Homological Algebra and Algebraic Surfaces

Fabrizio Catanese

Contents of the paper ^{1}

0- Introduction and notation 1- Basic set up and theorems of homological algebra 2- Symmetric (quadratic) modules and sheaves, and self linkage 3- Coherent sheaves and vector bundles on projective space 4- Pfaffian subschemes and canonical surfaces in ℙ^{5} 5- Canonical projections in 3-dimensional space 6- Canonical surfaces in ℙ^{4} 7- Finite covers and homological algebra.

0. Introduction and Notation

As the title suggests, our main concern is the problem of classification of algebraic surfaces of general type; in particular, in view of the exponential computational complexity of the problem (cf. [Cat9], [Man2], [Ch4]), of those with small numerical invariants.

We try to see how recent advances in homological algebra can be applied to this purpose: needless to say, the methods we shall outline have a much wider range of application, and this is the reason why we hope that this article may turn out to be of some interest to a broader audience.

I would like though to make clear to the potential reader who might wish to be introduced to some concrete applications of homological techniques that I made a quite personal choice, and that indeed there is a huge literature concerning this by now old tradition.

I tried to render some justice by including in the final references also papers which are not directly cited in the text, but which played an important role concerning these developments. Even so, our list is far from complete and I apologize to the authors who have not been properly credited.

On the other hand, the reasons to confine our examples and applications to the case of surfaces are threefold: to keep the material presented to a reasonable size, to be allowed to dwell on topics about which we have some experience, and to be able to locate these calculations into the framework of a well developed theory, where the coarse classification has been achieved.

1Acknowledgement: The present paper was written while the author was Professore distaccato at the Accademia dei Lincei, and in the framework of the projects AGE HCM, contract ERBCHRXCT 940557, and 40 % M.U.R.S.T.

We would also like to express our indebtedness to Frank Schreyer for many useful conversations (which will hopefully lead to a further joint work clarifying many issues touched here only very briefly).

Our style will be often the same as the one of several oral expositions we gave on the subject, and as the reader can see from the table of contents, we start on quite basic grounds of homological algebra of polynomial rings, notably the Hilbert-Burch and Buchsbaum-Eisenbud structure theorems of varieties of small codimension (≤ 3). We then proceed to describe their generalizations to the global case (bundle case) and their extensions to the relative case (finite birational covers of subvarieties), and only after a while we focus on some application to surfaces. We then end by seeing that also the case of finite covers of small degree is influenced by this circle of ideas.

It seems convenient to explain soon here what is the basic underlying problem we are confronted with.

It was shown by Castelnuovo and Enriques that if an algebraic surface S′ (over an algebraically closed field ℂ) is not birationally ruled (i.e., not birational to ℙ^{1}×C) then S′ is birational to a unique smooth complete surface S which is minimal, that is, is not obtained from another smooth surface by blowing up a point. Such an S is called the minimal model, and the problem of birational classification is reduced to the problem of biregular classification of the minimal models.

At the foreground of the classification lie the numerical invariants of S, namely if the canonical divisor K = K_{S} is the divisor such that O_{S}(K_{S}) = Ω_{S}^{2} (the sheaf of regular 2-forms), we have the holomorphic Euler Poincarè characteristic χ(O_{S}) and the self intersection K^{2} = K_{S}^{2} of K_{S}.

The surfaces for which both numerical invariants are ≥ 1 are exactly the surfaces of general type and the rational ones (those are the ones birational to ℙ^{2} and, cf. e.g. [Bea1], [BPV], one can shortly list them).

For the surfaces of general type the canonical ring of S, which is the graded ring R = R(S) = ⊕H^{0}(O_{S}(mK)), is a finitely generated ring of transcendence degree equal to 3, and the field of the fractions of R which are homogeneous of degree zero coincides with the field of rational functions on S.

The dimension of H^{0}(O_{S}(mK)) is denoted by P_{m} for m ≥ 2, and called the m-th plurigenus, whereas P_{1} is called the geometric genus and also denoted by p_{g}.

For a surface of general type, when char(ℂ) ≠ 2 at least (cf.[Ek], [S-B]), we have the following Kodaira formula

P_{m} = χ + \frac{1}{2}m(m − 1)K^{2}, for m ≥ 2

and therefore the plurigenera do not play here the same fundamental role that they play in the classification of the special surfaces (those not of general type).

As we mentioned above, knowing the canonical ring of S, we recover its function field, whence its birational class, but indeed we have more, namely if we let as usual P roj of a graded ring R be the subset of SpecR given by the homogeneous prime ideals of R which are not maximal, then X = P roj(R) is called the canonical model of S and there is a birational morphism π : S ⟶ X, called the multicanonical morphism, which contracts exactly the curves C with KC = 0 (these are called (−2) curves, since they have C^{2} = −2, and they are ≅ ℙ^{1}). These curves are responsible for the singularities of X, which are double points of embedding dimension ≤ 3 such that, if K_{X} = a canonical divisor on X, we have π^{∗}(K_{X}) = K_{S} (these singularities are called Rational Double Points, cf. [BPV], and for many purposes X behaves like S, except that curves on X are only ℚ-Cartier divisors, in particular their intersection numbers may be rational numbers which are not integral, cf. [Mum]). The major advantage of using X is that the pluricanonical maps φ_{mK} : S ⟶ ℙ(H^{0}(O_{S}(mK))^{∗}) (here V^{∗} denotes the dual vector space of V, and ℙ^{n} = ℙ(V) is the projective space of 1-dimensional vector subspaces of V) factor through π : S ⟶ X and rational maps ψ_{mK} : X ⟶ ℙ(H^{0}(O_{S}(mK))^{∗}) which become finite morphisms whenever the sections of H^{0}(O_{S}(mK)) have no common zeros (cf. [Cat6], this is almost always the case for m ≥ 2, the only known exception being the case when K^{2} = χ = 1).

The study of the pluricanonical maps is an essential tool in the classification of surfaces of general type, and I refer the reader to the article by Ciliberto on these Proceedings for recent progress in this direction.

On the other hand, if the main theoretical problem of classification is summarized as: ‘Find all the surfaces S which have given invariants χ and K^{2} ’, the first basic questions is the following ([VdV])

\underline{Van de Ven’s geographical question:} For which pair of invariants x, y ≥ 1 does there exist a surface of general type S with those invariants (i.e. χ(O_{S}) = x, K_{S}^{2} = y)?

More than one century ago M. Noether (father of a daughter who gave birth to Noetherian rings) showed ([Noe]) the lower bound

\underline{Noether’s inequality:} K_{S}^{2} ≥ 2p_{g} − 4, whence K_{S}^{2} ≥ 2χ − 6,

and in the late 70’s the upper bound conjectured by Van de Ven was proved through the efforts of Bogomolov, Miyaoka and Yau (cf. [Miy], [Yau1, 2])

\underline{B-M-Y inequality:} K_{S}^{2} ≤ 9χ.

In the allowed region, thanks to results of Persson, Chen and others ([Per1], [Chen1, 2], cf. [Per2]) the geographical question has a positive answer for almost all points. But for instance Persson fills most of the region where K_{S}^{2} ≤ 8χ with surfaces admitting a fibration f : S ⟶ B whose fibres are genus 2 curves.

Now, for those (cf. Ciliberto’s article in these Proceedings) not only the canonical, but also the bicanonical map fail to be birational, whence one can ask to fill the region with surfaces such that the bicanonical map is birational. More specifically, an interesting question is

\underline{Existence question about birationally-canonical surfaces:} For which pairs (x, y) does there exist a surface S with those invariants, and moreover such that its canonical map is birational onto its image?

In this direction S has then to satisfy the obvious inequality p_{g} ≥ 4 and the important (cf. [Cas])

\underline{Castelnuovo inequality}: K_{S}^{2} ≥ 3p_{g} − 7.

In fact, if K_{S}^{2} ≤ 3p_{g} − 8 then the canonical map is a double cover of a ruled surface, thus S carries a pencil of hyperelliptic curves. Castelnuovo also showed how to fill the Castelnuovo line K_{S}^{2} = 3p_{g} − 7 (his work was later clarified by several other authors, e.g. [Bea2], [Mir2], [A-K]) and I know of no systematic work except the one of Konno ([Ko1]) on the upper parallel line K_{S}^{2} = 3p_{g} − 6 (this still relies on Castelnuovo’s bound of the genus of a curve in projective space).

From Castelnuovo’s inequality follows the other inequality K_{S}^{2} ≥ 3χ − 10, and T. Ashikaga ([Ashi]) has shown how to obtain, in the range 3x − 10 ≤ y ≤ 8x − 78, surfaces with invariants (x, y) and with birational canonical map (he obtains these as fibrations of nonhyperelliptic curves of genus 3).

Unfortunately, for low value of the invariant x (x ≤ 7 as we are interested here) the result is empty (although the method can still provide many interesting examples). Moreover, the way we are interested to the problem is not according to the geographic vein (try to fill the region over the Castelnuovo line by constructing certain specific classes of surfaces for which φ_{1} is birational and then verifying that we have thus hit all of the integer points (x, y)) but from the more general point of view of moduli.

Namely, we look at the surfaces with given invariants and given behaviour of the (pluri)canonical maps as equivalence classes of points in certain parameter spaces. These parameters spaces are locally closed algebraic sets defined by explicit equations and inequalities, and we ask when these parameter spaces are non empty.

One possible way to set up the problem is as follows:

Definition 0.1. φ : S → Y ⊂ ℙ^{n} is an almost generic canonical projection if 1. φ is given by a base point free (n+1)-dimensional subspace V^{∗} of H^{0}(O_{S}(K))

2. φ is a morphism which is birational onto its image Y.

The above implies that the canonical model X of S is just the normalization of Y. Let {y_{0}, y_{1}, y_{2}, …, y_{n}} be a basis of V^{∗} ⊂ H^{0}(O_{S}(K)), let

V^{∗} ⊕ W^{∗} ≅ H^{0}(O_{S}(K)) ≅ H^{0}(O_{X}(K))

be a fixed splitting, and denote the graded polynomial ring ℂ[y_{0}, y_{1}, y_{2}, …, y_{n}] = Sym(V^{∗}) by A.

The first method consists in viewing the canonical ring as an algebra over the polynomial ring A, whereas the second consists in studying S through the coherent sheaf on ℙ^{n} given by the direct image φ_{∗}O_{S} = ψ_{∗}O_{X}.

Clearly the two approaches are quite interrelated, as we have been taught by Serre in FAC ([Ser1]). To an A-module M one associates a sheaf M^{∼} on ℙ^{n}, and to a sheaf F on ℙ^{n}, one associates the module Γ_{∗}F = ⊕_{m∈ℤ}H^{0}(ℙ^{n}, F(m)). In our particular case the fact that φ is given by sections of the canonical bundle implies that R^{∼} = φ_{∗}O_{S} = ψ_{∗}O_{X} and Γ_{∗}(φ_{∗}O_{S}) = R.

The interest of this setting goes beyond the other existence questions (which become rather hard for K^{2} large with respect to χ):

1. For which (x, y) there is a surface S with an almost generic canonical projection (equivalently, when is the canonical system without base points and yielding a birational map)?

2. When is there a surface S which is embedded by the canonical system? (cf.

also [Cat3]). In fact one can further generalize the theory to the following more general setting which includes all surfaces of general type.

Let q_{0}, q_{1}, q_{2}, …, q_{n} be positive integers and consider the weighted projective space ℙ(q_{0}, q_{1}, q_{2}, …, q_{n}): it is the P roj of the graded ring ℂ[y_{0}, y_{1}, y_{2}, …, y_{n}] where y_{i} has degree equal to q_{i}.

A rational mapping of S to ℙ(q_{0}, q_{1}, q_{2}, …, q_{n}) is provided by (s_{0}, s_{1}, s_{2}, …, s_{n}) if s_{i} is a section of H^{0}(O_{S}(q_{i}K)), and we can analogously define the notion of a \underline{weighted almost generic canonical projection}.

From the results on pluricanonical maps (cf. [Cil4]) follows that there exists such a weighted almost generic canonical projection (cf. [Cat3]) to a 3-dimensional ℙ(q_{0}, q_{1}, q_{2}, q_{3}) with q_{0}, q_{1} ≤ 2, q_{2} ≤ 3, q_{3} ≤ 5.

It would make therefore sense to consider all the theory in this more general setting, but we will not pursue this here for the sake of clarity and brevity.

As the reader can see from the table of contents, we also dwelled on other topics not directly related to the above basic problem. This surface-story continues in the sections 4 and following, especially the second part of section 4 and sections 5, 6. Section 7 is rather independent (although deeply interelated), and one could also in principle read it immediately after section 1.

The rest of the paper is a mixture between a survey paper and a personal notebook containing some new calculations, hints and results.

The deadline being over justifies in part the ‘unfinished’ appearance of the paper. We say so not in order to dream of ourselves as sitting next to Michelangelo or Schubert, but to ask once again for the reader’s patience (with respect to possible inaccuracies, ...).

1. Basic set up and theorems of homological algebra.

Let ℂ be any algebraically closed field, and let X_{d}^{m} ⊂ ℙ^{N}(ℂ) be a m-dimensional subvariety of degree d.

To X uniquely corresponds the cone C(X) over X, C(X) ⊂ ℂ^{N+1} and the homogeneous coordinate ring of X, denoted by ℂ[X], is defined to be the ring of functions on C(X) which are restrictions of polynomial functions on our vector space ℂ^{N+1}. Whence, ℂ[X] = the graded ring ℂ[y_{0}, y_{1}, y_{2}, …, y_{N}]/I(X), I(X) being the ideal generated by the homogeneous polynomials vanishing on X.

We denote instead by ℂ(X) the function field of X, i.e., the ring of fractions of ℂ[X] which are homogeneous of degree 0. Its elements are written as f = P/Q, with P, Q in ℂ[X] homogeneous of the same degree m > 0, and are called the rational functions on X.

Using n of these functions, f_{1}, …, f_{n}, we obtain a rational map φ : X ⟶ ℙ^{n} which is said to be birational if f_{1}, …, f_{n} generate the function field of X, and a morphism if φ is defined everywhere.

A particular case to which the general case can be reduced is the one of a linear projection ℙ^{N} ⟶ ℙ^{n} sending (y_{0}, y_{1}, y_{2}, …, y_{N}) to (y_{0}, y_{1}, y_{2}, …, y_{n}), and inducing φ : X ⟶ Y ⊂ ℙ^{n}.

The algebraic counterpart is a ring homomorphism φ^{∗} : ℂ[Y] ⟶ ℂ[X] which makes the ring ℂ[X] a module over the polynomial ring A = ℂ[y_{0}, y_{1}, y_{2}, …, y_{n}]. This module is not finitely generated in general (think of X = {y_{0}y_{1} = y_{2}^{2}} projected by (y_{0}, y_{2})), but it is so in the case where y_{0}, y_{1}, y_{2}, …, y_{n} have no common zeros on X (then φ is a finite morphism).

Let us assume then that we have a finitely generated graded A-module M = ⊕_{r∈ℤ}M_{r}. To be more precise in the discussion which follows, one should define a la Serre M(i) as the same module as M but with a different grading by which M(i)_{r} = M_{r+i}, and consider only graded homomorphisms of degree 0, even if for these general consideration we can temporarily forget about the gradings. Since M is finitely generated, M is a quotient of a free module L^{0} = ⊕_{r=1}^{h}A(i_{r}) by a submodule M^{0} called the module of relations of M; M^{0} is again finitely generated by the Hilbert basis theorem. Thus M^{0} is the quotient of another free module L^{1} by a submodule M^{1}, called the module of first syzygies.

This process terminates after a finite number of steps (in the sense that there is a c ≤ n + 1 such that the module M^{c−1} of (c − 1)-syzygies turns out to be free) by

Theorem 1.1 (Hilbert syzygies theorem, [Hilb], cf. also e.g. [Gr2]). A finitely generated module M over the ring A = ℂ[y_{0}, y_{1}, …, y_{n}] has a minimal finite free resolution of length c ≤ n + 1

0 ⟶ L^{c} ⟶ ⋯ L^{1} ⟶ L^{0} ⟶ M ⟶ 0. (1)

Here, c is called the homological dimension of M, although in our opinion it should be called the homological codimension of M, and c ≥ codimension (height) of M. Thus c ≥ n − m in our geometrical case. M is said to be \underline{arithmetically} \underline{Cohen-Macaulay} if equality holds: c = codimension(M).

The Cohen-Macaulay assumption is quite miraculous, at least in the particular case where N = n and X = Y (then M is in particular a principal module), as the following examples show:

1.1. Cohen-Macaulay in codimension 1. A Cohen-Macaulay subscheme X of codimension 1 is principal, i.e., the Hilbert resolution is given by a homogeneous polynomial α

⋯ 0 ⟶ A ⟶^{α} A ⟶ M ⟶ 0. (2)

1.2. Cohen-Macaulay in codimension 2 (Theorem of Hilbert-Burch, [Hilb][Burch]). A Cohen-Macaulay subscheme X of codimension 2 is determinantal of ranks r × (r + 1), i.e., the Hilbert resolution is given through a r × (r + 1) matrix of homogeneous polynomials α^{1} and its adjoint matrix α^{0} = Λ^{r}(α^{1})

0 ⟶ A^{r} ⟶^{α^{1}} A^{r+1} ⟶^{α^{0}} A ⟶ M ⟶ 0. (3)

More precisely, we have (^{∨} denoting the A-dual)

0 ⟶ L^{2} ⟶ L^{1} ≅ Λ^{r}(L^{1})^{∨} ⟶ Λ^{r}(L^{2})^{∨} ⟶ M ⟶ 0. (4)

The case r = 1 is the case of a complete intersection, the one where the resolution is self dual in the following sense:

Definition 1.2. An A module M is said to be \underline{Gorenstein} if it admits a minimal Hilbert resolution which is isomorphic to a twist of its dual.

As pointed out to by W. Bruns and D. Eisenbud, if M is \underline{Gorenstein} it is automatically also Cohen-Macaulay.

M is said to be \underline{strongly Gorenstein} if moreover there is a minimal Hilbert resolution L^{•} such that

• if the length of the resolution L^{•} is odd = 2k + 1, L^{•} is equal to a twist of its dual up to the sign (−1)^{k} in the middle homomorphism α^{k}

• if the length of the resolution L^{•} is even = 2k, L^{•} is either equal to a twist of its dual, or the middle module L^{k} has even rank and, J being the antisym-

metric matrix ( \begin{matrix}0 & −I \\ I & 0 \end{matrix} ), the homomorphism α^{k} equals the composition

J ∘^{t}(α^{k−1}), whereas α^{i} = ^{t}(α^{2k−i−1}) for i < k − 1.

Remark 1.3 (Buchsbaum and Eisenbud,[B-E3], thm. 1.5). When M is a quotient of the polynomial ring A, M is a Gorenstein ring (cf. also e.g. [Mat2]) iff it is Gorenstein in our sense; one of the main issues of the paper will be to prove the very strong Gorentein property in many circumstances.

With the Gorenstein hypothesis, one can go to higher codimension:

1.3. Gorenstein in codimension 3 (Theorem of Buchsbaum- Eisenbud, [B-E2]). A Gorenstein subscheme X of codimension 3 is Pfaffian, i.e., there exists an antisymmetric matrix of homogeneous polynomials α^{1} of size (2r + 1) × (2r + 1), whose (2r + 1) Pfaffians (the square roots of the determinants of the principal minors obtained by deleting the i-th row and column of α^{1}) give α^{0} and α^{2}

0 ⟶ A ⟶^{α^{2}} A^{2r+1} ⟶^{α^{1}} A^{2r+1} ⟶^{α^{0}} A ⟶ M ⟶ 0. (5)

Remark 1.4. Thus a Gorenstein subscheme of codimension 3 is in particular strongly Gorenstein. Kustin and Miller ([K-M]) erxtend this statement to codimension 4. The result should hold in any codimension, but we are not yet aware of an existing proof.

These last beautiful results are, though, not sufficient to understand algebraic varieties in general since

a) they only work for X = Y, and therefore we must have a variety X of small codimension

b) even in the case where X has small codimension, the Cohen-Macaulay assumption is, as we shall see, very restrictive. Whereas, the very general Hilbert syzygy theorem has the drawback that

c) also for Cohen-Macaulay rings, how does the Hilbert resolution encode not only the module but also the ring structure of ℂ[X]?

