}). Different pieces of the resolution contribute different pieces of the cohomology module H_{∗}^{0}(𝒪_{Z}). The submodule A is contributed by coker H_{∗}^{0}(β); the piece L/A by H_{∗}^{1}(ℱ); the piece A^{⊥}/L by H_{∗}^{2}(ℱ^{∗}(t − 4)); and the piece H_{∗}^{0}(𝒪_{Z})/A^{⊥} is contributed by ker H_{∗}^{3}(α).

The construction of this resolution and the verification of its properties can be done using the Horrocks correspondence by the same method as in Walter [32]. It is quite long and we omit the details.

Remark 7.4. The graded module A^{⊥}/A above can be thought of as the “intermediate cohomology” or deficiency module of (Z, t, η). To emphasize the dependence of this module on η, one could write it as (ηA)^{⊥}/A, where (ηA)^{⊥} ⊂ H_{∗}^{0}(𝒪_{Z}) means the orthogonal complement of ηA ⊂ H_{∗}^{0}(ω_{Z}) with respect to the Serre duality pairing. Now (ηA)^{⊥}/A is dual to (ω_{A}/ηA), and it is also self-dual with a shift. Consequently if η ∈ ω_{A} is of degree t, then the corresponding deficiency module is

(ηA)^{⊥}/A ≅ (ω_{A}/ηA)′ ≅ (ω_{A}/ηA)(t).

In Theorem 7.2 we split the deficiency module in half, and put a Lagrangian subhalf in ℱ and the quotient half in ℱ^{∗}(t − 4). The Pfaffian resolutions of 𝒪_{Z} are thus classified up to symmetric homotopy equivalence by pairs (η, L/A) with η ∈ ω_{A} generating the sheaf ω_{Z}, and with L/A ⊂ (ηA)^{⊥}/A a Lagrangian submodule.

An alternative strategy for dealing with this deficiency module is to construct a diagram of the form of (18) in Theorem 4.1 (we write 𝒪 := 𝒪_{ℙ^{3}} to try to stay inside the margins):

[figure: 0; 𝒪(t − 4); 𝒢; ψ; ⨁ 𝒪(−a_{i}); 𝒪; 𝒪_{Z}; 0; 𝒪(t − 4); ‖; ⨁ 𝒪(a_{i} + t − 4); ϕ; −ψ^{∗}; 𝒢^{∗}(t − 4); ϕ^{∗}; 𝒪; ‖; ω_{Z}(t); ≅ η] ; (32)

with ⨁ 𝒪(−a_{i}) corresponding to a minimal set of generators of the homogeneous ideal of Z, with H_{∗}^{2}(𝒢) ≅ (ηA)^{⊥}/A, the deficiency module, and with H_{∗}^{1}(𝒢) = 0.

We now give examples both of Pfaffian resolutions as in (30) which split the deficiency module, and of resolutions as in (32) in the form of Theorem 4.1 which gather the deficiency module up in one piece.

Example 1: one point. Consider a single rational point Q. Its geometry is very simple, but we can make its algebra surprisingly complex.

The canonical module of Q is ω_{A} ≅ ⨁_{n≥1} H^{0}(ω_{Q}(n)). If we pick a nonzero η of degree 1, then it generates ω_{A}, and its deficiency module vanishes. The constructions described above both lead unsurprisingly to the Koszul resolution

0 → 𝒪_{ℙ^{3}}(−3) → 𝒪_{ℙ^{3}}(−2)^{⊕3} → 𝒪_{ℙ^{3}}(−1)^{⊕3} → 𝒪_{ℙ^{3}} → 𝒪_{Q} → 0.

However, if we let η ∈ ω_{A} be a nonzero element of degree 2, then the deficiency module is k concentrated in degree −1, and the construction (32) yields a diagram

[figure: 0; 𝒪_{ℙ^{3}}(−2); Ω_{ℙ^{3}}^{2}(1); 𝒪_{ℙ^{3}}(−1)^{⊕3}; 𝒪_{ℙ^{3}}; 𝒪_{Q}; 0; 𝒪_{ℙ^{3}}(−2); ‖; 𝒪_{ℙ^{3}}(−1)^{⊕3}; Ω_{ℙ^{3}}(1); 𝒪_{ℙ^{3}}; ‖; ω_{Q}(2).; ≅ η]

More generally, if we let η ∈ ω_{A} be a nonzero element of degree t, then the deficiency module is ⨁_{n=−(t−1)}^{−1} H^{0}(𝒪_{Q}(n)), and the construction (32) yields

[figure: 0; 𝒪_{ℙ^{3}}(t − 4); ℱ_{t}^{∗}(t − 4); 𝒪_{ℙ^{3}}(−1)^{⊕3}; 𝒪_{ℙ^{3}}; 𝒪_{Q}; 0; 𝒪_{ℙ^{3}}(t − 4); ‖; 𝒪_{ℙ^{3}}(t − 3)^{⊕3}; ℱ_{t}; 𝒪_{ℙ^{3}}; ‖; 𝒪_{Q}; ‖]

with ℱ_{t} a rank 3 locally free sheaf which is the sheafification of the kernel of the presentation of the deficiency module:

0 → ℱ_{t} → 𝒪_{ℙ^{3}} ⊕ 𝒪_{ℙ^{3}}(t − 2)^{⊕3} → 𝒪_{ℙ^{3}}(t − 1) → 0.

If one lets η ∈ ω_{A} be a nonzero element of degree 3, then applying the methods of Theorem 7.2 yields a resolution which one recognizes as the Koszul complex associated to the zero locus of a section of the rank 3 bundle 𝒯_{ℙ^{3}}(−1):

0 → 𝒪_{ℙ^{3}}(−1) → Ω_{ℙ^{3}}^{2}(2) → Ω_{ℙ^{3}}(1) → 𝒪_{ℙ^{3}} → 𝒪_{Q} → 0.

Example 2: three points. If Z is the union of three noncollinear rational points, then the module ω_{A} has two generators of degree 0, and Z is not arithmetically Gorenstein. If we pick a general η ∈ ω_{A} of degree 0, then the deficiency module is k, concentrated in degree 0, and the construction (32) yields a diagram (in which we again write 𝒪 := 𝒪_{ℙ^{3}} in order to simplify the notation):

[figure: 0; 𝒪(−4); 𝒪(−3) ⊕ Ω_{ℙ^{3}}^{2}; 𝒪(−2)^{⊕3} ⊕ 𝒪(−1); 𝒪; 𝒪_{Z}; 0; 𝒪(−4); ‖; 𝒪(−3) ⊕ 𝒪(−2)^{⊕3}; Ω_{ℙ^{3}} ⊕ 𝒪(−1); 𝒪; ‖; 𝒪_{Z}; ‖]

If we pick a general η ∈ ω_{A} of degree 1, then the deficiency module (ω_{A}/ηA)(1) is of length 4, concentrated in degrees 0 and −1, and the methods of Theorem 7.2 yield a symmetric resolution (with alternating middle map ψ):

0 → 𝒪(−3) → Ω_{ℙ^{3}}^{2}(1)^{⊕2} ⊕ 𝒪(−2) ⟶^{ψ} Ω_{ℙ^{3}}^{⊕2} ⊕ 𝒪(−1) → 𝒪 → 𝒪_{Z} → 0.

8. Some weakly subcanonical subschemes

In this section we give some examples of weakly subcanonical subschemes. These are examples of subschemes Z ⊂ X which satisfy conditions A-B of the introduction but fail one or both of conditions C-D. Thus the Serre construction (in codimension 2) and our Theorem 6.1 (in codimension 3) fail for these subschemes.

A weakly subcanonical curve. We construct a subcanonical curve C ⊂ ℙ^{1} × ℙ^{n} for n ≥ 2 which fails the lifting condition D of the introduction.