Moreover, the structure of the Hilbert resolutions in higher codimension is amply unknown. For instance, the first question concerning the degrees of the polynomials matrices giving the Hilbert resolution is not yet answered even in the case of canonical curves. There we have the basic conjecture by Green which relates those degrees with the geometry of the algebraic curve, notably with the Clifford index of C, which is the maximum over the effective divisors D contained in a canonical divisor of Cliff(D) = deg(D) − 2dim|D| (cf. [Gr1], [G-L], and [Sch1, 2] for evidence to the conjecture).

A first approach to overcome difficulty b) is to sheafify everything. I.e., we replace the coordinate ring ℂ[X] by the sheaf O_{X} whose stalk at x is the subring O_{X,x} of the functions f in ℂ(X) which can be written as P/Q with Q not vanishing at x, we replace A locally by the sheaf O_{ℙ^{n}} and use the analogues of the above results in the local case (since O_{ℙ^{n},x} is a regular local ring).

But in order to get meaningful global result, we need to replace the free modules L^{j} by locally free sheaves (vector bundles) on ℙ^{n}. In the next section we analyze a particular issue of this procedure.

2. Symmetric (quadratic) modules and sheaves, and self linkage

Assume that M is a module of codimension 1 which is strongly Gorenstein. Therefore, M admits a Hilbert resolution where α is a symmetric matrix of homogeneous polynomials (where the entry (i, j) has degree 1/2(d_{i} + d_{j}))

0 ⟶ L^{1} ⟶ (L^{1})^{∨} ⟶ M ⟶ 0. (6)

The module M, if h is the rank of L^{1}, has a corresponding minimal set of generators v_{1}, …, v_{h}, subject to h relations

∑_{j=1}^{h} α_{ij}(y)v_{j} = 0 (fori = 1, …, h). (7)

M has codimension 1 since if y is a point where the matrix α(y) is invertible, then the localization of M at y is 0.

In particular, M is annihilated by the determinant f(y) of α, and if x is a point of the hypersurface X = f = 0 where rankα(x) = (h − 1), then the localization of M yields a locally free O_{X,x} module of rank 1.

In the Grothendieck correspondence between coherent sheaves on X and vector space fibrations, to the sheafification of M corresponds the subspace L (of the vector bundle on ℙ^{n} associated to L^{1}) defined by the equations (7).

On the other hand, set B(y) = Λ^{h−1}(α(y)): then B is also symmetrical, and for the points of X (where det(α) = 0) the rank of B is 1, and moreover, by Cramer’s rule

αB = Bα = fId.

Thus the columns of B give, for x in X, elements of kerα, and when rank(α) = h−1, B is proportional to the matrix v_{i}v_{j}.

Since the y’s are homogeneous coordinates, we can always adjust in such a way that

v_{i}v_{j} = B_{ij}(y). (8)

Forgetting the geometry behind what we have just explained, formula (8) gives a bilinear symmetric (quadratic) map

M × M ⟶ ℂ[X] = A/(f), (9)

which is well defined just by Cramer’s rule. If we keep track of the grading, as we should, we see that we can shift the grading of M in such a way that we get a homogeneous bilinear map

M × M ⟶ ℂ[X](−δ), (10)

where δ is either 0 (\underline{even case}) or 1 (\underline{half even case}).

By (1) we see that (10) establishes a perfect duality

M ≅ Hom_{ℂ[X]}(M, ℂ[X](−δ)). (11)

This is easy to explain via the couple of exact sequences (6) and (forgetting the grading)

0 ⟶ A ⟶ A ⟶ ℂ[X] ⟶ 0.

In fact, the former yields an isomorphism between M and Ext^{1}(M, A), whereas the latter, since f kills Ext^{1}(M, A), provides an isomorphism of Hom_{ℂ[X]}(M, ℂ[X]) = Hom_{A}(M, ℂ[X]) with Ext^{1}(M, A).

Remark 2.1. Observe moreover, for further use, and since we have just considered the Ext groups, that the Cohen-Macaulay property can be written in terms of Ext groups. Let M as usual be graded, and let ℂ = A/(y_{0}…. , y_{n}): then M is Cohen-Macaulay iff all the groups Ext^{i}(ℂ, M) vanish except the one with i = dim(M). A similar definition holds for F a module over a local ring O, setting ℂ = O/m (cf. [Mat1,2]). Another characterization of M being Cohen-Macaulay (cf. [Ser1], lemma 1, p. 269) is that Ext^{i}(M, A) = 0 for i > dim(M), and similarly if O = O_{ℙ^{n}}, a regular local ring, F is Cohen-Macaulay iff Ext^{i}(F, O) = 0 for i > dim(F).

Let us then sheafify:

Definition 2.2 (Quadratic sheaves, [C-C]). Let X be a hypersurface in ℙ^{n} of degree d, and let F be a coherent sheaf on X.

F is said to be a δ/2-quadratic sheaf (δ = 0 or 1) if

1) Hom_{O_{X}} (F, O_{X}(−δ)) ≅ F under a symmetric isomorphism, (i.e., such that the associated bilinear map F × F → O_{X}(−δ) is symmetric).

2) F is locally Cohen-Macaulay, i.e. Ext_{O_{ℙ^{n}}}^{i} (O_{ℙ^{n}}/𝔪_{ℙ^{n},x}, F) = 0 for all i < dim F = d − 1, and for all x ∈ X.

Note that the condition that F is a local Cohen-Macaulay O_{ℙ^{n}} module says that, at least locally, F is the cokernel of a map α : (O_{ℙ^{n}})^{r} ⟶ (O_{ℙ^{n}})^{r} such that F = coker(α) ≅ coker(^{t}α).

This local property, as we mentioned, is also equivalent to the validity of the following property 3):

3) Ext_{O_{ℙ^{n}}}^{i} (F, O_{ℙ^{n}}) = 0 for i ≥ 2.

1) is a much stronger global property, which indeed implies:

Theorem 2.3 (Casnati-Catanese, [C-C]). A δ/2-quadratic sheaf F on a hypersurface X of degree d in ℙ^{n} is bundle symmetric, i.e., there exists a vector bundle E on ℙ^{n} and an exact sequence ( ^{∨} denoting the dual)

0 ⟶ E^{∨}(−d − δ) ⟶^{α} E ⟶ F ⟶ 0

given by a symmetric map α = ^{t}α.

Moreover, F is split symmetric (this means that E is a direct sum of line bundles) if and only if F is arithmetically Cohen-Macaulay, indeed iff H^{i}(F(m)) = 0 for each m in ℤ and 0 < i ≤ (n − 1)/2.

Remark 2.4 (Historical note on quadratic sheaves). The theorem on quadratic sheaves has a long history, starting in 1906 when Dickson ([Di]) proved that a smooth plane curve X of degree d can be written as the determinant of a symmetric matrix of linear forms if and only if X contains a divisor D, not effective, with 2D ≡ K, K being the canonical divisor of X (since K equals (d − 3) times the hyperplane divisor H, to give such a D is equivalent to give a D′ with 2D′ ≡ δH, δ = 0 or 1, δ ≡ d − 1(mod 2): then we get a quadratic sheaf setting F = O_{X}(D′), and the exact sequence above determines the dimensions h^{0}(F(m))). Only much later, as far as I know, and in connection with plane spectral curves, was remarked that there is a correspondence between general divisors D on C of degree g − 1, g being the genus of C, and expressions of the equation of C as the determinant of a matrix of linear forms (up to elementary row and column operations). More precisely, Kravitsky and Livsic (in [Kra], [Li]) observed that the required condition is that both D and K − D should not be effective: in this context the involution on matrices (with linear forms as coefficients) sending α to ^{t}α corresponds to the involution sending D to K − D.

Slightly earlier, the interest in this type of questions had been revived by Barth ([Bar2]) who had extended Dixon’s result to singular curves, in the course of his research on the moduli spaces of vector bundles on the projective plane.

The curve case is special in two ways: firstly, because the global Cohen-Macaulay property is automatically satisfied, and secondly because no singularities are imposed by the symmetric determinantal expression of the equation of X. In fact, the locus ∑(2) where the rank of α drops by 2 has codimension at most 3 in the ambient space ℙ^{n} and, if α is sufficiently general, the local picture at the general poit of ∑(k) is the one of the variety of singular (k×k) symmetric matrices. That is, at the generic point of ∑(2) there are local parameters x, y, z such that X = {xy − z^{2} = 0} ⊃ ∑(2) = {x = y = z = 0}.

In the case of X ⊂ ℙ^{3}, for α general, the singular set of X is exactly ∑(2), a set of \underline{nodes} (ordinary quadratic singularities as above).

This set of nodes has a special topological property (we restrict for simplicity to the even case where δ = 0): there exists a double covering Z of X which is only ramified on ∑(2).

To see how this holds true in the case of any dimension, it suffices to take up again the pairing F × F ⟶ O_{X} which exists by the very definition of a 0-even quadratic sheaf. Using this pairing, we can give an algebra structure to the sheaf O_{X} ⊕ F and set Z = Spec(O_{X} ⊕ F). Conversely, such a finite double covering p : Z ⟶ X with Z normal (and, if n ≥ 3, Z locally Cohen-Macaulay) determines a quadratic sheaf F = p_{∗}(O_{Z})/O_{X}.

Classically (cf. e.g. the article of Gallarati, [Gal]) the emphasis was set more on the contact of surfaces: in fact a general section of a twist F(m) yields a surface G everywhere tangent to X along a smooth curve (to be more precise, the cycle intersection of X and G equals 2C, moreover, if degG > d = degX, the only singularities of G will be along C, where G will have a finite number of nodes (distinct from those of X)).

The theory of even sets of nodes has been successfully used by Beauville ([Bea3]) and, more recently, by Jaffe-Ruberman ([J-R]) in order to find, for d = 5, 6, the maximum number of nodes μ(d) that a nodal surface of degree d can have (μ(5) = 31, μ(6) = 65).

The above result ([C-C]) on quadratic sheaves allows a classification of even sets of nodes for d ≤ 6, using which J. Wahl ([Wah]) gave a simpler proof of the result by Jaffe-Ruberman stating that μ(6) = 65.

We can show an application of the theory of quadratic sheaves.

2.1. Quadratic sheaves and self-linkage. :

Definition 2.5. Let C be a reduced curve in ℙ^{3}: C is said to be self-linked if there are hypersurfaces F, G such that the cycle intersection F · G = 2C.

C, F, G being as above, if m = deg(G) ≥ d = deg(F), we can replace G by G + λF (λ a homogeneous polynomial of degree m − d), and assume that G is smooth outside C, and indeed at the generic points of C, so that G is a normal surface.

From the sheaf theoretical point of view, we let C′ be the scheme complete intersection of F and G, I be the ideal sheaf of C, I′ the ideal sheaf of C′. We have an exact sequence

0 ⟶ N ⟶ O_{C′} ⟶ O_{C} ⟶ 0 (12)

where N = I/I′ = I/(F O + GO) coincides at the smooth points of G with the normal bundle to C in G. Notice that the ideals I, I′ satisfy:

Property 2.6. I = Hom(I, I′).

In fact, clearly I^{2} ⊂ I′. Moreover, since I, I′ coincide with O = O_{ℙ^{3}} outside C and are torsion free, every φ in Hom(I, I′) is given by a rational function which is regular by the normality of O; if φ does not vanish on C, then at some general point of C we would have I = I′, a contradiction. q.e.d.

Definition 2.7. Let F_{G} be the sheaf I/GO, and F_{F} = I/F O.

By the property 2.6 we have pairings

F_{G} × F_{G} ⟶ I′/GO = (F O + GO)/GO = O_{G}(−d),

and analogously

F_{F} × F_{F} ⟶ O_{F} (−m).

To verify that a twist of F_{G} yields a quadratic sheaf on G (similarly for F_{F} on F) we have to show that the pairing is perfect, i.e. F_{G} = Hom(F_{G}, O_{G}(−d)). Let ψ be in Hom(F_{G}, O_{G}(−d)), so it induces ψ′ : I ⟶ I′/GO. If ψ′ is in the image of a φ in Hom(I, I′), then, by property 2.6, φ is in I, and clearly φ induces a zero ψ iff φ is in the principal ideal sheaf GO. Thus, we are left with the verification that

Hom(I, I′) ⟶ Hom(I, I′/GO)

is surjective. By the Ext_{O}^{i} - exact sequence, the above surjectivity is equivalent to the injectivity of Ext_{O}^{1}(I, GO) ⟶ Ext_{O}^{1}(I, I′).

Now, Ext_{O}^{1}(I, GO) ≅ Ext_{O}^{2}(O_{C}, O(−d)) ≅ ω_{C}(4 − d), which is a torsion free sheaf of rank 1. Thus injectivity is equivalent to the map μ : Ext_{O}^{1}(I, GO) ⟶ Ext_{O}^{1}(I, I′) being non zero at every generic point.

But indeed, reversing the argument, at each such generic point there are local parameters G, y such that I = (G, y), F = Gz − y^{2}, and clearly F_{G} is a principal O_{G} - module generated by y, while I′/GO = O_{G}(−d) is the principal O_{G} - module generated by y^{2}: thus at each generic point the pairing is perfect and the map μ is non zero.

The argument for F_{F} is the same if z does not vanish on the given generic point of C, otherwise one can easily prove that F_{F} occurs indeed locally as a cokernel

0 ⟶ O^{2} ⟶^{α} O^{2} ⟶ F_{F} ⟶ 0,

where the matrix

α = ( \begin{matrix}z & −y \\ −y & G \end{matrix} ) .

Then obviously F_{F} is a quadratic sheaf on F = {det(α) = 0}. The local picture is also important: we get F as the determinant of a square symmetric matrix, whereas G is the determinant of the submatrix α′′ obtained by deleting the first row and first column of α, and C is defined by the determinants of the maximal minors of the matrix α′ obtained by deleting the first row of α.

It looks at first sight that this relation is not symmetric between F and G, but this is not true: it suffices to consider the matrix

A = ( \begin{matrix}0 & 1 & 0 \\ 1 & z & −y \\ 0 & −y & G \end{matrix} ) .

This local calculation helps us to understand the relation occurring between the two vector bundles E′_{F}, E′_{G} which exist according to the main theorem on quadratic sheaves.

Notice in fact that, if we set d = 2d′ + δ, m = 2m′ + μ, then F_{G}(d′) is a δ/2 quadratic sheaf, and similarly F_{F} (m′) is a μ/2 quadratic sheaf. Set E_{F} = E′_{F} (−m′), E_{G} = E′_{G}(−d′), then there are symmetric resolutions (i.e., α = ^{t}α, A = ^{t}A):

0 ⟶ E_{F}^{∨}(−d−m) ⟶^{α} E_{F} ⟶ F_{F} ⟶ 0, 0 ⟶ E_{G}^{∨}(−d−m) ⟶^{A} E_{G} ⟶ F_{G} ⟶ 0.

Let us consider the following exact sequence:

0 ⟶ GO(−m) ⟶ I ⟶ F_{G} ⟶ 0;

from it we obtain H_{∗}^{1}(I) = H_{∗}^{1}(F_{G}), and similarly we get H_{∗}^{1}(I) = H_{∗}^{1}(F_{F}). Moreover, H_{∗}^{0}(I) maps onto H_{∗}^{0}(F_{G}) and H_{∗}^{0}(F_{F}). We can therefore assume E_{F} = E_{G}, but please notice that E_{F} , E_{G} are not

uniquely determined!

We can clearly lift E_{G} → F_{G} to E_{G} → I since H_{∗}^{2}(E_{G}) = 0, thus we obtain a resolution of I of Hilbert Burch type:

0 ⟶ E_{G}^{∨}(−d − m) ⟶ O(−m) ⊕ E_{G} ⟶ I ⟶ 0.

On the other hand, by the exact sequence

0 ⟶ F O(−d) ⟶ I ⟶ F_{F} ⟶ 0

and again since H_{∗}^{2}(E_{G}) = 0 we can lift the inclusion of F O(−d) to a map to E_{G} ⊕ O(−m), whence we get an exact sequence

0 ⟶ O(−d) ⊕ E_{G}^{∨}(−d − m) ⟶ O(−m) ⊕ E_{G} ⟶ F_{F} ⟶ 0.

We claim we can make this resolution symmetric: this can be shown in several ways, e.g. repeating the argument of [C-C] (that we will illustrate in the sequel of this paper).

Simply let us repeat the construction leading from a resolution of F_{G} to one of F_{F}, interchanging the roles of F and G.

We then obtain

0 ⟶ O(−m) ⊕ O(−d) ⊕ E_{G}^{∨}(−d − m) ⟶ O(−d) ⊕ O(−m) ⊕ E_{G} ⟶ F_{G} ⟶ 0,

and we can make this symmetric by applying [C-C] claim 2.1, mutatis mutandis, since H_{∗}^{1}(O(−d) ⊕ O(−m) ⊕ E_{G}) ≅ H_{∗}^{1}(E_{G}).

Finally, we can erase the first row and column and we are left with a symmetric matrix.

We have therefore proven the following result (a more general result will appear elsewhere).

Theorem 2.8. Let C be a reduced curve in ℙ^{3} which is self linked through hypersurfaces F, G. Then, I being the ideal sheaf of C, there is a symmetric resolution of F_{F} = I/F O

0 ⟶ O(−d) ⊕ E_{G}^{∨}(−d − m) ⟶^{α} O(−m) ⊕ E_{G} ⟶ F_{F} ⟶ 0

such that if α′ is the minor of α obtained by deleting the first column of α, and α′′ is obtained by also deleting the first row, α′ yields a resolution of I

0 ⟶ E_{G}^{∨}(−d − m) ⟶ O(−m) ⊕ E_{G} ⟶ I ⟶ 0,

and α′′ yields a symmetric resolution of F_{G}

0 ⟶ E_{G}^{∨}(−d − m) ⟶ E_{G} ⟶ F_{G} ⟶ 0.

Conversely, C is linked through the hypersurfaces F = det(α), and G = det(α′′).

\underline{Proof}. There remains only to show the converse assertion. But this follows by reversing the chain of arguments given to prove the other statement: in fact the pairing F_{G} ×F_{G} ⟶ I′/GO(−m) being perfect is equivalent to property 2.6 saying that I = Hom(I, I′). q.e.d.

One could also, since the statement is of local nature, reduce it to the Kronecker determinantal identity (cf. [Cat1] 1.2).

In our previous notation, where α is a symmetric matrix and f = det(α), B = Λ^{h−1}(α), let D_{i,k}^{j,l} be the determinant of the minor obtained by deleting the rows i, k and columns j, l of α. Then the formula is f · D_{i,k}^{j,l} = B_{kj}B_{il} − B_{kl}B_{ij}.

f is the local expression of F, G = B_{11}, and I is generated by B_{11}, …, B_{1h}. The above formula immediately shows (set k = j = 1) that I^{2} ⊂ (f, B_{11}). Moreover, if we assume C reduced, by the symmetry of F and G, we may assume B_{11} to be a local parameter at a generic point ξ of C, and also we will have that some D_{i,1}^{1,l} will not vanish at ξ. Whence f · unit ≡ B_{1l}B_{1i}(mod B_{11}), thus (f) ≡ I^{2}(mod B_{11}) and we are done.

Remark 2.9. i) The case where C is arithmetically Cohen-Macaulay is thus the case where the bundle E_{G} is split. This case was already known, as it follows via the above argument from theorem 2.19 of [Cat1], characterizing (with slightly different language) arithmetically Cohen Macaulay quadratic sheaves. The first examples of self linkage through symmetric matrices appeared in [Gal] and [Cat1], as we already mentioned, and this example was independently rediscovered by Valla ([Va]). A more general formulation assuming V an arithmetically Cohen-Macaulay subscheme of codimension 2 in ℙ^{n}, but not reduced (thus, assuming property 2.6) was proven by Rao ([Rao3]) and appears in an algebraic context in [K-L-U], theorem 2.3). A first example of self linkage in the non arithmetically C-M case was given by Barth in [Bar4].

ii) Ferrand ([Fer]) proved that any local complete intersection curve C is ”self linked” through a scheme C′ which is the scheme of zeros of a section of a rank 2 bundle (that is, property 2 holds).It would be interesting to try to obtain a structure theorem in this situation.