Let C be a nonsingular projective curve of genus 2 over an algebraically closed field k, let P be one of its Weierstrass points, and let D be a divisor of degree 4 on C. A base-point-free pencil in the linear system of divisors |D| defines a map f : C → ℙ^{1}, and a base-point-free net in |D + P | defines a map g : C → ℙ^{2}. Composing g with a linear embedding ℙ^{2} ↪ ℙ^{n} gives a map h : C → ℙ^{n}. Let i := (f, h) : C → ℙ^{1} × ℙ^{n}. If the linear systems are chosen sufficiently generally, then i is an embedding.

The restriction to C of a line bundle 𝒪_{ℙ^{1}×ℙ^{n}}(a, b) is 𝒪_{C}((a + b)D + bP). So the canonical bundle ω_{C} ≅ 𝒪_{C}(2P) is the restriction of 𝒪_{ℙ^{1}×ℙ^{n}}(−2, 2). If the class of D − 4P in Pic^{0}(C) is not torsion, then 𝒪_{ℙ^{1}×ℙ^{n}}(−2, 2) is the only line bundle on ℙ^{1} × ℙ^{n} whose restriction is ω_{C}. Hence the subcanonical curve C ⊂ ℙ^{1} × ℙ^{n} will definitely fail such structural theorems as the Serre construction or Theorem 6.1 if the lifting condition D of the introduction fails for the isomorphism η : ω_{C} ≅ 𝒪_{ℙ^{1}×ℙ^{n}}(−2, 2)|_{C}.

By (3) this failure is equivalent to the nonvanishing of the composite map

H^{1}(ℙ^{1} × ℙ^{n}, 𝒪(−2, 2)) ⟶^{rest} H^{1}(C, 𝒪(−2, 2)|_{C}) ⟶_{≅}^{η} H^{1}(C, ω_{C}) ⟶_{≅}^{tr} k. (33)

Now the image of g : C → ℙ^{2} is a singular quintic plane curve. If we resolve the singularities, then g factors as an embedding followed by the blowdown C ↪ ℙ̃^{2} → ℙ^{2}. The composite map of (33) now factors through the diagram

[figure: H^{1}(ℙ^{1} × ℙ^{n}, 𝒪(−2, 2)); H^{1}(ℙ^{1} × ℙ^{2}, 𝒪(−2, 2)); ≅; H^{1}(ℙ^{1} × ℙ̃^{2}, 𝒪(−2, 2)) ≅ k^{6}; k ≅ H^{1}(C, ω_{C}); α; γ; H^{1}(ℙ^{1} × C, 𝒪(−2, 2(D + P))) ≅ k^{9}; β]

The lifting condition D fails if and only if α is surjective, hence if and only if im(β) ⊄ ker(γ).

Now γ is part of the long exact sequence of cohomology for

0 → 𝒪_{ℙ^{1}×C}(−3, D + 2P) → 𝒪_{ℙ^{1}×C}(−2, 2D + 2P) → ω_{C} → 0.

So γ is surjective, and ker(γ) ⊂ k^{9} is a hyperplane.

The map g : C → ℙ^{2} is defined using a three-dimensional subspace U_{3} ⊂ V_{4} := H^{0}(C, 𝒪_{C}(D + P)). The complete linear system embeds C ↪ ℙ^{3} as a curve of degree 5 and genus 2 contained in a unique quadric surface Q. Then β is the natural map from S^{2}U_{3} ≅ k^{6} to S^{2}V_{4}/⟨Q⟩ ≅ k^{9}. Now we have a range of choices for the subspace U_{3} ⊂ V_{4} which vary in a Zariski open subset of ℙ^{3} = ℙ(V_{4}^{∗}). Hence we have a family of possible subspaces S^{2}U_{3} ⊂ S^{2}V_{4} whose different members are not all contained in any fixed hyperplane of S^{2}V_{4}. So if we choose a general U_{3} ⊂ V_{4}, then S^{2}U_{3} = im(β) is not contained in the hyperplane ker(γ) ⊂ S^{2}V_{4}/⟨Q⟩. In that case, C ⊂ ℙ^{1} × ℙ^{n} is a subcanonical curve which fails the lifting condition D.

Singular points. Examples can easily be given of subcanonical subschemes Z ⊂ X which are not covered by our construction because the finite projective dimension condition C of the introduction breaks down. This may happen at the same time that the lifting condition D breaks down, or it may happen independently. If D holds but C breaks down, 𝒪_{Z} will still have resolutions fitting into diagrams such as (18) of Theorem 4.1, except that ℰ or ℱ will not be locally free.

For instance if X ⊂ ℙ^{4} is a singular hypersurface of degree d, and P ∈ X is a singular point, then P is indeed subcanonical, but condition C fails because 𝒪_{P} is of infinite local projective dimension over 𝒪_{X}. There exist isomorphisms η : 𝒪_{P} ≅ ℰxt_{𝒪_{X}}^{3} (𝒪_{P} , 𝒪_{X}(ℓ)) for all ℓ ∈ ℤ, but these satisfy condition D if and only if ℓ ≥ d − 4.

Similarly, if D is a line in ℙ^{5}, and Y ⊂ ℙ^{5} a hypersurface containing D which is singular in at least one point of D, then condition C fails for D ⊂ Y, but all the other conditions hold (since H^{3}(Y, ω_{Y} (2)) = 0). So although D ⊂ Y may be obtained as a degeneracy locus of a pair of Lagrangian subsheaves of a twisted orthogonal bundle on Y, at least one of the Lagrangian subsheaves is not locally free.

A nonseparated example. We now give an example where there is no real choice about the η (because H^{0}(ω_{Z}) = k and there are no twists), where conditions A-B hold, but where condition D fails. The real reason for the failure in this example is that we are doing something silly on a nonseparated scheme. But the interesting thing is that the cohomological obstruction D is able to detect our misbehavior.

Let X be the nonseparated scheme consisting of two copies 𝔸^{3} glued together along 𝔸^{3} −{0}. In other words, X is 𝔸^{3} with the origin doubled up. Let P′ ∈ X be one of the two origins. It is a subcanonical subscheme of X of codimension 3 of finite local projective dimension, i.e. it satisfies conditions A-B of the introduction. We claim that it does not satisfy condition D.

The problem is to compute the map

Ext_{𝒪_{X}}^{3} (𝒪_{P′}, 𝒪_{X}) → Ext_{𝒪_{X}}^{3} (𝒪_{X}, 𝒪_{X}) = H^{3}(X, 𝒪_{X}). (34)

We will use the following notation: U′, U′′ ⊂ X are the two copies of 𝔸^{3}; for α = 1, 2, 3 let U_{α} ⊂ X be the open locus where x_{α} ≠ 0; and let U_{αβ} := U_{α} ∩ U_{β}, and U_{123} := U_{1} ∩ U_{2} ∩ U_{3}. For any inclusion of an affine open subscheme U ⊂ X, we denote by i_{!}𝒪_{U} the extension by zero of 𝒪_{U} to all of X. We will use the same letter i_{!} whatever the U.

Then 𝒪_{X} and 𝒪_{P′} have resolutions of the form

[figure: 0; i_{!}𝒪_{U_{123}}; ⨁_{α<β} i_{!}𝒪_{U_{αβ}}; ⨁_{α} i_{!}𝒪_{U_{α}}; i_{!}𝒪_{U′} ⊕ i_{!}𝒪_{U′′}; 𝒪_{X}; 0; i_{!}𝒪_{U′}; i_{!}𝒪_{U′}^{⊕3}; i_{!}𝒪_{U′}^{⊕3}; i_{!}𝒪_{U′}; 𝒪_{P′}]

The horizontal maps in the first row are more or less taken from a Čech resolution, while those from the second row are from a Koszul resolution. The vertical maps are, from left to right,

(\frac{1}{x_{1}x_{2}x_{3}} ) , (\begin{matrix}\frac{1}{x_{1}x_{2}} & 0 & 0 \\ 0 & \frac{1}{x_{1}x_{3}} & 0 \\ 0 & 0 & \frac{1}{x_{2}x_{3}} \end{matrix} ) , (\begin{matrix}\frac{1}{x_{1}} & 0 & 0 \\ 0 & \frac{1}{x_{2}} & 0 \\ 0 & 0 & \frac{1}{x_{3}} \end{matrix} ) , (1 0) .