The theorem ([C-C]) showing that quadratic sheaves are bundle symmetric was proven using new ideas introduced by Walter ([Wal]) to sheafify the codimension 3 theorem of Buchsbaum-Eisenbud, and on which we will report in the next sections. It relates, as we saw, even sets of nodes to vector bundles on ℙ^{3}, and permits a classification of even sets in degree ≤ 6.

These theorems though, so to say, cut the original problem to a half of its size: instead of studying coherent sheaves, we are reduced to study vector bundles.

What do we know about both these type of objects?

3. Coherent sheaves and vector bundles according to Horrocks, Beilinson, Bernshtein-Gelfand-Gelfand and Walter

As a consequence of Hilbert’s syzygy theorem for A-modules, any coherent sheaf F on projective space ℙ^{N} = ℙ has a minimal resolution

0 ⟶ L^{c} ⟶ ⋯ L^{1} ⟶ L^{0} ⟶ F ⟶ 0 (13)

where the sheaves L^{i} are direct sums of invertible sheaves O_{ℙ^{N}} (j) (this is called the Hilbert resolution): but of course it is often a non trivial matter to calculate it explicitly in concrete cases.

Alexandre Beilinson ([Be]) introduced a semi-explicit way of calculating it and, more importantly, another way of expressing F as the cohomology of a complex of locally free sheaves whose terms can be determined completely once one knows certain cohomology groups of F. We start with this other approach:

Theorem 3.1 (Beilinson’s theorem (version 1)). For any coherent sheaf F on a projective space ℙ there is a complex

⋯ ⟶ C^{i}(F) = ⊕_{p+q=i}H^{q}(F(p)) ⊗_{ℂ} Ω_{ℙ}^{−p}(−p) ⟶ ⋯

whose cohomology is concentrated in degree 0 and yields F.

In particular one has (cf. [O-S-S]) the following weaker

Corollary 3.2. For any coherent sheaf F on a projective space ℙ there is a spectral sequence, converging to the graded quotient of a suitable filtration of F, with E_{1}^{p,q} equal to H^{q}(F(p)) ⊗_{ℂ} Ω_{ℙ}^{−p}(−p), and with d_{1}^{p,q} : E_{1}^{p,q} ⟶ E_{1}^{p+1,q} given by the identity tensor

𝕀 = (∑_{j} y_{j} ⊗ y_{j}^{∗}) ∈ V^{∗} ⊗_{ℂ} V = H^{0}(O_{ℙ}(1)) ⊗_{ℂ} Hom(Ω_{ℙ}^{1}(1), O_{ℙ})

which acts according to the tensor rule

(w ⊗ v)(s ⊗ ω) = (ws) ⊗ (v⌟ω),

⌟ denoting contraction of a vector with a contravariant tensor.

The idea of proof is based on a marvellously simple trick, namely to consider the diagonal Δ in ℙ × ℙ, the projections on the two factors, and to note that F is the direct image under p_{2} of the restriction to Δ of the pull back of F under p_{1}. The first key step is that the diagonal is the locus of zeros of a section of a vector bundle, namely the dual of p_{1}^{∗}(O_{ℙ}(−1)) ⊗_{ℂ} p_{2}^{∗}(Ω_{ℙ}^{1}(1)). This section, under the above mentioned isomorphism

H^{0}(p_{1}^{∗}(O_{ℙ}(1)) ⊗_{ℂ} p_{2}^{∗}(T_{ℙ}(−1))) = H^{0}(O_{ℙ}(1)) ⊗_{ℂ} H^{0}(T_{ℙ}(−1)) = V^{∗} ⊗_{ℂ} V

corresponds to the identity tensor.

One gets therefore a Koszul resolution of O_{Δ}, i.e., a complex quasiisomorphic to O_{Δ} (its cohomology is zero, except at the extreme right, where we get O_{Δ})

⋯ p_{1}^{∗}(O_{ℙ}(−2)) ⊗ p_{2}^{∗}(Ω_{ℙ}^{2}(2)) ⟶ p_{1}^{∗}(O_{ℙ}(−1)) ⊗ p_{2}^{∗}(Ω_{ℙ}^{1}(1)) ⟶ O_{ℙ}

where all the addenda are locally free. Tensoring with p_{1}^{∗}(F) it is not hard to verify that we obtain a complex quasiisomorphic to p_{1}^{∗}(F) ⊗_{ℂ} O_{Δ}.

Now, applying the full direct image functor, and since

R^{q}p_{2∗}(p_{1}^{∗}(F(−j)) ⊗ p_{2}^{∗}(Ω_{ℙ}^{j}(j))) = H^{q}(F(−j)) ⊗_{ℂ} Ω_{ℙ}^{j}(j)

it is not difficult to obtain the weaker version of the corollary.

The stronger version of the theorem (cf. also [A-O]) is based on the following result which can be easily shown by using the Euler exact sequence.

Lemma 3.3 (Beilinson). Ext^{r}(Ω_{ℙ}^{i}(i), Ω_{ℙ}^{j}(j)) = 0 for r ≥ 1, and

Hom(Ω_{ℙ}^{i}(i), Ω_{ℙ}^{j}(j)) = \{ \begin{matrix}Λ^{i−j}(V) & for i ≥ j, \\ 0 & else. \end{matrix}

In fact, taking an injective resolution on ℙ × ℙ and then applying p_{2∗}, we only get a complex of sheaves whose cohomology is given by the higher direct image sheaves: the theorem then follows, roughly speaking, since any extension made with the ‘pieces’ H^{q}(F(−j)) ⊗_{ℂ} Ω_{ℙ}^{j}(j) splits.

Of course, the interesting next question is: how to determine, as explicitly as possible, the maps in the complex whose cohomology yields F?

The appropriate setting to answer this question seems to be the language of derived categories. We recall the standard definitions:

Definition 3.4. The homotopy category K (of an Abelian category, say of modules, or of sheaves of modules) is the category whose objects are complexes A^{•} = (A^{i}, d_{A}^{i}) and whose morphisms are homotopy equivalence classes of homomorphisms of complexes f : A^{•} ⟶ B^{•} (i.e., f = (f^{i}), with f^{i} : A^{i} ⟶ B^{i} commuting with the differentials, and f, g are equivalent if they are homotopic, which means that there exists k^{i} : A^{i+1} ⟶ B^{i} such that (f^{i} − g^{i}) = k^{i}d_{A}^{i} + d_{B}^{i−1}k^{i−1}).

f is a quasi-isomorphism if it induces an isomorphism between the cohomology groups H^{i}(A^{•}) and H^{i}(B^{•}).

The derived category D is obtained by inverting quasi-isomorphisms: it has the same objects, but a morphism is the equivalence class of the composition of the formal inverse of a quasi isomorphism s : C^{•} ⟶ A^{•} with a morphism g : C^{•} ⟶ B^{•}. Here, morphisms are taken in the homotopy category K, and the equivalence relation is generated by the direct equivalence for which fs^{−1} = f′s^{′−1} if there is a morphism a : C^{′.} ⟶ C^{•} such that s′ = sa, f′ = fa.

D^{−} stands for the derived category of complexes bounded from above (A^{i} = 0 for i sufficiently big), D^{b} for the derived category of bounded complexes.

Remark 3.5. A morphism in the derived category is an isomorphism iff it induces isomorphism of cohomology groups (i.e., it is of the form fs^{−1} with f a quasi-isomorphism).

Beilinson considers (we dualize his notation) a vector space V of dimension (N + 1) and the graded exterior algebra Λ(V) = Λ (where V is the (−1)-graded piece).

The category Mod(Λ) of finite dimensional graded Λ-modules admits a subcategory Mod_{[0,N]}(Λ) whose objects are direct sums of twists Λ(i) with 0 ≤ i ≤ N : let K_{Λ} be the homotopy category K^{b}(Mod_{[0,N]}(Λ)).

Similarly, one can consider the graded commutative algebra A = Sym(V^{∗}) and define K_{A} to be the homotopy category K^{b}(Mod_{[0,N]}(A)).

Theorem 3.6 (Beilinson’s theorem (version 2)). Let D^{b}(Coh(ℙ^{N})) be the derived category of bounded complexes of coherent sheaves on ℙ^{N}: then there exist functors F_{Λ} : K_{Λ} ⟶ D^{b}(Coh(ℙ^{N})), and F_{A} : K_{A} ⟶ D^{b}(Coh(ℙ^{N})), which are equivalence of categories and such that i) F_{Λ}(Λ(i)) = Ω_{ℙ}^{i}(i) ii) F_{A}(A(i)) = O_{ℙ}(−i).

Obviously the inverse of the functor F_{A} is provided by exchanging the role of the two projections, but keeping the same resolution of O_{Δ}, so that F is isomorphic in the derived category to a complex of the form

⋯ ⟶ C_{A}^{i}(F) = ⊕_{−p+q=i}H^{q}(F ⊗ Ω_{ℙ}^{p}(p)) ⊗_{ℂ} O_{ℙ}(−p) ⟶ · · · .

A somehow (how exactly?) related description of Hilbert resolutions via Koszul complexes was later given by M.Green in [Gr].

Instead, exactly at the same time and independently, Bernshtein, I.M. Gelfand and S.I. Gelfand formulated a similar result.

Let as above Mod(Λ) be the category of graded Λ - modules, and consider the subcategory P of free Λ - modules.

Definition 3.7. The stable category Mod(Λ)/P is the quotient category whose objects are Λ - modules but morphisms are stable equivalence classes of morphisms: M is said to be stably equivalent to M′ if there exist free modules P, P′ such that M ⊕ P ≅ M′ ⊕ P′, whereas a morphism f is stably equivalent to zero if it factors through a free module.

The situation is algebraically not complicated since (cf. [BGG]):

1. A free Λ - module P is injective and projective.

2. Any finitely generated Λ - module M can be uniquely written, up to isomorphism, as a direct sum P ⊕ M′ of a free module P and of a torsion module M′ (i.e., Ann(M′) is non trivial).

3. Whence stable equivalence classes amount to isomorphism classes of modules over Λ/soc(Λ) (soc(Λ), the socle of Λ is here the top exterior power of our vector space V).

To any graded module M can be associated the complex of vector bundles L(M) such that

L(M)_{j} = O_{ℙ^{n}}(j) ⊗ M_{−j}

and where the differential is induced by contraction with the identity tensor 𝕀 = (∑_{j} y_{j} ⊗ y_{j}^{∗}) ∈ V^{∗} ⊗_{ℂ} V.

If M is a free module, say M = Λ, then the above complex is exactly the Koszul complex, which is an exact complex of sheaves (warning: the complex of modules associated to it via the Serre correspondence is a resolution of the module ℂ). Therefore, it follows that in this way one defines a functor L′:

Mod(Λ)/P ⟶ D^{b}(Coh(ℙ^{N})).

The result of Bernshtein-Gelfand-Gelfand is somehow a mixture of the two results of Beilinson, and provides the following characterization of vector bundles on projective space.

Definition 3.8. A Λ - module M is said to be \underline{faithful} if the complex L(M) is an exact complex of vector bundle maps in degree ≠ 0, and therefore is such that its cohomology H^{i}(L(M)) is zero for i ≠ 0, and yields a vector bundle H^{0}(L(M)).

Theorem 3.9 (Theorem of Bernshtein-Gelfand-Gelfand:).

L′ : Mod(Λ)/P ⟶ D^{b}(Coh(ℙ^{N}))

is an equivalence of categories. Moreover, any vector bundle on ℙ^{N} is of the form H^{0}(L(M)) for a faithful Λ - module M, unique up to stable equivalence.

The essential argument in the proof goes, via the Serre correspondence which identifies coherent sheaves with equivalence classes of finitely generated graded A - modules (where a module is equivalent to zero iff it is of finite dimension over ℂ), through the definition of an analogue of L′.

Namely we define L : D^{b}(Λ) ⟶ D^{b}(A) in such a way that L(M^{•}) = A ⊗ M^{•}, and the differential d_{L} is the sum of (id_{A} ⊗d_{M^{•}}) with the map given by contraction with the identity tensor (bidegrees have then to be defined suitably).

Its inverse is provided by the functor G, such that G(N^{•}) = Hom_{ℂ}(Λ, N^{•}) and the differential is the adjoint of d_{L}.

An earlier and very important characterization of vector bundles on projective space, although slightly less effective, had been given by Horrocks ([Hor]) in 1964: we shall follow in the presentation of this result the modern version of the theorem given recently by Walter ([Wa]).

In order to do so, we need to recall the Serre correspondence in greater detail. Let F be a coherent sheaf on ℙ^{N}: to F we associate the graded A - module

Γ_{∗}(F) = ⊕_{m∈ℤ}H^{0}(ℙ^{N}, F(m)). This is called the Serre - module of F, but the only trouble is that (take e.g. F to be the structure sheaf of a point, where the module Γ^{∗}(F) is non zero in any degree) it need not be finitely generated.

One can instead consider the module Γ_{≥0}(F) = ⊕_{m≥0}H^{0}(ℙ^{N}, F(m)), and the modules H_{∗}^{i}(ℙ^{N}, F) = ⊕_{m∈ℤ}H^{i}(ℙ^{N}, F(m)), nowadays called Hartshorne-Rao modules, even if they were introduced and studied by Serre ([Ser1]).

By Serre’s theorem B(m) H^{i}(ℙ^{N}, F(m)) = 0 for m >> 0, whence not only the modules H_{∗}^{i}(ℙ^{N}, F) are zero in high degree, but also the modules Γ_{≥0}(F) are finitely generated (by theorem A(m) any F is a quotient sheaf of a sheaf L^{0} which is a direct sum of line bundles O_{ℙ}(r), by theorem B(m) applied to H = kerL^{0} ⟶ F, for m >> 0 H^{0}(ℙ^{N}, L^{0}(m)) ⟶ H^{0}(ℙ^{N}, F(m)) is surjective).

\underline{Serre’s correspondence}: the functor F ⟶ Γ_{∗}(F) induces an equivalence between the category Coh(ℙ^{N}) and the category T F/C (T F stands here for Finite Type and C is the subcategory of modules M asymptotically equivalent to zero, i.e., such that M_{m} = 0 for m >> 0) whose objects are the asymptotic equivalence classes of graded A - modules which can be represented by finitely generated modules.

The functor F ⟶ Γ_{≥0}(F) induces instead an equivalence between Coh(ℙ^{N}) and the quotient category of the category Mod(A) of finitely generated graded A - modules by the subcategory of finite length modules.

The inverse of the Serre functor is given by localization of a given finitely generated graded module M.

The first meaning of the modules H_{∗}^{i}(ℙ^{N}, F) is in terms of depth (this is a consequence of duality and B(m)):

Proposition 3.10 ([Ser1], Thm 1, p. 268). H^{i}(ℙ^{N}, F(−m)) = 0 for m >> 0 (i.e., for i > 0, H_{∗}^{i}(ℙ^{N}, F) is a module of finite length) iff Ext_{O}^{N−i}(F, O) = 0 (O = O_{ℙ}).

Since, for any point x in ℙ^{N}, O_{x} = O_{ℙ,x} is a regular local ring, we have the Auslander-Buchsbaum extension ([A-B]) of Hilbert’s syzygy theorem, saying that F_{x} has a minimal resolution

0 ⟶ L^{c} ⟶ ⋯ ⟶ L^{1} ⟶ L^{0} ⟶ F_{x} ⟶ 0. (14)

if and only if Ext^{j}(F_{x}, O_{x}) = 0 for j > c (cf. also [Ser1], Lemma 1, p. 269).

Corollary 3.11. F is locally free if and only if H^{i}(ℙ^{N}, F(−m)) = 0 for m >> 0 for i < N or, equivalently, if and only if Γ_{∗}(F) is finitely generated and, for N > i > 0, H_{∗}^{i}(ℙ^{N}, F) is a module of finite length.

The beautiful result of Horrocks says roughly speaking that the finite-length intermediate cohomology modules H_{∗}^{i}(ℙ^{N}, F) (0 < i < N) determine a locally free sheaf E up to stable equivalence, that is, up to direct sums of line bundles. Also here a weaker result is more known, which follows right away from the full statement

Corollary 3.12 (of Horrocks’ theorem). A coherent sheaf F on ℙ^{N} is a direct sum of line bundles if and only if Γ_{∗}(F) is finitely generated and the cohomology groups H^{i}(ℙ^{N}, F(m)) vanish for each m and for 0 < i < N.

The statement of the theorem is more technical, and requires some further notation:

Definition 3.13. Given a complex (C^{•}) = ⋯ → C^{i−1} → C^{i} → C^{i+1} → ⋯, the (non-naive) truncation at level < r + 1 is the new complex

τ_{<r+1}(C^{•}) = ⋯ ⟶ C^{r−2} ⟶ C^{r−1} ⟶ ker(d_{r}) ⟶ 0

while

τ_{>r}(C^{•}) = 0 ⟶ C^{r}/ker(d_{r}) ⟶ C^{r+1} ⟶ ⋯

The Serre functor Γ_{∗} gives rise to a right derived functor

RΓ_{∗} : D^{b}(Coh(ℙ^{N})) ⟶ D^{b}(Mod(A)),

and in this context the second meaning of the functors H_{∗}^{i} is that they are the cohomology functors associated to RΓ_{∗}.

They are therefore defined through hypercohomology, namely we replace (C^{•}) by a quasi-isomorphic complex (D^{•}), where the D^{i}’s are injective (or acyclic), we apply Γ_{∗}, and then we take the cohomology of the complex (Γ_{∗}D^{•}).

Definition 3.14. The stable category of vector bundles on ℙ^{N}, denoted by \underline{V B} is the category whose objects are vector bundles but morphisms are stable equivalence classes of morphisms: again a morphism f is stably equivalent to zero if it factors through a direct sum of line bundles (two vector bundles V, V′ are thus stably equivalent iff there exist two direct sums of line bundles, L, L′ such that V ⊕ L ≅ V′ ⊕ L′).

Definition 3.15. The finite length category is the full subcategory F inL of D^{b}(A) consisting of the complexes (C^{•}) of A-modules such that H^{i}(C^{•}) has finite length for 1 ≤ i ≤ N − 1, and is otherwise = 0.

Given a complex (C^{•}) in F inL, one takes its minimal projective resolution

(P^{•}) = 0 ⟶ P^{−N} ⟶ ⋯ ⟶ P^{0} ⟶ P^{1} ⟶ ⋯ ⟶ P^{N−1} ⟶ 0

(which is quasi-isomorphic to (C^{•})) and then definines Syz(C^{•}) to be the sheafification E of (kerd_{0}). Since localization is exact and finite length modules yield a 0 sheaf, by sheafification we get an exact sequence of sheaves, and therefore E is quasi-isomorphic to the complex of sheaves localization of the naively truncated complex

P^{0} ⟶ P^{1} ⟶ ⋯ ⟶ P^{N−1} ⟶ 0.

Since by applying Γ_{∗} to the localization of a free module P we reobtain P, the consequence is that the intermediate cohomology groups of E are of finite length and E is a vector bundle by the criterion we recalled above.

In order to understand the following compact statement of the theorem, especially the functor τ_{>0}τ_{<N}RΓ_{∗}, notice that this makes sense after associating to a vector bundle E its class in the derived category of O-modules. One replaces e.g. E by an injective resolution (H^{•}) of E made of O-modules and then one can apply Γ_{∗} followed by the two truncations in order to get a bounded complex of A-modules.

Theorem 3.16 (Horrocks - Walter). The functor τ_{>0}τ_{<N}RΓ_{∗} : \underline{V B} → F inL is an equivalence of categories. Its inverse is the syzygy bundle functor Syz.

Remark 3.17. i) In practice, Horrocks gives a more concrete recipe, namely, to E he associates the dual bundle E^{∨} = Hom(E, O), then lets E^{∨} = Γ_{∗}(E^{∨}). Then taking a minimal projective resolution P^{∨ .} of E^{∨}, the dual complex P^{•} = Hom(P^{∨ .}, A) yields the desired complex in F inL. ii) There are of course some relations between the BGG construction and the theorem of Horrocks. Grassi (private communication) has shown to us how the BGG construction yields the faithfulness of the functor τ_{>0}τ_{<N}RΓ_{∗} : \underline{V B} ⟶ F inL.