If we apply Hom_{𝒪_{X}} (−, 𝒪_{X}) to the resolutions, we get complexes which compute the Ext_{𝒪_{X}}^{p} (𝒪_{P′}, 𝒪_{X}) and the H^{p}(X, 𝒪_{X}). (This is because 𝒪_{X} is quasicoherent, and the i_{!}𝒪_{U} are extensions by zero of locally free sheaves on affine open subschemes.) Writing R := k[x_{1}, x_{2}, x_{3}], these complexes are

[figure: 0; R; R^{⊕3}; R^{⊕3}; R; 0; 0; R ⊕ R; ⨁_{α} R[x_{α}^{−1}]; ⨁_{α<β} R[x_{α}^{−1}x_{β}^{−1}]; R[x_{1}^{−1}x_{2}^{−1}x_{3}^{−1}]; 0]

with Koszul and Čech horizontal arrows. The vertical arrows are as before, but transposed and in the reverse order.

The map (34) which we wish to compute may now be identified as k ↪ H_{𝔪}^{3}(R). This map sends a nonzero η to a nonzero multiple of socle element x_{1}^{−1}x_{2}^{−1}x_{3}^{−1} of H_{𝔪}^{3}(R). So P′ ⊂ X fails condition 3) of the introduction.

However, even if condition D of the introduction had held, we could not have gone any farther. For reasons of depth, any map from a locally free sheaf ℰ on X to 𝒪_{X} is determined by what happens outside the two origins. So the images of a map ℰ → 𝒪_{X} at the two origins must be identical. As a result, there can be no surjection from a vector bundle ℰ on X onto ℐ_{P′}.

9. Codimension one sheaves

In this section we consider the analogues of the results in the previous sections for (skew)-symmetric sheaves of codimension 1. We include necessary and sufficient conditions for such sheaves on ℙ^{N} to have locally free resolutions which are genuinely (skew)-symmetric, similar to those in Walter [32]. We also prove that any such sheaf on a quasi-projective variety has a resolution which is (skew)-symmetric up to quasi-isomorphism, in analogy with Theorem 6.1. We finish the section with several examples.

In this section we will suppose that the characteristic is not 2, although all the theorems have variants which are valid in characteristic 2.

Symmetric sheaves of codimension 1. Suppose ℱ is a coherent sheaf on a scheme X which is of finite local projective dimension and perfect of codimension 1. This means that locally ℱ has free resolutions 0 → ℒ_{1} → ℒ_{0} → ℱ → 0 such that the dual complex 0 → ℒ_{0}^{∗} → ℒ_{1}^{∗} → ℰxt_{𝒪_{X}}^{1} (ℱ, 𝒪_{X}) → 0 is also exact. The operation

ℱ ⇝ ℱ^{∨} := ℰxt_{𝒪_{X}}^{1} (ℱ, 𝒪_{X})

provides a duality on the category of such sheaves. A symmetric sheaf of codimension 1 is a pair (ℱ, α) where ℱ is a sheaf as above, and α : ℱ → ℱ^{∨}(L) is an isomorphism which is symmetric in the sense that α = α^{∨}. (Here L is some line bundle on X.) Skew-symmetric sheaves of codimension 1 on X are defined similarly.

Symmetric resolutions in codimension 1. Resolutions of codimension 1 symmetric sheaves on ℙ^{3} have been studied fairly extensively in by Barth [3], Catanese [7] [8], and Casnati-Catanese [6] in the context of surfaces with even sets of nodes and by Kleiman-Ulrich [21] in the context of self-linked curves. The next theorem, conjectured by Barth and Catanese, was proven by Casnati-Catanese for symmetric sheaves on ℙ^{3} ([6] Theorem 0.3). They also remarked ([6] Remark 2.2) that essentially the same proof works for codimension 1 symmetric sheaves on any ℙ^{n}, which is true as long as one remembers to include in one’s statement a parity condition analogous to that in Walter [32] Theorem 0.1. For a case where the parity condition fails, see Example 9.4 below.

Theorem 9.1 ([6] with correction). Let k be an algebraically closed field of characteristic different from 2. Suppose that (ℱ, α) is a symmetric sheaf of codimension 1 on ℙ_{k}^{n}, with α : ℱ ⟶^{∼} ℱ^{∨}(ℓ − n − 1). Then ℱ has a symmetric resolution, i.e. a locally free resolution of the form

0 → 𝒢 ⟶^{f} 𝒢^{∗}(ℓ − n − 1) → ℱ → 0

with f symmetric, if and only if the following parity condition holds: if n ≡ 1 (mod 4) and ℓ is even, then χ(ℱ(−ℓ/2)) is also even.

A higher-codimension generalization of this theorem is proven in our paper [12].

As the parity condition indicates, symmetric sheaves do not always possess symmetric resolutions. The following structure theorem, analogous to Theorem 6.1, shows that they do still have locally free resolutions which are symmetric up to quasi-isomorphism.

Theorem 9.2. Let A be a noetherian ring, and X ⊂ ℙ_{A}^{N} a locally closed subscheme. Suppose that (ℱ, α) is a symmetric sheaf of codimension 1 on X, with α : ℱ ⟶^{∼} ℱ^{∨}(L) for some line bundle L. Then ℱ has symmetrically quasi-isomorphic locally free resolutions

[figure: 0; 𝒢; ψ; ℋ; ℱ; 0; 0; ℋ^{∗}(L); ϕ; ψ^{∗}; 𝒢^{∗}(L); ϕ^{∗}; ℱ^{∨}(L); ≅ α; 0] ; (35)

with ϕ^{∗}ψ : 𝒢 → 𝒢^{∗}(L) a symmetric map.

Proof. Because ℱ is locally Cohen-Macaulay of codimension 1, it has a locally free resolution 0 → 𝒫_{1} → 𝒫_{0} → ℱ → 0. The symmetric isomorphism α corresponds to a morphism in the derived category

[figure: S^{2}(𝒫_{▪}) :; 0; Λ^{2}𝒫_{1}; 𝒫_{1} ⊗ 𝒫_{0}; S^{2}𝒫_{0}; 0; L[1] :; α; . . . . . . . .; 0; 0; . . . . . . . . .; L; . . . . . . . . .; 0; . . . . . . . . .; 0]

This α is a member of the hyperext 𝔼xt_{𝒪_{X}}^{1} (S^{2}(𝒫_{▪}), L), which in turn is the abutment of the hyperext spectral sequence

E_{1}^{pq} = Ext_{𝒪_{X}}^{q} ((S^{2}(𝒫_{▪}))_{p}, L) ⟹ 𝔼xt_{𝒪_{X}}^{p+q}(S^{2}(𝒫_{▪}), L).

The differentials d_{1} define complexes (indexed by p = 0, 1, 2)

0 → Ext_{𝒪_{X}}^{q} (S^{2}𝒫_{0}, L) ⟶^{d_{1}} Ext_{𝒪_{X}}^{q} (𝒫_{1} ⊗ 𝒫_{0}, L) ⟶^{d_{1}} Ext_{𝒪_{X}}^{q} (Λ^{2}𝒫_{1}, L) → 0

whose cohomology groups are the E_{2}^{pq}. In particular, E_{2}^{10} is the space of homotopy classes of chain maps S^{2}(𝒫_{▪}) → L[1]. Hence α will be the class of an honest chain map if and only if it comes from E_{2}^{10}. However, according to the 5-term exact sequence

0 → E_{2}^{10} → 𝔼xt_{𝒪_{X}}^{1} (S^{2}(𝒫_{▪}), L) → E_{2}^{01} → ⋯,

the obstruction lies in E_{2}^{01} ⊂ Ext_{𝒪_{X}}^{1} (S^{2}𝒫_{0}, L). As in the proof of Theorem 6.1, this obstruction may be nonzero, but it can be killed by pulling back along a suitable epimorphism ℋ ↠ 𝒫_{0} (cf. Lemma 6.2). The proof may now be completed with arguments taken from the proof of Theorem 6.1.