Of course, in spite of the above results, the knowledge of vector bundles on projective space is still at a preliminary stage. For instance, the famous Hartshorne’s conjecture ([Har2]) that any rank 2 bundle on ℙ^{n}, when n ≥ 7, is a direct sum of line bundles is still unproven. In the same article Hartshorne points out that for n ≥ 5 there is no known bundle E of rank r, where 2 ≤ r ≤ n − 2, which is indecomposable (i.e., E is not a non trivial direct sum).

4. Pfaffian subschemes and canonical surfaces in ℙ^{5}

Walter ([Wal4]) used the Horrocks equivalence in order to answer a conjecture raised by Okonek ([Ok]), concerning a bundle analog of the Buchsbaum-Eisenbud theorem on Gorenstein subvarieties of codimension 3.

Definition 4.1. A subscheme X of ℙ^{N} is said to be (bundle) Pfaffian if there exists a vector bundle E on ℙ^{N} of odd rank 2k + 1 and an antisymmetric map α : E^{∨}(−t) ⟶ E such that X is defined by the size 2r Pfaffians of α. X is said to be split Pfaffian if moreover one can also have that E is a direct sum of line bundles.

In other words, if X is Pfaffian, we have the following self dual resolution of the ideal sheaf of X, where α^{1} = α, and α^{0} is given by the Pfaffians of α:

0 ⟶ O(−t − 2s) ⟶^{α^{2}} E^{∨}(−t − s) ⟶^{α^{1}} E(−s) ⟶^{α^{0}} I_{X} ⟶ 0. (15)

We know already that if X has the expected codimension, namely 3, then X is locally Gorenstein: moreover the self duality of (15) implies that ω_{X} = O_{X}(t + 2s − N − 1), which is referred to as X being subcanonical of level t + 2s − N − 1.

Theorem 4.2 (Walter). Let ℂ be an algebraically closed field of characteristic ≠ 2, and let X ⊂ ℙ^{N} be a locally Gorenstein subscheme of pure codimension 3 which is subcanonical of level p (i.e., ω_{X} = O_{X}(p)). Then X is Pfaffian if and only if the following parity condition holds: if N − 3 ≡ 0(mod 4) and p is even, then χ(O_{X}(p/2)) is even.

To understand the parity condition, note that if p is an even integer, by Serre duality on X, H^{i}(O_{X}(p/2)) is dual to H^{N−3−i}(ω_{X}(−p/2)) = H^{N−3−i}(O_{X}(p/2)).

Thus χ(O_{X}(p/2)) is zero if N − 3 is odd, and automatically even if N − 3 ≡ 2(mod 4), since then H^{(N−3)/2}(O_{X}(p/2)) carries a non degenerate alternating bilinear form.

When N − 3 ≡ 0(mod 4) the above bilinear form is symmetric, thus the rank of H^{(N−3)/2}(O_{X}(p/2)) need not be even.

If though X is Pfaffian, this parity condition has to hold true because of Serre duality on ℙ^{N} and in view of the exact sequence

0 → O_{ℙ}(−N−1−p/2) → E^{∨}(s−p/2−N−1) → E(p/2−s) → O_{ℙ}(p/2) → O_{X}(p/2) → 0. (16)

Putting the sequence (16) together with the obvious exact sequence

0 ⟶ I_{X} ⟶ O_{ℙ} ⟶ O_{X} ⟶ 0,

and taking the associated long exact cohomology sequences we obtain

0 ⟶ I_{X} ⟶ A ⟶ ℂ[X] ⟶ 0,
H_{∗}^{1}(ℙ^{N}, I_{X}) ≅ R(X, O_{X}(1)/ℂ[X]

(where R(X, O_{X}(1) = H^{0}(O_{X}(m)))

H_{∗}^{i+1}(ℙ^{N}, I_{X}) ≅ H_{∗}^{i}(ℙ^{N}, O_{X})

In particular we observe, for later use, the fact that we have the following criterion (already part of the Buchsbaum-Eisenbud theorem)

\underline{Criterion}: X of pure codimension 3 in ℙ^{N} is split Pfaffian if and only if it is locally Gorenstein, subcanonical, projectively normal (R(X, O_{X}(1) = ℂ[X]), and with H_{∗}^{i}(ℙ^{N}, O_{X}) = 0 for 1 ≤ i ≤ dimX − 1.

Remark 4.3. If X is smooth of dimension ≥ 5, then by Barth-Lefschetz’ theorem ([Bar1]), automatically X is subcanonical. Thus, if the parity condition holds, X is Pfaffian. Let Y be the locus defined by the vanishing of the Pfaffians of order 2k − 2: Y is contained in the singular locus of X and has expected codimension equal to 10 when the rank of E is at least 5 (k ≥ 2). The Hartshorne conjecture predicts that X should be a complete intersection: thus, if N ≥ 10, either

i) rankE ≥ 5 and Y should be non empty; equivalently, its class (computable according to [J-L-P]) should be non zero or ii) rankE = 3 and E is a direct sum of line bundles.

To explain the main idea of proof, we need to observe that in general the bundle E is not unique: therefore, whereas in the case of A modules the concept of a minimal resolution is rather clear, here one has to search for the correct extension of this concept.

This is achieved by Walter first observing that if X is Pfaffian one has a long exact sequence of cohomology modules

H_{∗}^{i+1}(E(−s)) ⟶ H_{∗}^{i+1}(I_{X}) (= H_{∗}^{i}(O_{X}) for i ≥ 1) ⟶ H_{∗}^{i+2}(E^{∨}(−t − s))

and deciding that for i < (N − 3)/2 the left map should be an isomorphism, for i > (N − 3)/2 the right map should be an isomorphism, whereas for i = (N − 3)/2 the left map should be injective with image equal to a certain subspace W, whereas the right map should be surjective.

The symmetrical way in which W is chosen is as follows: choose a maximal isotropic subspace U in H^{(N−3)/2}(O_{X}(p/2)) for the pairing in H^{(N−3)}(O_{X}(p)) = ℂ, and let

W = U ⊕ (⊕_{h>p/2}H^{(N−3)/2}(O_{X}(h))).

Therefore, under the Horrocks correspondence, E is obtained in the derived category by truncating τ_{>0}τ_{<N}RΓ_{∗}(I_{X}) up to dimension (N − 3)/2 + 1 but replacing the ((N − 3)/2 + 1)-th module by the submodule, contained in ker(d(N − 3)/2 + 1) and containing Im(d(N − 3)/2), inverse image of the subspace W of H_{∗}^{(N−3)/2+1}(I_{X}). Roughly speaking, E(−s) carries 1/2 of the intermediate cohomology of I_{X}, and by duality E^{∨}(−t − s) carries the (quotient) other 1/2.

Now, this truncation can be made for any X satisfying the assumptions of the theorem, and by the Horrocks correspondence one obtains therefore a vector bundle E′ on ℙ^{N}.

Since H_{∗}^{0}(E′(−s)) ⟶ H_{∗}^{0}(I_{X}) needs not be surjective, Walter adds to E′ a direct sum of line bundles in order to obtain a minimal E which moreover gives an epimorphism H_{∗}^{0}(E(−s)) ⟶ H_{∗}^{0}(I_{X}).

As mentioned above, Walter’s construction can also be used to prove that quadratic sheaves are bundle-symmetric ([C-C]). Therefore, in order to illustrate the rest of the arguments, we shall sketch the proof in this context (where the length of the resolution is 1 instead of 2) and in the simpler case where X ⊂ ℙ^{3} of degree d and F is a (0-)quadratic sheaf on X.

As before, we start by taking a maximal isotropic subspace U in H^{1}(F((d − 4)/2)), by letting W = ⊕_{m<(d−4)/2}H^{1}(F(m)) ⊕ U, and then letting E be stably equivalent to Syz(W) such that A) H_{∗}^{0}(E) ⟶ H_{∗}^{0}(F) B) H_{∗}^{1}(E) ≅ W ⊂ H_{∗}^{1}(F) C) H_{∗}^{2}(E) = 0 By A), E surjects onto F, whence we have an exact sequence

0 ⟶ K ⟶ E ⟶ F ⟶ 0,

where K is automatically locally free, since H_{∗}^{1}(K) = 0, H_{∗}^{2}(K) ≅ H_{∗}^{1}(F)/W. Recall that we have the isomorphism Ext_{O_{ℙ}}^{1}(F, O_{ℙ}(−d)) ≅ F: so we can try to lift it to the two (dual) exact rows

0 ⟶ K ⟶ E ⟶ F ⟶ 0

0 ⟶ E^{∨}(−d) ⟶ K^{∨}(−d) ⟶ Ext_{O_{ℙ}}^{1}(F, O_{ℙ}(−d)) ⟶ 0.

The first crucial step is to show that any lift is given by isomorphisms. The second crucial step consists in proving that a lifting s : E ⟶ K^{∨}(−d), exists, and indeed satisfies the condition that its restriction to K (which lands into E^{∨}(−d)) coincides with the dual s^{∨}(−d) of s.

It is rather clear that, once we know this, we simply take as α : E^{∨}(−d) ⟶ E the composition of s^{∨}(−d)^{−1} with the inclusion of K into E, and then α = α^{∨}(−d), i.e., α = ^{t}α.

It is worthwhile to explain how the careful choice of W is relevant: it is so precisely in step two. Again we have two exact rows

0 ⟶ Λ^{2}K ⟶ K ⊗ E ⟶ S^{2}E ⟶ S^{2}F ⟶ 0
0 ⟶ 0 ⟶ O_{ℙ}(−d) ⟶ O_{ℙ} ⟶ O_{X} ⟶ 0

and we claim that the multiplication map ϵ : S^{2}F ⟶ O_{X} lifts to η : S^{2}E ⟶ O_{ℙ}, yielding the desired s by ‘restriction’.

In turn, to show that η exists, it suffices to show that, ψ being the composition S^{2}E ⟶ S^{2}F ⟶ O_{X}, ∂(ψ) = 0, where ∂ : Hom(S^{2}E, O_{X}) → Ext^{1}(S^{2}E, O_{ℙ}(−d)) is the coboundary of the Ext long exact sequence. One then notices that

Ext^{1}(S^{2}E, O_{ℙ}(−d)) ≅ H^{2}(S^{2}E(d − 4))^{∨} ≅ Λ^{2}(H^{1}(E(\frac{d − 4}{2})))^{∨},

and that in this way ∂(ψ) calculated on an exterior product of two vectors in H^{1}(E(\frac{d−4}{2})), factors through the composition of the maps

Λ^{2}(H^{1}(E(\frac{d − 4}{2}))) ⟶ Λ^{2}(H^{1}(F(\frac{d − 4}{2}))) ⟶ H^{2}(ω_{X}) = ℂ.

Since H^{1}(E(\frac{d−4}{2})) maps to the maximal isotropic subspace U, this composition is the zero map, as claimed. Let us return now to the case of Pfaffian subschemes, and let us concentrate on the particular case of surfaces in ℙ^{5}.

For reasons about which we shall dwell in the next section, the more natural case to consider is when X is locally Gorenstein and canonical, i.e., ω_{X} = O_{X}(1). Thus we have: t + 2s = 7, whence (we can assume s = 0 by replacing E(−s) by E) we have a self dual resolution

0 ⟶ O_{ℙ}(−7) ⟶ E^{∨}(−7) ⟶ E ⟶ O_{ℙ} ⟶ O_{X} ⟶ 0. (17)

\underline{Example:} Assume that the bundle E is O_{ℙ}(−2)^{⊕3} ⊕ O_{ℙ}(−4)^{⊕2}, so that α is an antisymmetric matrix which can be written in block form

( \begin{matrix}0 & A \\ −^{t}A & B \end{matrix} ) ,

where B is an antisymmetric (3 × 3) matrix with entries cubic forms, and A is a (2 × 3) matrix with entries linear forms.

It is easy linear algebra to see that when rankA = 2, then rank(α) = 4, thus X is contained in the determinantal variety Y = {rank(A) ≤ 1}. When the 6 entries of A are independent, Y is smooth of dimension and degree 3, and is well known to be the Segre embedding of ℙ^{1} × ℙ^{2}.

In this case X, being canonical in ℙ^{5}, is the divisor on ℙ^{1} × ℙ^{2} of a polynomial F (t_{0}, t_{1}, x_{0}, x_{1}, x_{2}) of bidegree (3, 4). The 5 Pfaffians are the 3 quadrics Q_{0}, Q_{1}, Q_{2} defining Y (the entries of Λ^{2}(A)), and two quartics inducing F t_{0}, F t_{1} on Y. The first two rows give the relations among the Q_{i}’s, while the last 3 give the three relations (F t_{0})(t_{1}x_{j}) ≡ (F t_{1})(t_{0}x_{j})(mod(Q_{0}, Q_{1}, Q_{2})).

Since on Y (1, 1)^{2}(3, 4) = 11, and X is split Pfaffian, thus projectively normal, X has p_{g} = 6, and K^{2} = 11, which is the minimum allowed by the \underline{Castelnuovo inequality:} If the canonical map is birational, then K^{2} ≥ 3p_{g} − 7.

We are going to determine all the cases where X is a canonical surface which is split Pfaffian (i.e., the irregularity q = h^{1}(O_{X}) = 0, and the homogeneous coordinate ring ℂ[X] equals the full canonical ring R(X) = ⊕H^{0}(O_{X}(mK))).

We know from general principles (and the Bogomolov-Miyaoka-Yau inequality) that we have a finite number of families. We will get it directly as follows: given a sheaf F, denote by P_{F}(t) the Poincaré series of F, P_{F}(t) = ∑ χ(F(m))t^{m}. By additivity, using (17), the Poincaré series P of O_{X} is determined by the Chern classes of E.

On the other hand, X being canonical, d = degree X = K_{X}^{2} and

P = χ/(1 − t) − t^{2}d/(1 − t)^{3}

by Riemann-Roch on X (χ = χ(O_{X})). Since the Poincaré series of O_{ℙ}(−a) = t^{a}(1 − t)^{−(N+1)}, and here N = 5, if we assume that E = ⊕O_{ℙ}(−a_{i}), then (17) yields:

t^{7} + ∑ t^{a_{i}} + χ(1 − t)^{5} − t^{2}d(1 − t)^{3} ≡ 1 + ∑ t^{−a_{i}} + 7(mod (1 − t)^{6}).

Therefore we evaluate the Taylor series at t = 1, setting t = 1 + s. The value being 0, we get the first condition by looking at the coefficient of s:

∑(7 − 2a_{i}) = 7

more generally we get 7(r − 1) + 2c_{1}(E) = 0, r being the rank of E) If this equation is satisfied then i) also the coefficient of s^{2} turns out to be 0 ii) the coefficient of s^{3} yields the equality:

6d = 6K^{2} = 2σ_{3} − 21σ_{2} + 49σ_{1},

where σ_{i} is the sum of the i-th powers of the ”roots” a_{i}. Assume now that E is split, and that X is not contained in a hyperplane (since then X is a complete intersection): then a_{i} ≥ 2 for each i.

By minimality and since no row of α is zero, follows that for each i there exists j such that a_{j} − 7 ≤ −a_{i} + 1, i.e., a_{j} + a_{i} ≤ 6, whence 2 ≤ a_{j} ≤ 4. Therefore E = O_{ℙ}(−4)^{⊕a} ⊕ O_{ℙ}(−3)^{⊕b} ⊕ O_{ℙ}(−2)^{⊕c}, moreover c ≥ a + 1, else the rank of α drops rank by 2 in codimension ≤ 1. The solutions to 7 = 3c + b − a ≥ 2c + b + 1 are therefore only the following 5, yielding d = K^{2} between 11 and 14.

c = 3, b = 0, a = 2, K^{2} = 11 (K^{2} = 35 − 10c − 3b + 3a)

c = 2, b = 2, a = 1, K^{2} = 12

c = 2, b = 1, a = 0, K^{2} = 12 (complete intersection of type 2, 2, 3)

c = 1, b = 4, a = 0, K^{2} = 13 c = 0, b = 7, a = 0, K^{2} = 14.

It is an interesting exercise on determinants to show that the case c = 2, b = 2, a = 1, K^{2} = 12 must be discarded. In fact, in this case we have an alternating map of

E^{∨}(−7) = O_{ℙ}(−3) ⊕ O_{ℙ}(−4)^{2} ⊕ O_{ℙ}(−5) ⟶ E = O_{ℙ}(−4) ⊕ O_{ℙ}(−3)^{2} ⊕ O_{ℙ}(−2)^{2},

and if there is a non zero constant in the matrix α, it yields an isomorphism of the addendum O_{ℙ}(−3) of E^{∨}(−7) with an addendum of E, thus we may mod out this addendum and reduce to the case of a complete intersection of type (2, 2, 3).

If all the constants in the matrix α are zero (as they must in a minimal resolution), the matrix α has the following shape:

( \begin{matrix}0 & 0 & 0 & y_{1} & y_{3} \\ 0 & 0 & y_{2} & Q_{1} & Q_{3} \\ 0 & −y_{2} & 0 & Q′_{1} & Q′_{3} \\ −y_{1} & −Q_{1} & −Q′_{1} & 0 & G \\ −y_{3} & −Q_{3} & −Q′_{3} & −G & 0 \end{matrix} )

(y_{1}, y_{2}, y_{3} linear forms, Q_{1}, Q_{3}, Q′_{1}, Q′_{3} quadratic forms, G a cubic form). The corresponding 5 Pfaffians equations are:

y_{1}y_{2} = 0, y_{3}y_{2} = 0,
y_{1}Q_{3} − y_{3}Q_{1} = 0, y_{1}Q′_{3} − y_{3}Q′_{1} = 0,
(Q′_{1}Q_{3} − Q′_{3}Q_{1}) − y_{2}G = 0.

Looking carefully at these equations, we see that if the entries are general the locus they define is the union of a surface Y of degree 8 in the hyperplane y_{2} = 0, defined by the 22 minors of the submatrix formed by the first 3 rows and last 2 columns, and the quartic surface F in the ℙ^{3} = {y_{1} = y_{3} = 0} of equation F = (Q′_{1}Q_{3} − Q′_{3}Q_{1}) − y_{2}G = 0. Y and F meet along the plane quartic curve y_{1} = y_{2} = y_{3} = F = 0.

There is an obvious explanation for the bound K^{2} ≤ 14 if we require X to be projectively normal and with q = 0: since then p_{g} = 6, thus χ = 7 and then 21 = h^{0}(O_{ℙ}(2)) ≥ h^{0}(O_{X}(2K)) = K^{2} + χ = K^{2} + 7 (and with this remark one can also calculate more easily K^{2} as 14 − K^{2} = h^{0}(ℙ^{N}, I_{X}(2))).

When we want to proceed to higher degree canonical surfaces, at least in order to construct them, a reasonable assumption is the following \underline{Maximal rank assumption}: H^{0}(O_{ℙ}(m)) ⟶ H^{0}(O_{X}(mK)) has maximal rank.

Under the above, since h^{0}(O_{X}(mK)) = \frac{m(m−1}{2}K^{2} + χ, h^{0}(O_{ℙ}(m)) = C(m + 5, m) (C(a, b) being the binomial coefficient), if we assume χ = 7, the restriction map is onto for m = 2, K^{2} ≤ 14, m = 3 and K^{2} ≤ 16, m = 4 and K^{2} ≤ 19, m = 5 and K^{2} ≤ 24.

The systematic way of proceeding, which was also previously known to Schreyer and was formalized by Walter as above is in this case the following.

Assume for simplicity q = 0: then H_{∗}^{1}(ℙ^{N}, I_{X}) ≅ R(X, O_{X}(K))/ℂ[X] and this module, if p_{g} = 6, is non zero only in degree 2 for 15 ≤ K^{2} ≤ 16 (its dimension being K^{2} − 14), also in degree 3 for 17 ≤ K^{2} ≤ 19 (then the dimension of the graded piece of degree 3 equals 3K^{2} − 49), and also in degree 4 for 20 ≤ K^{2} ≤ 24 (then the dimension of the graded piece of degree 4 equals 6K^{2} − 119).