The diagram (35) has a geometric interpretation in terms of a pair of Lagrangian subbundles of a twisted symplectic bundle, in analogy with the construction of Theorem 4.1.

Theorem 9.3. Let 𝒢 be a vector bundle and L a line bundle on a locally noetherian scheme X. Let 𝒢 ⊕ 𝒢^{∗}(L) be the hyperbolic twisted symplectic bundle and that

ℰ ^{⊂}^{( \begin{matrix}ψ \\ ϕ \end{matrix} )} 𝒢 ⊕ 𝒢^{∗}(L) (36)

is a Lagrangian subbundle. If the determinant of ψ is nowhere a zero-divisor on X, then ψ is injective, and ℱ := coker(ψ) is a codimension 1 symmetric sheaf with locally free resolutions fitting into a commutative diagram with exact rows

[figure: 0; ℰ; ψ; 𝒢; ℱ; 0; 0; 𝒢^{∗}(L); ϕ; ψ^{∗}; ℰ^{∗}(L); ϕ^{∗}; ℱ^{∨}(L); ≅ α; 0] ; (37)

We now use this theorem to construct an example of a symmetric codimension 1 sheaf on ℙ^{5} for which the parity condition of Theorem 9.1 fails. The construction is similar to the main examples of [11].

Example 9.4. Let V = H^{0}(𝒪_{ℙ^{5}}(1))^{∗}. The exterior product defines a symplectic form on the 20-dimensional vector space Λ^{3}V, which makes the trivial bundle Λ^{3}V ⊗ 𝒪_{ℙ^{5}} into a symplectic bundle. If W, W^{∗} ⊂ Λ^{3}V are general Lagrangian subspaces, then we may identify the symplectic vector space Λ^{3}V with hyperbolic symplectic vector space W ⊕ W^{∗}. Moreover, one may see that Ω_{ℙ^{5}}^{3}(3) is a Lagrangian subbundle of Λ^{3}V ⊗ 𝒪_{ℙ^{5}} (cf. [11], §5). The construction of Theorem 9.3 then produces a symmetric codimension 1 sheaf ℱ on ℙ^{5} with resolutions

[figure: 0; Ω_{ℙ^{5}}^{3}(3); ψ; W ⊗ 𝒪_{ℙ^{5}}; ℱ; 0; 0; W^{∗} ⊗ 𝒪_{ℙ^{5}}; ϕ; ψ^{∗}; Ω_{ℙ^{5}}^{2}(3); ϕ^{∗}; ℱ^{∨}; ≅ α; 0.] ; (38)

The sheaf ℱ fails the parity condition of Theorem 9.1 because ℓ = 6 and χ(ℱ(−3)) = 1.

The geometry of the sheaf ℱ is best explained using the degeneracy loci of the Lagrangian subbundles Ω_{ℙ^{5}}^{3}(3) and W^{∗} ⊗ 𝒪_{ℙ^{5}} of Λ^{3}V ⊗ 𝒪_{ℙ^{5}}:

D_{i} := {x ∈ ℙ^{5} | dim [Ω_{ℙ^{5}}^{3}(3)(x) ∩ W^{∗}] ≥ i}.

The sheaf ℱ is supported on the sextic fourfold D_{1}. If W^{∗} is general, then D_{1} is smooth (cf. [11], Theorem 2.1 and the discussion following it) except along the surface D_{2} where it has A_{1} singularities with local equations x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 0. The surface D_{2} is of degree 40 according to the formulas of Fulton-Pragacz [15] (6.7).

Now choose a general 9-dimensional subspace U of the 10-dimensional space W^{∗}. Then the composite map U ⊗ 𝒪_{ℙ^{5}} ↪ Λ^{3}V ⊗ 𝒪_{ℙ^{5}} ↠ Ω_{ℙ^{5}}^{2}(3) degenerates in codimension 2 along a threefold Y of degree 18. Since

Y = {x ∈ ℙ^{5} | dim [Ω_{ℙ^{5}}^{3}(3)(x) ∩ U] ≥ 1},

we have D_{2} ⊂ Y ⊂ D_{1}. Moreover, ℱ ≅ ℐ_{Y/D_{1}}(6). In addition, Y is selflinked by the complete intersection of D_{1} and of another sextic hypersurface corresponding to another Lagrangian subspace of Λ^{3}V containing U.