Let 0 ⟶ P^{6} ⟶ P^{5} ⟶ ⋯ ⟶ P^{0} ⟶ H_{∗}^{1}(ℙ^{5}, I_{X}) be a free resolution of our A-module, and take the first syzygy module F^{1} such that we have an exact sequence

0 ⟶ F^{1} ⟶ P^{1} ⟶ P^{0} ⟶ H_{∗}^{1}(ℙ^{5}, I_{X}) ⟶ 0.

Let E^{1} be obtained sheafifying F^{1}: then we have an exact sequence of sheaves (by sheafifying the above sequence of modules)

0 ⟶ E^{1} ⟶ L^{1} ⟶ L^{0} ⟶ 0,

and by construction H_{∗}^{1}(ℙ^{5}, I_{X}) ≅ H_{∗}^{1}(ℙ^{5}, E^{1}), so that E^{1} is stably equivalent to the E we are looking for.

Remember that we get E simply by adding a direct sum of line bundles in order to get a minimal generation for the quotient module Γ_{∗}(ℙ^{5}, I_{X})/Γ_{∗}(ℙ^{5}, E^{1}).

\underline{K^{2} = 15, p_{g} = 6, q = 0.} In this case H_{∗}^{1}(ℙ^{5}, I_{X}) is of length 1, thus it is ℂ(−2), whose minimal resolution is the Koszul complex. Therefore the first syzygy module, yields, when sheafified, the vector bundle Ω_{ℙ}^{1}(−2).

Since h^{0}(ℙ^{5}, I_{X}(3)) = 4 by the maximal rank assumption, we need to set E = Ω_{ℙ}^{1}(−2) ⊕ O_{ℙ}(−3)^{⊕4} ⊕ L, where L is a direct sum of line bundles of degree ≤ −4. Then r = rank(E) = 9 + 2t, where t = rankL, but on the other hand −c_{1}(E) ≥ 28 + 8t. Since 7(r − 1) = −2c_{1}(E), 28 + 7t ≥ 28 + 8t, which is only possible if t = 0. Therefore E = Ω_{ℙ}^{1}(−2) ⊕ O_{ℙ}(−3)^{⊕4}, and conversely if X is a corresponding Pfaffian surface, we get

h^{0}(ℙ^{5}, I_{X}(3)) = 4,
h^{0}(ℙ^{5}, I_{X}(m)) = 0 for m = 1, 2,
h^{1}(ℙ^{5}, I_{X}(2)) = 1,
h^{1}(ℙ^{5}, I_{X}(1)) = 0,

thus p_{g} = 6 and K^{2} = 15.

F. Schreyer has pointed out to us that this X is a linear section Y ∩ℙ^{5}, where Y is a 6-dimensional Fano manifold of degree 15 in ℙ^{9}, occurring as the Pfaffian locus of an antisymmetric map Ω_{ℙ}^{8}(8) ⟶ Ω_{ℙ}^{1}(1). Recall that, by Beilinson’s lemma,

Hom(Ω_{ℙ}^{8}(8), Ω_{ℙ}^{1}(1)) = Λ^{7}(V) ≅ Λ^{3}(V)^{∨},

and this isomorphism is well understood once we consider the two Euler sequences

0 ⟶ Ω_{ℙ}^{1}(1) ⟶ V^{∨} ⊗ O_{ℙ} ⟶ O_{ℙ}(1) ⟶ 0
0 ⟶ O_{ℙ}(−2) ⟶ V ⊗ O_{ℙ}(−1) ⟶ Ω_{ℙ}^{1}(1)^{∨}(−1) ⟶ 0,

since Hom(Ω_{ℙ}^{8}(8), Ω_{ℙ}^{1}(1)) maps to Hom(V ⊗ O_{ℙ}(−1), V^{∨} ⊗ O_{ℙ}) via composition. Indeed this point of view shows that any such homomorphism is antisymmetric.

\underline{K^{2} = 16, p_{g} = 6, q = 0.} In this case H_{∗}^{1}(ℙ^{5}, I_{X}) is of length 2, thus it is ℂ(−2)^{⊕2}, on the other hand h^{0}(ℙ^{5}, I_{X}(3)) = 1, whence by an analogous reasoning we infer that

E = Ω_{ℙ}^{1}(−2)^{⊕2} ⊕ O_{ℙ}(−3).
\underline{K^{2} = 17, p_{g} = 6, q = 0.}

In this case H_{∗}^{1}(ℙ^{5}, I_{X}) is of length 3 in degree 2, and of length 2 in degree 3. This module belongs to a class of modules among which ℂ(−2)^{⊕3} ⊕ ℂ(−3)^{⊕2} is the only one which is killed by the maximal graded ideal of A. If the module M is generic, again in the sense that multiplication is of maximal rank, we obtain an isomorphism of the degree 1 graded piece of A with the space of linear maps between H^{1}(ℙ^{5}, I_{X}(2)) = U′ and H^{1}(ℙ^{5}, I_{X}(3)) = W′.

h^{0}(ℙ^{5}, I_{X}(4)) = 17, whence the minimal free resolution of M runs as follows:

A(−4)^{⊕33} ⟶ A(−3)^{⊕16} ⟶ A(−2)^{⊕3} ⟶ M ⟶ 0,

where indeed these maps can be very explicitly described. We get E already as the first syzygy sheaf, since h^{0}(ℙ^{5}, E(4)) = 33.

The generic module M has a very precise presentation: M has generators e_{1}, e_{2}, e_{3} in degree 2 and A has generators x_{i,j} (i = 1, 2, j = 1, 2, 3) such that x_{i,j}e_{j} = x_{i,1}e_{1} for each i, j ≠ 1 (4 relations), while x_{i,j}e_{h} = 0 for j ≠ h (12 relations).

It is in fact easy to see that M has dimension 2 in degree 3 and is zero in degree 4 (every monomial containing x_{i,j} kills e_{h} if j ≠ h, whereas if we take any j ≠ 1, then x_{i,1}x_{i′,1}e_{1} = x_{i,1}x_{i′,j}e_{j} = x_{i′,j}x_{i,1}e_{j} = 0).

By our choice, our E sits as a kernel, E^{∨}(−7) as a cokernel:

0 ⟶ E ⟶ O_{ℙ}(−3)^{16} ⟶^{^{t}i} O_{ℙ}(−2)^{3} ⟶ 0

0 ⟵ E^{∨}(−7) ⟵ O_{ℙ}(−4)^{16} ⟵^{i} O_{ℙ}(−5)^{3} ⟵ 0.

By composition, giving α is equivalent to giving its lift

a : O_{ℙ}(−4)^{16} ⟶ O_{ℙ}(−3)^{16},

such that

i) ^{t}i ∘ a = 0, a ∘ i = 0. Moreover, then ii) α = −^{t}α iff a = −^{t}a. But then, if a = −^{t}a, from ^{t}i ∘ a = 0 follows also a ∘ i = 0. We seek therefore for antisymmetric matrices a of linear forms such that ^{t}i ∘ a = 0: our surface X will then be defined by the (12 × 12) Pfaffians of a.

We are going to show that there are plenty of such matrices a, omitting for the time being the verification that for a general such matrix the Pfaffian locus is a smooth surface (which is then canonical).

We let a basis of O_{ℙ}(−3)^{16} be given by F_{i,jh} (j ≠ h) F_{i,jj} (j ≠ 1), thus, using the dual basis we can write a as:

a = ∑_{i,jh≠11;i′,j′h′≠11} a[i, jh; i′, j′h′](x)F_{i,jh} ⊗ F_{i′,j′h′}^{∨}.

Let E_{1}, E_{2}, E_{3} be a basis of O_{ℙ}(−2)^{3} corresponding to the generators e_{1}, e_{2}, e_{3} for M.

Then the condition ^{t}i ∘ a = 0 reads out as (for each i′, and j′h′ ≠ 11)

∑_{i,jh≠11} a[i, jh; i′, j′h′](x)x_{ij}E_{h} − ∑_{i,j≠1} a[i, jj; i′, j′h′](x)x_{i1}E_{1}.

In turn, this means that (for each h ≥ 2, for each i′, and j′h′ ≠ 11)

∑_{i,j} a[i, jh; i′, j′h′](x)x_{ij} = 0 (18)

(for each i′, and j′h′ ≠ 11)

∑_{i,j≠1} a[i, j1; i′, j′h′](x)x_{ij} − a[i, jj; i′, j′h′](x)x_{i1} = 0. (19)

Luckily enough, both (18) and (19) are Koszul relations. Namely, recall that ∑_{β} a_{β}x_{β} = 0, where the a_{β}’s are linear forms, if and only if there is an antisymmetric matrix of constants, c_{αβ}, such that a_{α} = ∑_{β} c_{αβ}x_{β}.

Then there are constants c_{i,j|i”j”}[h; i′, j′h′] which are alternating in the pair of (i, j) and (i”, j”) and in the pair of [i, jh] and [i′, j′h′], and constants d_{i,j|i”j”}[i′, j′h′] such that

a[i, jh; i′, j′h′](x) = ∑_{i”,j”} c_{i,j|i”j”}[h; i′, j′h′]x_{i”j”} (20)

a[i, j1; i′, j′h′](x) = ∑_{i”,j”} d_{i,j|i”j”}[i′, j′h′]x_{i”j”} − a[i, 22; i′, j′h′](x) − a[i, 33; i′, j′h′](x)

= ∑_{i”,j”} d_{i,1|i”j”}[i,′ j′h′]x_{i”j”}. (21)

It is tedious but not difficult to see that the antisymmetry of a imposes few more linear equations on the above constants, and we get an explicit vector space of solutions, but in order to show that some corresponding Pfaffian locus is a smooth surface requires harder work.

F. Schreyer has a slightly different approach to this case, using the higher order syzygies of M.

\underline{K^{2} = 14, p_{g} = 7, q = 0.} h^{1}(ℙ^{5}, I_{X}(1)) = 1, h^{1}(ℙ^{5}, I_{X}(2)) = K^{2} − 13 = 1, whence the generic rank case is the one where we have a module M with 1 generator e in degree 1, and subject to relations x_{i}e = 0 for i = 1, …, .5, while x_{6}^{2}e = 0.

Let E be the bundle kernel of the relations map (first-syzygy bundle):

0 ⟶ E ⟶ O_{ℙ}(−2)^{5} ⊕ O_{ℙ}(−3) ⟶^{^{t}i} O_{ℙ}(−1) ⟶ 0,

thus

0 ⟵ E^{∨}(−7) ⟵ O_{ℙ}(−5)^{5} ⊕ O_{ℙ}(−4) ⟵^{i} O_{ℙ}(−1) ⟵ 0.

Since h^{0}(ℙ^{5}, E(3)) = 10, h^{0}(ℙ^{5}, I_{X}(3)) = 6, from the maximum rank assumption we expect E to be the required odd rank bundle.

It suffices as before to find an antisymmetric matrix a:

a : O_{ℙ}(−5)^{5} ⊕ O_{ℙ}(−4) ⟶ O_{ℙ}(−2)^{5} ⊕ O_{ℙ}(−3)

such that ^{t}i ∘ a = 0 .

If we try the same maximal rank assumption for the next case K^{2} = 15, we get a bundle E of rank 6, so this assumption cannot be made.

5. Canonical projections in 3-dimensional space.

Recall the situation we outlined in the introduction: φ : S ⟶ Y ⊂ ℙ^{3} is an almost generic canonical projection, i.e. φ is given by a (4)-dimensional subspace V^{∗} of H^{0}(O_{S}(K)) without base points, so φ is a morphism which we assume to be birational onto Y (there is also an analogous theory if φ has degree 2, cf. [Cat3] and [C-S]).

Thus φ is obtained by composing the canonical map φ_{1} with a linear projection (assuming of course p_{g} ≥ 4), and if we know that φ_{1} is an embedding, we can make the

Classical assumption (cf. [En]): Y has ordinary singularities.

Recall that the ordinary singularities are exactly

1) a double curve Γ which is smooth except for t triple points with independent tangents

2) ν pinch points

3) t triple points for Y (which are also triple for Γ)

1) at the smooth points of Γ there are two local parameters x, y such that Γ = {x = y = 0}, while the equation f of Y is formally equivalent to xy = 0 (i.e., in the completion of the local ring it can be written as u · xy, where u is a unit)

2) at the pinch points there are local parameters x, y, z such that f is formally equivalent to x^{2} − zy^{2} = 0

3) at the pinch points the formal equation is xyz = 0 (thus Γ is formally equivalent to {xy = yz = xz = 0}).

Let d be the degree of Y (thus d = K^{2}): then the basic subadjunction formula (which, in this particular case, follows right away by blowing up first the triple points and then the proper transform ofΓ) is

φ_{∗}K_{S} = I_{Γ}O_{Y} ((d − 4)).

Let H be the hyperplane divisor, so that φ^{∗}(H) = K. The basic calculation done by Enriques goes as follows.

Let D be φ^{−1}(Γ) (note that D maps to Γ 2:1), and reread the subadjunction formula as

|K_{S}| = |φ^{∗}(d − 4)H − D|.

Since φ^{∗}|H| ⊂ |K_{S}|, the above equation implies that D + φ^{∗}|H| is a subseries of |φ^{∗}(d − 4)H|, thus

\underline{Existence of a unique minimal adjoint:} There exists a unique surface F of degree (d − 5) such that φ^{∗}(F) = D. F will be referred to as the \underline{adjoint surface}.

In the classical situation, since F pulls back precisely to D, follows that F is smooth at the smooth points of Γ.

Moreover, at the triple points, we get (multiplying x, y, z by a suitable non zero constant) that F ≡ xy(mod z), F ≡ xz(mod y), F ≡ yz(mod x).

Thus F ≡ xy + yz + xz(mod xyz), whence F has a node (ordinary quadratic singularity) at the triple point of Y. Moreover, since

2Γ = F · Y

follows that deg(Γ) = \frac{d(d−5)}{2}. Let p_{1}, …, p_{t} be the triple points and let us blow up them, so that we get ℙ^{∼} with exceptional divisors P_{i} ≡ ℙ^{2}: the proper transform F^{∼} of F is linearly equivalent to (d − 5)π^{∗}H − 2 ∑ P_{i}. Moreover, F^{∼} intersects the ℙ^{2}P_{i} in a conic A_{i}: whence A_{i} ≅ ℙ^{1} and with self intersection −2.

Moreover, since the full transform of Y is dπ^{∗}H − 3 ∑ P_{i}, we get on F^{∼} the linear equivalence (Γ^{∼} = proper transform of Γ)

dπ^{∗}H ≡ π^{∗}Y ≡ 2Γ^{∼} + 3 ∑ A_{i}.

Setting δ ≡ d(mod 2), δ = 0 or 1, we get

δπ^{∗}H + ∑ A_{i} ≡ 0(mod 2)

and therefore there exists a double cover of F^{∼} branched on δπ^{∗}H + ∑ A_{i}.

We therefore find that the set N = {p_{1}, …, p_{t}} is a δ/2-even set of nodes according to the definition given in section 2. If d is even, there is therefore a double cover M of Y branched exactly on N.

A particular case occurs when d = 10, because then M also has K^{2} = 2·5 = 10. Enriques pointed out that considering the pencil λY + μF^{2}, for λ tending to zero, we obtain that the double cover M is a deformation of Y, and indeed the first construction of surfaces with K^{2} = 10, p_{g} = 4, q = 0, and φ_{1} birational was done deforming the double cover M ([Cat1]).

But one can indeed do more, and construct Y starting from F and the δ/2-even set of nodes N. In fact, if we have Y with ordinary singularities and Γ as double curve, and Y · F = 2Γ, then for the normalization S of Y (which is smooth), we obtain:

K_{S} = φ^{∗}(d − 4)H − F = φ^{∗}H,

and φ is an almost generic canonical projection.

Ciliberto ([Cil1]) proceeded to construct canonical surfaces with K^{2} = 6, …, 10, p_{g} = 4, q = 0, and with ordinary singularities using the following method.

Start from F and look for a section of the line bundle O_{F^{∼}}(L + \frac{d−δ}{2} ), where 2L ≡ δπ^{∗}H − ∑ A_{i}, which vanishes on ∑ A_{i}. Since L · A_{i} = 1, and A_{i} ≅ ℙ^{1}, we have to impose 2t linear conditions, and then we obtain a section whose divisor can be written as Γ^{∼} + ∑ A_{i}. Squaring this section, and observing that the sections of dπ^{∗}H are induced by the ambient space, we obtain a surface Y′ such that π^{∗}Y′ ≡ 2Γ^{∼} + 3 ∑ A_{i}.

Since Y and Y′ cut the same divisor on F, it follows right away that our surface Y must be of the form Y′ + F · B, where B is a degree 5 surface (warning about the argument: if Y exists, with double curve the given Γ, then the above holds; but, for d ≥ 11, it may happen that Y exists only for very special choice of Γ).

There remains the final issue of having Y with ordinary singularities, and this is split into two steps:

• (Closed conditions) Show that the B’s such that Y = Y′ + F · B is singular along Γ form an algebraic set C of positive dimension.

• (Open conditions) Verify that a non empty open set of C yields Y ’s with ordinary singularities.

Remark 5.1. i) The dimension of C can be calculated simply observing that if Y has more than a certain number k of double points along Γ, then Y has Γ as a double curve (k is the degree of a line bundle on the normalization of Γ, corresponding to the ‘double structure’, namely to I_{Γ}/(I_{Y′} + I_{F})). ii) A very degenerate solution is provided by any surface Y′′ whose equation is divisible by F^{2} (here the condition d ≤ 10 obviously comes in), and looking in a neighbourhood of Y′′ in C one can show that the general Y has ordinary singularities.

5.1. Going beyond the assumption of ordinary singularities needs some algebraic counterpart of the picture.

Case of regular surfaces q = H^{1}(O_{S}) = 0

The almost generic canonical projection φ : S ⟶ Y ⊂ ℙ^{3} determines a ring homomorphism

φ^{∗} : A = ℂ[y_{0}, y_{1}, y_{2}, y_{3}] = Sym(V^{∗}) ⟶ R = ∑ H^{0}(O_{S}(mK)),

which obviously factors through the quotient ring A/(Y), Y being an equation of Y. We view now R as an A- module (whose associated sheaf is φ_{∗}O_{S}), and the condition that S is regular entails, by Kodaira vanishing for m ≥ 2, the vanishing of H^{1}(O_{S}(mK)) for all m, therefore it follows easily that

R is a Cohen-Macaulay A-module.

As such, once we take a minimal set of generators, 1 = v_{1}, v_{2}, v_{3}, …, v_{h}, we find that R has a length 1 minimal free resolution

0 ⟶ L^{1} ⟶^{α} L^{0} ⟶ R ⟶ 0.

Moreover, the fact that φ^{∗}O(1) = O_{S}(K) implies, with some work, that R is Gorenstein and we can take α to be a symmetric matrix.

Thus if φ is birational onto Y:

(1) α = ^{t}α

(2) det(α) = Y, which is an irreducible polynomial of degree K^{2}. Now, we have to use the fact that R is a ring, thus the relations given by α, keeping in mind that 1 = v_{1}, can be written a system of non-homogeneous linear equations

∑_{j=2}^{h} α_{ij}(y)v_{j} = −α_{i1}(y) (for i = 1, …, h). (22)

It is not entirely straightforward, but it holds true as in linear algebra over a field (so called theorem of Rouché-Capelli) that then holds true the following \underline{Rank Condition = RC:} The rank of the incomplete matrix α′ = (α_{ij}(y)) (i = 1, …, h, j = 2, …, h) equals the rank of α, in the precise sense that, setting β = Λ^{h−1}(α),

(3) the ideal (β_{ij}(y)) = Ideal(Λ^{h−1}(α)) equals the a priori smaller ideal I = Ideal(Λ^{h−1}(α′))

I = (β_{11}(y), β_{12}(y), …, β_{1h}(y)).

More concretely, there do exist polynomials λ_{ij}^{k}(y) such that (3’) β_{ij}(y) = ∑_{k=1}^{h} λ_{ij}^{k}(y)β_{1k}(y). Moreover (see [Cat3, 4] for details)

(4) I = (β_{11}(y), β_{12}(y), …, β_{1h}(y)) is the ideal of the double curve Γ

(5) β_{11}(y) = F, the unique adjoint surface of minimal degree

(6) The canonical model X of S is the blow up of Y in the ideal I which, more concretely, means

(7) R is generated by y_{0}, y_{1}, y_{2}, y_{3}, and v_{2}, v_{3}, …, v_{h}, subject to the relations

∑_{j=2}^{h} α_{ij}(y)v_{j} = −A_{i1}(y)

and

v_{i}v_{j} = ∑_{k=1}^{h} λ_{ij}^{k}(y)v_{k}(y)

(here v_{1} = 1). In particular, the above are the equations which define X = P roj(R).

The upshot is that a converse holds (in fact, as we noticed earlier, the theorem holds more generally for an almost generic canonical projection to a 3-dimensional weighted projective space)

Theorem 5.2 (Theorem of canonical projections, [Cat3], [Cat4]). To give an almost generic canonical projection φ : S ⟶ Y ⊂ ℙ^{3}, together with a choice of a minimal system of generators of R as an A-module, is equivalent to giving a matrix α of polynomials such that (1), (2), (3) hold, and moreover such that the scheme X defined as in (7) has at most R.D.P.’s as singularities.

De Jong and van Straten ([dJ-vS]) noticed that a similar result holds also in the Cohen-Macaulay, but not necessarily Gorenstein case (then α is not symmetric). \underline{RC = Ring condition:} Set A_{Y} = A/(Y) (Y = det(α)), and let C ⊂ A_{Y} be the conductor ideal C := Hom_{A_{Y}} (R, A_{Y}).

Lemma 5.3. Let A_{Y} ⟶ R be an injection of integral domains having the same field of fractions. Then, if C ⊂ A_{Y} is the conductor ideal C := Hom_{A_{Y}} (R, A_{Y}), then Hom_{A_{Y}} (C, A_{Y}) = Hom_{A_{Y}} (C, C).

\underline{Proof}. It suffices to show that if φ(C) ⊂ A_{Y}, then φ(C) ⊂ C. I.e., that if γ is in C, then φγ is in C; to show this, take f in R: then (γf) is in A_{Y}, but indeed, C being an ideal in R, (γf) is in C, thus φγf is in A_{Y}. q.e.d.

Under the relative Cohen-Macaulay assumption, Ext_{A}^{i}(R, A) = 0 except for i = dimA − dimR = 1, it follows that R ≅ Hom_{A_{Y}} (C, A_{Y}), and if the matrix α yields a resolution for the A-module R, then ^{t}α yields a resolution for the module C, and the relatively Gorenstein case (where moreover Ext_{A}^{1}(R, A) ≅ R) is the one where α can be chosen to be symmetric.

Another way of putting the structure theorem is that R.C. is equivalent, given an A-module R, which is relatively Cohen-Macaulay of codimension 1, to the condition

Hom_{A_{Y}} (C, A_{Y}) = Hom_{A_{Y}} (C, C).

Since R ≅ Hom_{A_{Y}} (C, A_{Y}), if R.C. holds, the above isomorphism gives a commutative ring structure to R.

Remark 5.4. The theorem on canonical projections works in the weighted case and also in higher dimension (cf. [Ca5]), provided we get vanishing of the pluri irregularities, H^{i}(X, O) = 0 for 0 < i < dimX. In fact Reid ([Reid1]) pointed out that the above is a structure theorem for a ring map A ⟶ R which is relatively Gorenstein of codimension 1 (i.e., dimA − dimR = 1), for A a polynomial ring or also a regular local ring. These assumptions on A are indeed not needed, as later shown by M. Grassi ([Gra]) and Kleiman-Lipman-Ulrich ([K-L-U]). In the next section we shall deal with the relatively Gorestein maps of codimension 2, following [Gra].

There remains, though, to fully describe the geometrical analogues of the determinantal description, and to illustrate the algebraic analogue of Ciliberto’s geometric method. Let α′ be the incomplete matrix α′ = (α_{ij}(y)) (i = 1, …, h j = 2, …, h) whose rows are (by the Hilbert-Burch theorem) the relations among the polynomials β_{11}(y), β_{12}(y), …, β_{1h}(y) which generate the ideal I of the ”Double curve” subscheme Γ. The minor α′′ = (α_{ij}(y)) (i = 2, …, h, j = 2, …, h) has as determinant the minimal adjoint surface F.

We have three further determinantal subschemes:

{x| rank(α) ≤ h − 3} ⊂ {x| rank(α′) ≤ h − 3} ⊂ {x| rank(α′′) ≤ h − 3}.

In the case of ordinary singularities, these are exactly the triple points of Y, the triple points of Γ, and the nodes of F, and we know already that in this case the three subschemes are the same.

Indeed this holds more generally provided R.C. holds, and provided the ideal I defines a codimension 2 pure subscheme (cf. [Cat3], where the result was proven under the assumption that the above 3 subschemes are of codimension 3, and [M-P]).

5.2. Example: surfaces with K^{2} = 7, p_{g} = 4, q = 0, and |K| without base points. Here the canonical map provides, 7 being prime, the almost generic canonical projection, and the matrix α belongs to a parameter space which is irreducible and rational. There is a Zariski open set where the matrix α takes the following form:

( \begin{matrix}d_{1}d_{2}x + (d_{3}d_{4} + d_{2}^{2})y + (d_{2}d_{3} + d_{1}d_{4})z & d_{4}y & d_{1}x + d_{2}y + d_{3}z \\ d_{4}y & x & z \\ d_{1}x + d_{2}y + d_{3}z & z & y \end{matrix} )

Here, d_{1}, d_{2}, d_{3} and d_{4} are arbitrary quadratic forms in the homogeneous coordinates (x, y, z, w). In fact, from the above equality of subschemes (where the rank is 0) follows that all entries belong to the ideal (x, y, z), and then by row operation one can achieve that the lower matrix α′ is in the given form. Finally, R.C. forces α_{11} = c_{1}x + c_{2}y + c_{3}z to be in the given form. By examining further the case where |K| has base points (this has been done by I. Bauer [Ba] and F. Zucconi [Zu]) one arrives at the important consequence that the moduli space of the above surfaces is irreducible, unirational of the expected dimension 36. Problem: Do there exist canonical surfaces with p_{g} = 4, q = 0, and K^{2} ≥ 11? Do they exist such that their canonical image has ordinary singularities?

Here, one possible strategy is the one followed by Ciliberto, and which, in this more algebraic context, reads out as considering the following maps

{α| α satisfies R.C. } → {α| A satisfies R.C.2} → {α′| α′ satisfies R.C.2} → {α′′}.

Here, R.C.2 stands for the condition upon α (resp. α′):

(R.C. 2) Ideal(Λ^{h−2}(α)) = Ideal(Λ^{h−2}(α′′)).

The key point why Ciliberto was able to construct the surfaces up to K^{2} = 10 is that in this range all the above maps are dominant, and with fibres linear spaces.

Of course, the interest of the above fibration lies in the fact that it provides a tool to study all the canonical surfaces with fixed invariants: for a mere existence result it may still be useful (although cumbersome, cf.[Kod]) to consider deformations of surfaces with a canonical map which is not a birational morphism (cf.[C-S]). In turn, those can be constructed in many ways, for instance as the desingularization of singular covers of known surfaces (e.g., as in [En], as ‘double planes’).

5.3. Example: A surface with K^{2} = 22, p_{g} = 7, q = 0. Let us consider the following configuration of 4 circles in the real plane, C, C_{0}, C_{1}, C_{2} which, using a complex variable z and a primitive 3-rd root ϵ of 1 can be described as follows. Set C = {z| |z| = 2}, and, for i = 0, 1, 2, C_{i} = {z| |z − rϵ^{i}| = 2 − r}, so that C and C_{i} are tangent to each other in the point P_{i} = 2ϵ^{i}. If moreover we let r = 4(2 + 3^{1/2})^{−1}, then r|1 − ϵ| = 2(2 − r) and C_{i−1} is tangent to C_{i} at the point P′_{i} = \frac{r}{2}ϵ^{i−1}(1 + ϵ) = −\frac{r}{2}ϵ^{i+1}.

We may now take a general conic Q_{i} in the pencil spanned by C_{i} and the double line joining P_{i} and P′_{i}. Upon considering the projective closure, we obtain a curve B of degree 14 having 6 points of type (3, 3) (the P_{i}’s and P′_{i}’s) and two ordinary 4-uple points (at the two cyclic points at infinity).

The same construction specializes to the union of C, C_{0}, C_{1}, C_{2} with a circumscribed triangle T (at the 3 points P_{i}) and the star R of three lines joining the origin with P′_{i}.

Let S be the minimal desingularization of the double cover Z of ℙ^{2} branched on B. Z has ω_{Z} = O_{Z}(4H), where H is the divisor yielding the map to ℙ^{2}. Thus ω_{Z}^{2} = 32, thus K_{S}^{2} = 32 − 2 · 2 − 6 = 22; moreover the canonical system of S is given by the system of plane quartics passing through the 8 singular points of B. \underline{Claim}: the above system has projective dimension 6.

\underline{Proof}: Else, there would be a quartic containing Q_{1} as a component, thus a circle passing through the 4 remaining points P_{0}, P_{2}, P′_{0}, P′_{2}. q.e.d.

In particular, the canonical map of S has 6 base points (lying above the points of type (3, 3) and corresponding to their first infinitesimal neighbourhood), and yields a double cover of a rational surface of degree 8 in ℙ^{6}. Since our construction depends upon 3 parameters, while the dimension of the Kuranishi family of deformations of S has dimension at least 10χ − 2K^{2} = 36, a general deformation of S should be a canonical surface.

It is an amusing exercise to play with these configurations, and try to see to which numerical characters they give rise (the important feature of these examples is that the canonical system has many base points), for instance a good ratio between K^{2} and p_{g} is achieved by the following configuration:

5.4. Example of a surfaces with K^{2} = 40, p_{g} = 10, 8 base points for |K|. Take in the plane the six lines xy(x+y −1)(x+y +1)(x−y +1)(x−y −1) = 0, forming a square with two diagonals. Take both the inscribed and the circumscribed circle to the square, and the 4 inscribed circles to the 4 triangles into which the square is parted. The branch curve of the double plane will be the union of all these curves, thus it has 2 6-tuple, 4 4-tuple points, 8 (3, 3) points (plus negligible singularities).

We have not being playing enough e.g. to find a surface with p_{g} = 4 and K^{2} ≥ 11, but to our surprise we have found 4 configurations yielding

5.5. Surfaces with K^{2} = 10, p_{g} = 4, 4 base points for |K|. i) keep from the previous configuration the lines xy = 0, the 4 inscribed circles and any other conic

ii) keep from the previous configuration the lines xy = 0, the 4 inscribed circles, the line at infinity, the two lines joining the origin with the two cyclic points, and any other line iii) consider the 4 circles of the first example, C, C_{0}, C_{1}, C_{2} and

two sides of the triangle T plus two sides of the star R. iv) (omitted)

The above examples shed some light on the expected complexity of the

5.6. Classification of the surfaces with p_{g} = 4, q = 0. Let us recall the status of the classification, as the value of K^{2} increases. Before doing so, it is convenient to remind the reader of the basic inequalities (implying in particular the Noether inequality K^{2} ≥ 2p_{g} − 4).

Let us write the canonical system |K| as F + |M|, where F is the fixed part of |K|. Then K^{2} ≥(since K is nef) KF + KM ≥ (since M is free from fixed part, whence nef) KF + F M + M2 = KF + F M + b + degφdegY, Y being the image of S, and b being the number of base points of |M|, counted with multiplicity (b = ∑_{p also infinitely near} m_{p}^{2}).

Proposition 5.5. If the canonical image Y is a curve, then K^{2} ≥ 3p_{g} − 3.

\underline{Proof.} Let d be the degree of Y, so that d ≥ p_{g} − 1. Then, in a blow up S′ of S we get a strict transform M′ ≅ (M −∑_{p} m_{p}E_{p}), M′ ≈ dM′′, with K′·M′′ ≥ 2, since (M′′)^{2} = 0 and the genus of M′′ is at least 2. Then K^{2} ≥ KM = K′M′ +∑_{p} m_{p} ≥ 2d + ∑_{p} m_{p}.

On the other hand M^{2} = ∑_{p} m_{p}^{2} ≥ d^{2} (since M is composed of d components), thus ∑_{p} m_{p} ≥ d, and K^{2} ≥ 3d ≥ 3p_{g} − 3. q.e.d.

Therefore, as K^{2} ≤ 8, Y has to be a surface. In this case degY ≥ p_{g} − 2 in general, and in the particular case p_{g} = 4 we get

K^{2} ≥ KF + F M + b + degφdegY ≥ 4.

K^{2} = 4: this case was analyzed by Noether and Horikawa ([Noe], [Hor1]), gives rise to a double cover of an irreducible quadric, and yields an irreducible moduli space.

K^{2} = 5: this case was analyzed by Enriques and Horikawa ([En], [Hor2]), gives rise to a connected but reducible moduli space, with one component corresponding to 5-ic surfaces with RDP’s, the other to (the canonical map having a simple base point) a double cover of a smooth quadric.

K^{2} = 6: this case was analyzed by Enriques and Horikawa ([En], [Hor3]), gives rise to a reducible moduli space, but to our knowledge it is not known whether this moduli space is connected.

K^{2} = 7: this case was analyzed by Enriques, Ciliberto, myself, Bauer and Zucconi, and yields an irreducible moduli space. However the possibilities to examine are several, since one can have up to 3 base points.

K^{2} = 8: in this case the moduli space is not connected, as follows from recent work of P. Oliverio ([Ol]). In fact, the property that the canonical divisor K be divisible by 2 is a topological property of S (equivalent to the intersection form being even), thus in view of Ciliberto’s construction of canonical surfaces of degree 8 the result follows from

Theorem 5.6 (Oliverio). Any canonical model of a surface with p_{g} = 4, q = 0, K even, K^{2} = 8 is a complete intersection of type (6, 6) in a projective space ℙ(1, 1, 2, 3, 3). Moreover, S is even if and only if the canonical map has a quadric cone as image.

A complete classification for K^{2} = 8 is probably rather complicated, and our above examples (where the canonical image is a double cover of a rational cubic surface) seem to indicate that for K^{2} = 10 the situation should become much worse.

In spite of the many things we said, we did not yet contemplate the case where q > 0. So this is about the right time!

5.7. Canonical projections of irregular surfaces. Here, the canonical ring R is not a Cohen Macaulay A-module, therefore we may want to consider instead the sheaf

F = ψ_{∗}O_{X} = φ_{∗}O_{S}.

We will sometimes also consider ψ_{∗}O_{X}(m), with m = 2 or 3. Notice in fact that the Beilinson theorem, once we replace a sheaf F by a twist F(m), does not give a twist of the complex whose cohomology is F, but indeed a different complex.

Note that

H^{3}(ψ_{∗}O_{X}(i)) = 0 for all i,
H^{2}(ψ_{∗}O_{X}(i)) = H^{2}(O_{X}(iK)) = H^{0}(O_{X}(−(i − 1)K))^{∗} = 0 for i ≥ 2,
H^{1}(ψ_{∗}O_{X}(i)) = H^{1}(O_{X}(iK)) = H^{1}(O_{X}(−(i − 1)K))^{∗} = 0 for i ≠ 0, 1.

By looking at the Beilinson complex for m = 3 we can prove ([C-S]) the following

Proposition 5.7. If p_{g} ≥ 4 and |K| has no base points R is generated in degrees ≤ 3 as an A-module.

In the case m = 2 the Beilinson table is a square which is completely symmetric around its centre: the symmetry takes each vector space to its Serre dual and each d_{1}-differential to its Serre dual map.

\begin{matrix}0 & 0 & 0 & 0 \\ H^{2}(O_{X}(−K)) & H^{2}(O_{X}) & H^{2}(O_{X}(K)) & 0 \\ 0 & H^{1}(O_{X}) & H^{1}(O_{X}(K)) & 0 \\ 0 & H^{0}(O_{X}) & H^{0}(O_{X}(K)) & H^{0}(O_{X}(2K)) \end{matrix}

One sees moreover (cf. [C-S]) that the d_{1}-differential from H^{0}(O_{X}) ⊗ Ω_{ℙ}^{2}(2) ⟶ H^{0}(O_{X}(K)) ⊗ Ω_{ℙ}^{1}(1) is an isomorphism onto a subbundle contained in V^{∗} ⊗ Ω_{ℙ^{1}}(1) according to the exact sequence given by the Beilinson complex for O_{ℙ}(2)

0 ⟶ Ω_{ℙ}^{2}(2) ⟶ V^{∗} ⊗ Ω_{ℙ}^{1}(1) ⟶ Sym^{2}(V^{∗}) ⊗ O_{ℙ} ⟶ O_{ℙ}(2) ⟶ 0.

Since (φ being assumed to be almost generic the image Y is not contained in a quadric, we infer that A_{2} = H^{0}(O_{ℙ}(2)) = Sym^{2}(V^{∗}) is a direct summand of H^{0}(O_{X}(2K)), so we can choose a splitting H^{0}(O_{X}(2K)) ≅ Sym^{2}(V^{∗})⊕U^{∗}. Moreover, we can replace the Beilinson complex by a homotopic one, which gives a new diagram.

\begin{matrix}0 & 0 & 0 & 0 \\ (U ⊗O_{ℙ}(−1))⊕O_{ℙ}(−3) & W ⊗Ω_{ℙ}^{2}(2) & 0 & 0 \\ 0 & H^{1}(O_{X})⊗Ω_{ℙ}^{2}(2) & H^{1}(O_{X}(K))⊗Ω_{ℙ}^{1}(1) & 0 \\ 0 & 0 & W^{∗}⊗Ω_{ℙ}^{1}(1) & (U^{∗}⊗O_{ℙ})⊕O_{ℙ}(2). \end{matrix}

Let (W^{∗} being a supplementary subspace to V^{∗} in H^{0}(O_{X}(K))) E be the vector bundle on ℙ^{3} such that:

E(2) = (U^{∗} ⊗ O_{ℙ}) ⊕ (H^{1}(O_{X}(K)) ⊗ Ω_{ℙ}^{1}(1)) ⊕ (W ⊗ Ω_{ℙ}^{2}(2)). (23)

Then F = φ_{∗}O_{S} admits a locally free resolution of length 1

0 ⟶ (O_{ℙ} ⊕ E)^{∨}(−5) ⟶ O_{ℙ} ⊕ E ⟶ F = φ_{∗}O_{S} ⟶ 0. (24)

Now, the two locally free terms of the resolution are dual to each other up to a twist, and we can see that one can indeed achieve (if charℂ ≠ 2) that the resolution itself is given by a symmetric map

α : (O_{ℙ} ⊕ E)^{∨}(−5) ⟶ O_{ℙ} ⊕ E (that is, α = α^{t} = α^{∗}(−5)).

Note that the matrix α is a block matrix with entries as indicated below:

entries of α O_{ℙ}(−3)⊕ U ⊗ O_{ℙ}(−1)⊕ H^{0,1} ⊗ Ω_{ℙ}^{2}(2)⊕ W^{∗} ⊗ Ω_{ℙ}^{1}(1)
O_{ℙ}(2)  S_{5}V^{∗}  S_{3}V^{∗}  H^{0}(Λ^{2}J_{ℙ})  H^{0}(J_{ℙ}(1))
⊕
U^{∗} ⊗ O_{ℙ} S_{3}V^{∗} V^{∗} ≅ Λ^{3}V Λ^{2}V V
⊕
H^{2,1} ⊗ Ω_{ℙ}^{1}(1) symm Λ^{2}V V 0
⊕
W ⊗ Ω_{ℙ}^{2}(2)  symm  V  0  0

Note that the h_{0,1} × h_{2,1} block is actually skew-symmetric, since it is induced from wedge-product

H^{1}(O_{X}) × H^{1}(O_{X}) ⟶ H^{2}(O_{X}) = H^{0}(O_{X}(K))^{∗}

composed with the projection

H^{0}(O_{X}(K))^{∗} ⟶ H^{0}(O_{ℙ}(1))^{∗} = V.