Skew-symmetric sheaves of codimension one. Analogues of Theorems 9.1, 9.2, and 9.3 hold for skew-symmetric sheaves of codimension 1. The only significant change is in the parity condition of Theorem 9.1, which in the skew-symmetric case has the form: “if n ≡ 3 (mod 4) and ℓ is even, then χ(ℱ(−ℓ/2)) is also even.” We leave the exact formulation of these results to the reader.

If S ⊂ ℙ^{3} is a smooth surface of degree d, then its cotangent bundle Ω_{S} is a skew-symmetric sheaf of codimension 1 on ℙ^{3} with twist ℓ = 0. Since χ(Ω_{S}) = −h^{11}(S) ≡ d (mod 2), this skew-symmetric sheaf fails the parity condition when d is odd.

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] C. Bănică, M. Putinar, On complex vector bundles on rational threefolds, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 2, 279–288.

[3] W. Barth, Counting singularities of quadratic forms on vector bundles, in: Vector bundles and differential equations (Proc. Conf., Nice, 1979), pp. 1–19, Progr. Math. 7, Birkhäuser, 1980.

[4] N. Bourbaki, Formes sesquilinéaires et formes quadratiques, Éléments de mathématiques, Livre II, Algèbre, Chap. 9, Hermann, 1959.

[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] G. Casnati, F. Catanese, Even sets of nodes are bundle symmetric, J. Differential Geom. 47 (1997), no. 2, 237–256. Corrigendum: J. Differential Geom. 50 (1999), no. 2, 415.

[7] F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465.

[8] , 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.

[9] J. A. Eagon, D. G. Northcott, On the Buchsbaum-Eisenbud theory of finite free resolutions, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, J. Reine Angew. Math. 262/263 (1973), 205–219.

[10] D. Eisenbud, S. Popescu, Gale duality and free resolutions of ideals of points, Invent. Math. 136 (1999), no. 2, 419–449.

[11] D. Eisenbud, S. Popescu, and C. Walter, Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3, preprint math.AG/******.

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

[13] W. Fulton, Schubert varieties in flag bundles for the classical groups, in: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), pp. 241–262, Israel Math. Conf. Proc. 9, Bar-Ilan Univ., 1996.

[14] , Determinantal formulas for orthogonal and symplectic degeneracy loci, J. Differential Geom. 43 (1996), no. 2, 276–290.

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

[16] P. Griffiths and J. Harris, Residues and zero-cycles on algebraic varieties, Ann. of Math. (2) 108 (1978), no. 3, 461–505.

[17] M. Grassi, Koszul modules and Gorenstein algebras, J. Algebra 180 (1996), no. 3, 918–953.

[18] J. Harris and L. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84.

[19] R. Hartshorne, Stable vector bundles of rank 2 on ℙ^{3}, Math. Ann. 238 (1978), no. 3, 229–280.

[20] T. Jozefiak, A. Lascoux, and P. Pragacz, Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Math. USSR, Izv. 18 (1982), 575–586.

[21] S. Kleiman, B. Ulrich, Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4973–5000.

[22] M-A. Knus, Quadratic and Hermitian forms over rings. With a foreword by Bertuccioni. Grundlehren der Mathematischen Wissenschaften, 294 Springer-Verlag, Berlin, 1991.

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

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

[25] D. G. Northcott, Finite free resolutions, Cambridge Tracts in Mathematics, No. 71, Cambridge University Press, Cambridge-New York-Melbourne, 1976.

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

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

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

[29] P. Pragacz, Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in algebra, Part 2 (Warsaw, 1988), 189–199, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.

[30] and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; Q̃-polynomial approach, Compositio Math. 107 (1997), no. 1, 11–87.

[31] J. Vogelaar, Constructing vector bundles from codimension-two subvarieties, Thesis Leiden, 1978.

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

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

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

E-mail address: de@msri.org

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

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

   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/