However each element of

V ≅ Hom(Ω_{ℙ}^{2}(2), Ω_{ℙ}^{1}(1)) ≅ Hom((Ω_{ℙ}^{1}(1))^{∗}, (Ω_{ℙ}^{2}(2))^{∗})

gives a skew-symmetric morphism of bundles. So the resulting morphism H^{0,1} ⊗ Ω_{ℙ}^{2}(2) ⟶ H^{2,1} ⊗ Ω_{ℙ}^{1}(1) is symmetric again.

We can also extend R.C. to the bundle case, namely given a morphism α = (α_{1}, α′) : G ⟶ O ⊕ E of vector bundles on a scheme Z

with r = rankE = rankG−1, we denote by I_{s}(α) the ideal sheaf of s×s minors of α. Suppose that det(α) = I_{r+1}(α) is a nonzero divisor and that depth(I_{r}(α′), O) ≥ 2. If Y ⊂ Z is the subscheme defined by det(α), then the following are equivalent (Ring Condition = Rank Condition):

(1) F = coker(α) carries the structure of a sheaf of commutative O_{Y^{−}} algebras with the identity given by the image of 1 ∈ Γ(Z, O) ⊂ Γ(Z, O ⊕ E) in F,

(2) I_{r}(α) = I_{r}(α′).

From all the above, and some work, follows the

Theorem 5.8. The datum φ : S ⟶ Y ⊂ ℙ^{3} = ℙ(V) of an almost generic canonical projection determines a morphism

(O_{ℙ} ⊕ E)^{∗}(−5) ⟶^{α} O_{ℙ} ⊕ E

where E = (K^{2} − q + p_{g} − 9)O_{ℙ}(−2) ⊕ qΩ_{ℙ}^{1}(−1) ⊕ (p_{g} − 4)Ω_{ℙ}^{2}, such that

1. α is symmetric,

2. det(α) is an irreducible polynomial defining Y,

3. α satisfies the rank condition R.C.

4. if F is equipped with the ring structure provided by R.C., then SpecF is a surface with at most rational double point singularities.

Conversely given α satisfying 1), 2), 3) and 4), then X = SpecF is the canonical model of a minimal surface S with the given invariants, and φ : S ⟶ Y ⊂ ℙ^{3} = ℙ(V) an almost generic (birational) canonical projection.

Notice that the new version of the theorem predicts a self dual resolution in the regular case q = 0 equivalent but different from the old one, which was clearly easier to handle.

Therefore we may restrict here to consider the case of an irregular surface. The first case to encounter is q = 1, when the Albanese map is a pencil a : S ⟶ E, with E elliptic.

Horikawa ([Ho5]) proved that for an irregular surface with K^{2} < 3χ the Albanese map has a curve as image, and the fibres are hyperelliptic of genus ≤ 3 (necessarily 2 if K^{2} < 8/3χ).

Thus, if we want S to be canonical, the first case will be K^{2} = 12, which can be completely classified ([C-S]). We are here just content of giving the main stream of those surfaces:

5.8. Example of canonical surfaces with K^{2} = 12, p_{g} = 4, q = 1. Let E be an elliptic curve of degree 4 in ℙ^{3}. Thus E is the complete intersection of two quadrics Q, Q′. There exists a degree 4 finite map λ from the symmetric cube E^{(3)} of E to the dual space (ℙ^{3})^{∨}, which associates to a degree 3 divisor on E, D = a + b + c, the 2-plane < a, b, c > spanned by it (in other words, the only hyperplane divisor H ≥ D, and therefore it is clear that for a general H there are exactly 4 degree 3 divisors D ≤ H).

For each point x in E, there is an embedding i_{x} : E^{(2)} ⟶ E^{(3)} (associating to a divisor D′ the divisor D′ + x), whose image surface will be denoted by Δ (notice that the choice of x alters Δ only up to an automorphism of E^{(3)}).

Finally, we choose our surface S inside the linear system |4Δ|. This choice is appropriate because by adjunction the canonical map φ of S is then just the composition of the inclusion of S in E^{(3)} with λ.

An interesting feature of this construction is that the map associating to S the set of triple points of its canonical image Y (up to projective equivalence) is generically 1-1.

Proceeding with higher irregularity, we encounter two cases: a1) the Albanese image is a curve a2) the Albanese image image is a surface

In case a1), simply by imposing that det(α) is an irreducible (non zero) polynomial, one obtains (since then the bilinear map H^{1}(O_{X}) × H^{1}(O_{X}) ⟶ H^{2}(O_{X}) is identically zero and we get a zero block in the matrix α) the inequality

K^{2} ≥ 2p_{g} + 4q − 3.

But, more generally, if f : S ⟶ B is a morphism to a curve B of genus b, with fibres of genus g, then necessarily K^{2} ≥ 8(b − 1)(g − 1) by Arakelov’s inequality (cf. [Be3]): we get thus a better inequality, at least for p_{g} = 4 (since g ≥ 3, we get 16 instead of 13 for q = 2, 32 for q = 3).

5.9. Example of canonical surfaces with K^{2} = 18, p_{g} = 4, q = 2. Let A be a principally polarized Abelian surface, and let D_{1}, D_{2}, D_{3}, be general translates of the theta divisor. Thus there exist divisors L_{i} (i = 1, 2, 3) such that, for each permutation i, j, k of {1, 2, 3}, 2L_{i} ≡ D_{j} + D_{k}. Therefore, as in [Ca5], we can take the (ℤ/2)^{2} Galois cover S of A branched on the three divisors D_{i}. We get a surface S with the above invariants, and in [C-S] is shown that the canonical map is birational onto its image.

Remark 5.9. The theory of canonical projections works also in the case where φ is of degree 2, e.g. when it is 2:1 onto a normal surface Y (cf. the theory of quadratic sheaves). An important example (discussed in [C-S]), with p_{g} = 4, q = 3, K^{2} = 12 is provided by the surfaces S in a polarization of type (1, 1, 2) of an Abelian 3-fold A.

6. Canonical surfaces in ℙ^{4}

We should warn the reader that the topic of canonical surfaces in ℙ^{4} is rather skew to the topic ”Surfaces in ℙ^{4}”, treated in the lecture by Schreyer: the only surfaces which belong here and there are just the smooth complete intersections of type (2, 4) or (3, 3).

In fact, the interest of articles like [E-P], [D-E-S] is rather oriented towards the classical problem of studying the smooth surfaces X contained in ℙ^{4}. These are linearly normal (i.e., h^{0}(O_{X}(1)) = 5), with the only exception of the Veronese surface, as proved by Severi in [Sev].

Moreover, being locally Gorenstein of codimension 2, they are the zero locus of the section of a rank 2 vector bundle if and only if they are subcanonical ([Ser2]).

In the non subcanonical case, they may be gotten as the degeneracy locus of a map of vector bundles u : E ⟶ V, where rank(V) = rank(E) + 1, and where (locally Gorenstein condition) the rank of u never drops by more than 2. In fact, it suffices to take V as a direct sum of line bundles surjecting onto I_{X}, and then the kernel will be a vector bundle, whose stable equivalence class up to twists will be an invariant of X (cf. [Rao2]: we refer to [D-E-S] for the relations occurring between E, V and the syzygy modules of the homogeneous ideal of X).

On the other hand, in the case of an almost generic canonical projection φ : S ⟶ Y, at best we can expect that Y has at most improper double points as singularities (two smooth branches intersecting transversally).

Their number δ is predicted by the Severi double point formula (cf. [Sev], [Cil2]), which states that, d being the degree of X:

δ = d(d − 5)/2 − 5(g − 1) + 6χ − K^{2},

where g is the genus of a hyperplane section of X, thus also

2δ = d(d − 10) − 5HK + 12χ − 2K^{2}.

If X is canonical, thus H = K, then

2δ = d(d − 17) + 12χ,

where now d = K^{2}. For the case δ = 0 to occur must therefore be d < 17 (and d ≥ 8 by Castelnuovo’s inequality) and 12χ = d(17 − d), what implies d ≡ 0, −1(mod 3), d ≡ 0, 1(mod 4). The only solutions (in view of Debarre’s improvement of Castelnuovo’s inequality K^{2} ≥ 3p_{g} + q − 7, cf. [De]) are: A) d = 8 or 9, q = 0, p_{g} = 5, B) d = 12, q = 1, p_{g} = 5. We shall soon see that case A) corresponds exactly to the complete intersections of type (2, 4) and (3, 3), whereas (in the old literature this was still considered a possible case)

Proposition 6.1. Case B) does not occur.

\underline{Proof}: Let F be a general fibre of the Albanese map: then, since K^{2} = 12 < 8/3(χ) = 40/3, it follows from the quoted result of Horikawa ([Ho5]) that F has genus equal to 2, whence φ cannot be birational onto its image. q.e.d.

Let us restrict henceforth to the case q = 0, p_{g} ≥ 5, and to the usual assumption that φ is almost generic; therefore R is a Gorenstein module, thus it has a length 2 resolution

0 ⟶ A(−6) ⊕ (L^{′0})^{∨}(−6) ⟶^{α_{1}} L^{1} ⟶^{α_{0}} A ⊕ L^{′0} ⟶ R ⟶ 0. (25)

where, if we set h − 1 = rank(L^{′0}), rank(L^{1}) = 2h. Since, as we saw (cf. also [Cat3], cor. 2.19), R is generated as a module in degree ≤ 3, we have that (assuming the resolution to be minimal, as usual)

L^{′0} = A(−1)^{p_{g}−5} ⊕ A(−2)^{n} ⊕ A(−3)^{k},

and (L^{1})^{∨}(−6) ≅ L^{1}, thus L^{1} is a direct sum of copies of A(−2), A(−3), A(−4).

For simplicity, and partly since we already mentioned several examples with p_{g} ≥ 6, let us make the

\underline{Assumption}: p_{g} = 5.

Then P_{2} = 1 + p_{g} + K^{2} = 6 + K^{2} = 15 + n if Y is not contained in a quadric, and = 14 + n otherwise (since this quadric is necessarily then unique).

\underline{Case 1)}: φ(S) = Y is contained in a quadric Q.

Then L^{1} = A(−2) ⊕ A(−3)^{a} ⊕ A(−4), and since α_{1} is injective and minimal, it follows that k ≤ 1. But indeed, if k = 1, then the rank of α_{1} (or of its dual) would drop on a hyperplane, whereas the rank drops exactly on Y: whence k = 0, and a = 2n. We look at the dimension of R_{3}, which we know to equal 1 + p_{g} + 3K^{2} = 6 + 3K^{2}: by the above sequence it also equals 35 + 5n − 5 = 5K^{2} − 10, therefore we obtain K^{2} = 8, and n = 0.

We have thus proven (the argument can be given more generally even without assuming φ to be birational):

Proposition 6.2. Let S be a surface with q = 0, p_{g} = 5, and |K| yielding a birational morphism φ. Then if the image Y of S is contained in a quadric Q, then Y is a complete intersection of type (2, 4), and is isomorphic to the canonical model of S.

\underline{Proof}: We have just shown that we have a surjection of A onto R, with kernel generated by A(−2)⊕A(−4), thus our given sequence is the Koszul sequence on two relatively prime homogeneous polynomials of respective degrees 2, 4. In particular Y is isomorphic to the canonical model X. q.e.d.

\underline{Case 2)} : φ(S) = Y is not contained in a quadric Q.

Then L^{1} = A(−3)^{a}, and the same argument as in case 1 shows that k = 0, a = 2n + 2, with n = K^{2} − 9.

Corollary 6.3. Let S be a surface with q = 0, p_{g} = 5, and |K| yielding a birational morphism φ. Then if the image Y of S is not contained in a quadric Q, then K^{2} ≥ 9, equality holding if and only if Y is a complete intersection of type (3, 3). In this last case Y is isomorphic to the canonical model of S.

We have in general the following resolution, which one can take to be self dual, as proven by M. Grassi ([Gra])

0 ⟶ A(−6) ⊕ A(−4)^{n} ⟶^{α_{1}} A(−3)^{2n+2} ⟶^{α_{0}} A ⊕ A(−2)^{n} ⟶ R ⟶ 0. (26)

The matrix α_{0} has a first row with entries cubic forms, while the other rows have linear forms as entries.

We shall write α_{0}, α_{1} as in [Gra], namely, α_{0} = (α, β), and α_{1} = ^{t}(−β, α), so that we have a complex as soon as γ = α ·^{t}β equals β ·^{t}α = ^{t}γ. The first interesting case is the case K^{2} = 10 (i.e., n = 1), which was considered by Ciliberto in [Cil2].

Here, we can write

α = ( \begin{matrix}A_{1} & A_{2} \\ a_{3} & a_{4} \end{matrix} ) , β = ( \begin{matrix}A_{3} & A_{4} \\ −a_{1} & −a_{2} \end{matrix} )

and the symmetry of γ amounts to the equality

a_{1} · A_{1} + a_{2} · A_{2} + a_{3} · A_{3} + a_{4} · A_{4} = 0. (27)

This equality says precisely that 2 of the six two by two minors of (α, β) yield two determinants whose sum is zero.

(**) Assume the linear forms a_{1}, a_{2}, a_{3}, a_{4} to be independent.

Then (27) is a Koszul relation which simply says that there exist 6 quadratic forms Q_{ij} (i < j) such that (setting Q_{ij} = −Q_{ji})

A_{i} = ∑_{j} Q_{ij}a_{j}. (28)

We want now to investigate the ring structure of R (ring condition). R is generated by generators 1 and v, and if B_{1}, B_{2}, B_{3}, B_{4} are just a_{3}, a_{4}, −a_{1}, −a_{2}, then we have relations

A_{i} + B_{i}v = 0.

Let C be the conductor of R (into A_{Y}): then C is generated by B_{1}, B_{2}, B_{3}, B_{4} as it is immediate to see (an element φ of A_{Y} lies in C iff there is a lift ψ of φ in A and an element η in A such that ψv + η = 0 in R).

Now v belongs to Hom(C, C) by (28), therefore we would get that R is a ring provided that R = Hom(C, A_{Y}).

We observe by the way that A_{Y} = A/(A_{i}B_{j} − A_{j}B_{i}).

In fact, A_{i} = −B_{i}v implies that A_{i}B_{j} = −B_{i}vB_{j} = A_{j}B_{i}. Moreover, if (P, 0) is a relation for 1, then there exist λ_{i} such that ∑_{i} λ_{i}B_{i} = 0, ∑_{i} λ_{i}A_{i} = P, whence λ_{i} = ∑_{j} μ_{ij}B_{j}, (with μ_{ij} = −μ_{ji}), and P = ∑_{i,j} μ_{ij}B_{j}A_{i}, as we wanted.

Now, we would obtain that the chain of inclusions

R ⊂ Hom(C, C) ⊂ Hom(C, A_{Y})

is a chain of equalities if R/A_{Y} would equal Hom(C, A_{Y})/A_{Y}.

This would be the case if we would know for instance that Y is a surface which has just an improper double point at the origin P the point which is the zero set of C (cf. (**)). In fact, the sheaf R^{∼} associated to R is different from O_{Y} and Hom(C^{∼}, O_{Y})/O_{Y} has length 1 (as we shall see in the following example 1A), thus firstly R^{∼} = Hom(C^{∼}, O_{Y}). Secondly R is Cohen Macaulay, and thus it equals the full module of sections of R^{∼}, which contains Hom(C, A_{Y}) and thus we have equality.

We draw from the above argument a first important conclusion: consider the complexes (26) with n = 1 and α, β as above satisfying (28). Then Y = supp(R) has codimension ≤ 2 and by a theorem of Buchsbaum and Eisenbud ([B-E1], cf. also [Ei], thm. 20.9 page 436) codimension Y = 2 iff the complex is exact. This condition defines an open set Ω in the parameter space for (α, β), namely {(Q_{ij}) | Q_{ij} = −Q_{ij}}. Certainly Y is singular at the origin and again the condition that the singularity is an improper double point is an open one (as we will see in the sequel).

Therefore we morally see that we get the ring conditions for free. We are going to show this more precisely by analyzing the situation in greater generality.

In fact, one can also write explicitly here the condition that R is a ring, since then one must have λ and μ such that

v^{2} = λ + μv

holds in the module R. This implies that v^{2}B_{i} = −A_{i}v = − ∑_{j} c_{ij}B_{j}v (where,

letting J the matrix ( \begin{matrix}0 & −I \\ I & 0 \end{matrix} ), we define the matrix (c_{ij}) by (c_{ij}) = (Q_{ij})J).

Whence

v^{2}B_{i} = ∑_{j} c_{ij}A_{j} = ∑_{jk} c_{ij}c_{jk}B_{k}

should equal λB_{i} − μ ∑_{j} c_{ij}B_{j}, i.e., the following holds in A_{Y}

(λI − μc − c^{2})^{t}(B_{1}, B_{2}, B_{3}, B_{4}) = 0. (29)

(Observe that if the matrix c = (c_{ij}), (or (Q_{ij})) were a matrix of constants, (29) would imply that c satisfies a second degree polynomial).

Indeed, one can normalize (29), replacing v by v − μ/2 (this replaces the upper row by a multiple of the second without destroying the basic relation (28), so that we might assume μ = 0. Then we would have

(λI − c^{2})^{t}(B_{1}, B_{2}, B_{3}, B_{4}) = 0 in A_{Y} . (30)

In any case, since (B_{1}, B_{2}, B_{3}, B_{4}) is a regular sequence, (29) means: there are antisymmetric matrices D^{i} = (D_{jk}^{i}) with entries cubic forms and G^{i} = (G_{ik}^{i}) with entries linear forms such that

(λI − μc − c^{2})_{ij} = ∑_{k} D_{jk}^{i} + ∑_{r} G_{jr}^{i}A_{r}

in out polynomial ring A. Equivalently,

(λI − μc − c^{2})_{ij} = ∑_{k} (D_{jk}^{i} + ∑_{r} G_{jr}^{i}c_{rk}) B_{k} (31)

(in our polynomial ring A).

The above equations describe a parameter space for the canonical rings of the given surfaces with K^{2} = 10; it is a projection of an open set of an explicit algebraic set, namely

{(Q_{ij}, λ, D_{jk}^{i}, G_{ik}^{i}) | (31) holds}.

Let us investigate (31) more carefully, writing (Q_{ij}) = ( \begin{matrix}aj & q \\ −^{t}q & bj \end{matrix} ), where j =

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Then

c = QJ = ( \begin{matrix}q & −aj \\ bj & ^{t} q \end{matrix} ) ,

c^{2} = ( \begin{matrix}q^{2} + abI & −a(qj + j^{t} q) \\ b(jq + ^{t} qj) & ^{t}(q^{2} + abI) \end{matrix} ) .

The stronger condition 0 = (−λI + μc + c^{2}) is equivalent to

\{ \begin{matrix}a(qj + j^{t}q + μj) = 0 \\ b(jq + ^{t} qj + μj) = 0 \\ q^{2} + μq + (ab − λ)I = 0. \end{matrix}

It is easy to find μ, λ which solve the above equations, since

i) qj + j^{t}q = qj − ^{t}(qj) is antisymmetric and equal to jq + ^{t} qj. Indeed both matrices equal (trace q) j, thus we simply take μ = −trace q. By Hamilton Cayley ii) q^{2}−(trace q)q + (det q)I = 0, thus it suffices to take λ = ab − det(q).

Therefore, for each choice of (Q_{ij}) in our affine space of parameters, we get a ring R defined by

\{ \begin{matrix}B_{i}v & = & − ∑_{j} c_{ij}B_{j} \\ v^{2} & = & (ab − det(q)) − (trace q)v. \end{matrix}

For (Q) in the open set Ω where the determinants of the corresponding entries of Λ^{2}(α, β) have no common divisor, we obtain a ring. We then obtain a surface of general type with K^{2} = 10, p_{g} = 5, q = 0 exactly when X = P roj(R) has at most rational double points as singularities. But indeed, there is an open set Ω′ ⊂ Ω such that X is smooth: this is an immediate consequence of Bertini’s theorem since our five equations define a variety of codimension 3 in Ω × ℙ(1, 1, 1, 1, 1, 2) whose singular set has big codimension.

There remains, for completeness, to consider the case where the four linear forms B_{1}, …, B_{4} are not linearly independent. We claim in this case that they generate a 3-dimensional vector space. Otherwise we can transform the matrix (αβ) in the form

( \begin{matrix}A_{1} & A_{2} & A_{3} & A_{4} \\ B_{1} & B_{2} & 0 & 0 \end{matrix} ) ,

and since Y is irreducible, non degenerate, of degree 10, Y ⊂ {A_{3} = A_{4} = 0}, thus A_{3} and A_{4} have a common quadratic factor. But then our ring R does not have a resolution as above.

In the remaining case we can combine α, β and obtain that for instance B_{2} = 0. Therefore we may assume that (αβ) has the form

( \begin{matrix}A_{1} & A_{2} & A_{3} & A_{4} \\ B_{1} & 0 & B_{3} & B_{4} \end{matrix} )

and (27) implies the Koszul relation (28), but just for indices i, j = 1, 2, 3.

One can analyze exactly as above the condition that R is a ring; after taking also care of the case where Y is contained in a quadric, one can reprove in this way the following

Theorem 6.4. (Ciliberto [Cil2]) Surfaces of general type with K^{2} = 10, p_{g} = 5, q = 0 and such that the canonical map is a birational morphism form an irreducible unirational moduli space of dimension 40.

Another possible algebraic approach to the above surfaces is to consider them as the split pfaffian locus in ℙ = ℙ(1, 1, 1, 1, 1, 2) of an antisymmetric 5 × 5-matrix. Namely, we get a resolution

0 → O_{ℙ}(−8) ⟶^{pf(α)} O_{ℙ}(−5)^{4} ⊕ O_{ℙ}(−4) ⟶^{α} O_{ℙ}(−3)^{4} ⊕ O_{ℙ}(−4) ⟶^{pf(α)} O_{ℙ} → O_{X} → 0

and α = −^{t}α. This is another explanation of the unirationality of the moduli space of our surfaces with K^{2} = 10, p_{g} = 5, q = 0.

The way Ciliberto obtained these surfaces was more geometric, using the method of linkage.

Observe in fact that det(α) = det(β) = 0 defines a complete intersection scheme Z which contains Y, and therefore splits as Z = Y ∪ F, where F is therefore a surface of degree 6. Since at the double point P of Y both determinants have degree 2, and since for a general choice in the projectivized tangent space we get a complete intersection of 2 quadrics containing 2 skew lines, we see that this complete intersection consists of 4 lines on a smooth quadric: therefore also F has an improper double point at P.

By the Cohen Macaulay property of Z, F ∩Y has codimension 1 in both Y and Z, and X is obtained by blowing up F ∩ Y in Y (this alters Y only at the point P where F ∩ Y is not Cartier). Ciliberto starts therefore from a rational surface F of degree 6 with an improper double point P. F is the image of the plane by the complete linear system |5L − 3P_{0} − P_{1} − ⋯ − P_{10}|, L being the class of a line, and the 11 points P_{0}, P_{1}, … P_{10} being general.

The linear system has clearly degree 6 and dimension 4 since on the blow up ℙ of the plane H^{1}(O_{ℙ}(5L−3P_{0}−P_{1}−⋯−P_{10})) = H^{1}(O_{ℙ}(−(8L−4P_{0}−2P_{1}−⋯−2P_{10}))) and there is a connected curve in |8L − 4P_{0} − 2P_{1} − ⋯ − 2P_{10}| (the above system has projective dimension equal to 4, and for general choice of the points the general element is irreducible).

The genus of the hyperplane section is 3, whence, by Severi’s formula, the image has one double point. Moreover the surface F is ruled by conics (image of |L−P_{0}|). Since the system |10L − 6P_{0} − 2P_{1} − ⋯ − 2P_{10}| has dimension 14 (the same as the system of quadrics in ℙ_{4}), if we show that it embeds ℙ, then it follows that F is contained in a quadric Q (singular at P).

We can also achieve the construction of F starting from this observation. In fact, since blowing up the point P we get a proper transform of F which intersects in 2 skew lines the exceptional divisor, it follows that the quadric Q has rank 4.

Thus Q is the image of the projective bundle ℙ′′ over ℙ^{1} × ℙ^{1}, given as ℙ(O ⊕ O(1, 1)), by the complete linear system |H|, where H is the hyperplane bundle of ℙ′′. Let us denote by L_{1}, L_{2} the pull backs of the classes on ℙ^{1} × ℙ^{1}, so that the exceptional divisor H_{∞} (since H · H = H · (L_{1} + L_{2})) is linearly equivalent to H − L_{1} − L_{2}.

Since F′′, the blow up of F, cuts on H_{∞} the divisor 2L_{2}, follows that F′′ ≡ aH + 2L2, and since H · H · F′′ = 6, we obtain a = 2, so that by adjunction the genus of the hyperplane section of F equals 3 (on ℙ^{1} × ℙ^{1} we get a curve of type (4, 2)).

Clearly the conic fibration is given by the system |L_{2}|. We can therefore start from a divisor F′′ in |2H + 2L_{2}|, take its image F in ℙ^{4}, which is a sextic rational surface with an improper double point at P, and then consider two general quartic hypersurfaces M′, M′′ containing F, and let Y be the residual component of the intersection.

The system of quartics containing F pulls back to the system on ℙ′′ |4H −F′′| = |2H −2L_{2}| = 2H_{∞} +|2L_{1}|, which has therefore projective dimension 2 and no fixed point outside H_{∞}. That Y is canonical follows from liaison arguments (cf. [P-S], [Cil2]): the crucial property is that M′, M′′ provide a map of sheaves N_{F} ⟶ O_{F} (4) ⊕ O_{F} (4) with cokernel of rank 1 concentrated on the curve C intersection of F with Y (thus, taking the first Chern class on F′′, we obtain C′′ ≡ 4H − 2L_{2}).

The final computation concerns the number of moduli of Y: first of all the linear system |F′′|, by the exact sequence |2H| ⟶ |2H + 2L_{2}| ⟶ |2H|_{2L_{2}}|, has projective dimension 14 + 6 + 6 − 1 = 25, whereas the automorphism group of Q has dim= 11.

On the other hand, the 4-ics containing Q are a projective space of dimension 17, and the 4-ics containing Y are a ℙ^{4} (look at the sequence (26) in degree 4, and notice that the inverse image of A_{4} under α_{0} has dimension 6, but, because of the symmetric form of α_{1} and α_{0}, it contains the image of A(−4)_{4}, whence it has vector dimension 5).

Therefore, the number of moduli is 14 (=25-11= moduli for F′′) + 32 (dimension of Grass(1, 17)= moduli for the choice of Y once F is given)-6(dimension of Grass(1, 4)=parameters for the F ’s yielding the same Y)= 40.

The above beautiful geometric construction cannot easily be repeated for K^{2} ≥ 11, since then Y is contained in the complete intersection of two hypersurfaces of degree K^{2} − 6, whence it is linked to a surface of higher degree than Y.

Equations completely analogous to the Ring Conditions (31) were provided by Grassi ([Gra]). Before stating Grassi’s characterization of Gorenstein morphisms in codimension 2, let us understand the underlying geometry in a few basic examples (this may be useful in order not to make wrong conjectures).

\underline{Example 1A:} Let R be the ring ℂ[x, y] ⊕ ℂ[z, t], which is the ring of the normalization of 2 planes meeting transversally in ℂ^{4}.

As a module over A = ℂ[x, y, z, t], it has two generators 1′ and 1′′, which are the identities of the two subrings yielding the direct sum.

A presentation of the module R is easily cooked up since we get relations x · 1′′ = y · 1′′ = z · 1′ = t · 1′ = 0, and the first syzygies are given by the transpose

of the matrix ( \begin{matrix}−y & x & 0 & 0 \\ 0 & 0 & −z & t \end{matrix} ).

To get the right presentation, we notice that 1 = 1′ +1′′, and we set v = 1′′ −1′ (we work in char ≠ 2), thus 2 · 1′′ = 1 + v. Now the matrix of relations has become

( \begin{matrix}x & y & t & z \\ x & y & −t & −z \end{matrix} ) and we can exchange columns till we get ( \begin{matrix}x & z & y & t \\ x & −z & y & −t \end{matrix} ),

and therefore the form (α, β) for the matrix of relations, and^{t}(−β, α) for the matrix of syzygies.

The ideal (det(α), det(β)) equals (xz, yt) which gives the union of Y (the union of the given two planes, with ideal (xz, yt, xt, yz) given by the 2 × 2 minors) with F, the union of the two planes x = t = 0 and y = z = 0.

The matrix γ = α ·^{t}β is symmetric, being ( \begin{matrix}xy + zt & xy − zt \\ xy − zt & xy + zt \end{matrix} ).

The ideal of (xy − zt, xy + zt), i.e. of (xy, zt) mod (xz, yt) yields precisely the ideal of F.

The ideal of Y ∩ F is the ideal (xy, xz, xt, yt, yz, tz) and we obtain the normalization X of Y by blowing it up, since e.g. we have 1′ = tz/(xy + tz).

The conductor C of R into A_{Y} = A/(xt, xz, yz, yt) is (x, y, z, t)/(xt, xz, yz, yt) = (x, y)ℂ[x, y] ⊕ (z, t)ℂ[z, t] and here R = Hom(C, A_{Y}) since if w = w′ + w′′ is in Hom(C, A_{Y}) w′x and w′y are regular, whence w′ is a polynomial, and similarly for w′′. It may be worthwhile to observe that Hom(ℂ[x, y], A/(xz, yt)) = (zt)/(xz, yt) = ℂ[z, t], showing that R is self linked through the complete intersection of the two hypersurfaces xz, yt.

\underline{Example 2A:} If more generally we take a 2 × 4 matrix of linear forms (α, β) satisfying the symmetry condition (27), and with the lower row given by 4 independent linear forms, whence such that (28) holds, with (Q_{ij}) an antisymmetric matrix of constants, we see that the zero locus of the determinants of its 2 × 2 minors define, for general choice of Q = (Q_{ij}), a codimension 2 subset which is the union of 2-planes.

The normalized rank condition (30) says that (λI − c^{2}) = 0, thus (since we have λ ≠ 0) up to multiplying the first row by a constant, that c^{2} = I.

Since c = QJ, if we write Q = ( \begin{matrix}aj & q \\ −^{t}q & bj \end{matrix} ) where j = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), then c =

( \begin{matrix}−q & ai \\ −bj & −^{t} q \end{matrix} ), and c^{2} = I amounts to b(jq +^{t}qj) = 0, a(j^{t}q +qj), q^{2} = (1−ab)I.

These equations amount to either trace(q) = 0 and (detq) = ab − 1, or q = ±I, a = b = 0 (if a = b = 0, then q^{2} = I, whence either trace(q) = 0, det (q) = −1, or q = ±I).

The latter case is the one where the two rows are proportional (thus it should be excluded), whereas in the former case the point a = b = 1, q = 0 again corresponds to the union of two transversal planes. We can go on with our linear algebra calculations, since if we do column operations setting A′ = diag{δ, γ}A, B′ = diag{δ, γ}B, with γ = ^{t}δ^{−1}, our basic relation (27) is preserved, and our matrix q is transformed into δqδ^{−1}.

Either q = 0, and then, since, by letting δ be = wI, b is transformed into bw^{−2}, we may assume we are in the previous case a = b = 1, q = 0.

Or, using the same transformation, if q ≠ 0, we may assume q = diag{u, −u} and b = 0, a = 1, u = 1, or b = 1 − u^{2}. The case b = 0, a = 1, u = 1 leads to

Y = \{ ( \begin{matrix}−B_{1} + B_{4} & +B_{2} − B_{3} & −B_{3} & B_{4} \\ B_{1} & B_{2} & B_{3} & B_{4} \end{matrix} ) = 0 \} ,

whence (B_{4} − 2B_{1})(2B_{2} − B_{3}) = B_{4}B_{3} = B_{4}(B_{4} − 2B_{1}) = B_{3}(2B_{2} − B_{3}), two planes.

The last yields Y as the degeneracy locus of the matrix

( \begin{matrix}+at & +a + 2uz & z & x + 2ut \\ x & −z & y & −t \end{matrix} )

where a = 1 − u^{2}. With some pain we get

\begin{matrix}(x + ut) + (1 − 2u^{2})t^{2} = 0 & (1 − u^{2})ty − xz = 0 \\ (z + uy)^{2} + (1 − 2u^{2})y^{2} = 0 & zt + xy + 2uty = 0 \end{matrix} ,

again the union of two planes.

\underline{Example 1P:} Let R be the coordinate ring of the normalization X of two transversal planes in ℙ^{4}. X is the union of {x = y = v − w = 0} with {z = t = v + w = 0}. Then R is generated as an A module by 1, v, with relations

( \begin{matrix}wx & wy & wt & wz & xz & xt & yz & yt \\ x & y & −t & −z & 0 & 0 & 0 & 0 \end{matrix} )

and the ring structure is provided by v^{2} = w^{2}. We will now state Grassi’s theorem

Theorem 6.5 (Gra). Let A be a local Cohen Macaulay domain such that 2 is invertible, and R a finite A-module with Ext_{A}^{i}(R, A) = 0 except for i = 2, and assume dim(A) = dim_{A}(R) + 2. Then the following are equivalent:

1) R is an algebra, Gorenstein over A (Ext_{A}^{2}(R, A) ≅ R)

2) R has two free length two resolutions, related by an isomorphism of complexes which is the identity in degrees 0 and 2, such that 2i) one resolution is Gorenstein symmetric, that is, α^{0} = (α, β), α^{1} = ^{t}(−β, α) (thus γ = α ·^{t}β is symmetric). 2ii) the other resolution satisfies the weak rank condition, i.e., if we write α^{′0} = (σ, τ), α^{′1} = (−τ′, σ′), φ = σ · τ′ = τ · σ′ is such that its adjoint ψ satisfies ψ_{ij} = ∑_{k=1}^{h} λ_{ij}^{k}ψ_{1k}(mod (detσ, detτ)), where λ_{1j}^{k} = δ_{jk}, and moreover the ring structure defined by the elements λ_{ij}^{k} is well defined and associative.

Remark 6.6. Of course it would be interesting to see whether one can achieve more, namely:

i) to have one resolution satisfying both properties ii) to have a strong rank condition ψ_{ij} = ∑_{k=1}^{h} λ_{ij}^{k}ψ_{1k}, iii) to obtain automatically (under some condition?) that the ring structure defined by the elements λ_{ij}^{k} is well defined and associative.

We have seen, when discussing the case of surfaces with K^{2} = 10, p_{g} = 5, q = 0, that the rank condition can in this case be explicitly solved. It would be interesting to analyze the cases K^{2} = 11, 12, concerning this point.

It is clear from the argument we gave earlier (R̃ = Hom_{O_{Y}} (C̃, O_{Y})) that the ring condition is quite automatical if one has control over the singularities of Y, e.g. one knows that they are improper double points.

In this vein, examples of canonical images of surfaces with K^{2} = 11, 12, p_{g} = 5, q = 0 were constructed by Rossberg ([Ros1, 2]) considering Y as the degeneracy locus of a morphism between reflexive sheaves F, G of respective ranks r, r + 1. Let us illustrate his method in the case K^{2} = 10.

We had the resolution

0 → A(−6) \underrightarrow{\binom{−^{t}β}{α}} A(−3)^{4} ⟶^{(αβ)}  A  → R → 0
⨁  ⨁
A(−4)  A(−2)

Let us write, as we did, (αβ) = \binom{A}{B}, \binom{−^{t}B}{α} = (A′B′), then we sheafify and we let

G = ker(O(−3)^{4} ⟶^{B} O(−2))/B′(O(−4)), which has rank 2.

Then A′ : O(−6) → G is a general section whose zero locus is Y.

7. Finite covers and homological algebra

In this last section we want to report rather briefly on recent important applications of homological algebra to the effective construction of algebraic varieties with given invariants.

Here, the globalization of the structure theorems for subvarieties of codimension equal to 1, 2, 3 are done ‘horizontally’ with respect to a finite (ramified) covering map.

To set up the notation, let f : Z ⟶ X be a finite flat morphism of degree d between algebraic varieties. Then we have on X a locally free sheaf f_{∗}(O_{Z}) of rank d: an analogous statement works if X is smooth and Z is Cohen Macaulay, but the example we considered in section 2 shows that for X singular flatness is not automatical, even in dimension 2. We get moreover an exact sequence

0 ⟶ O_{X} ⟶ f_{∗}(O_{Z}) ⟶ f_{∗}(O_{Z})/O_{X} = E ⟶ 0 (32)

which splits if d is prime to the characterisic of the base field ℂ.

We will make this assumption throughout in order to simplify the treatment (cf. [Par1] for a case where there is no apriori splitting): thus (32) splits and we can indeed identify E with the image of the functions on Z with trace zero on X.

Then O_{X} ⊕ E is a sheaf of O_{X} -algebras such that Z = Spec(O_{X} ⊕ E), and we can construct Z just by giving a symmetric bilinear map E ⊗ E ⟶ O_{X} ⊕ E which satisfies the associativity condition.

The double covering case is well known: in this case, if charℂ ≠ 2, we get a bilinear map s : E ⊗ E ⟶ O_{X} whose image is an invertible ideal sheaf O_{X}(−B), B being the branch divisor.

The structure theorem for double covers says then that giving the double cover is equivalent to giving E up to isomorphism, and the section s up to a constant (whence, the divisor B).

In the early 80’s Rick Miranda ([Mir1]) found an unexpected structure theorem for triple covers: to give them it is equivalent to give a rank 2 bundle E and a section s of Sym^{3}(E)^{∨} ⊗det(E) (here the open property, unlike the case d = 2, is less simple than : s ≠ 0).

Miranda found explicitly that by associativity, the component in Sym^{2}(E)^{∨} ⊗E of the bilinear map E ⊗E ⟶ O_{X} ⊕E determines the other component in Sym^{2}(E)^{∨}. On the other hand there is a contraction mapping Sym^{2}(E)^{∨} ⊗ E ≅ Sym^{2}(E^{∨}) ⊗ E ⟶ E^{∨} such that its kernel is isomorphic to the irreducible representation Sym^{3}(E)^{∨} ⊗ det(E).

The upshot is that this subspace corresponds to the associative multiplication maps.

A nice geometric interpretation came out later through the theory of Casnati and Ekedahl of Gorenstein covers of degree ≤ 5 ([C-E], [Ca]). Here f is said to be Gorenstein if f_{∗}(O_{Z}) is a Gorenstein O_{X}-algebra (of codimension 0) (this means that the relative dualizing sheaf ω_{Z|Y} (cf. [Kl]) is invertible and f_{∗}(ω_{Z|Y}) ≅ Hom_{O_{X}} (f_{∗}(O_{Z}), O_{X})): this holds true e.g. if both Z and X are smooth, or simply Gorenstein.

Ekedahl’s discovery was that the Gorenstein condition ensures the following: whereas in general Z = Spec(f_{∗}(O_{Z})) = Spec(O_{Z} ⊕E) embeds in the vector bundle E^{∨} whose sheaf of sections is the dual of E, in the Gorenstein case the natural relative canonical map provides an embedding of Z into ℙ = ℙ(E) = P roj(E^{∨}). Now, inside ℙ we have a relative Cartier divisor Z which is linearly equivalent to 3H + a pull back.

Now, the direct image f_{∗}(O_{ℙ}(3H)) = Sym^{3}(E^{∨}), and the interpretation of the section s in Sym^{3}(E)^{∨} ⊗ det(E) is that s is precisely the section corresponding to the section of O_{ℙ}(3H +f^{∗}det(E)) whose divisor on ℙ is exactly Z. In the Gorenstein case Z is embedded inside the projective bundle ℙ as a subvariety of codimension d − 2.

When d ≥ 4 we no longer have a divisor, and for instance for d = 4 we cannot expect Z to be a complete intersection. But over affine open sets U of the base X, we have that Z|_{U} has codimension (d − 2) in U × ℙ^{d−2}, whence we can apply the structure theorems for Gorenstein subvarieties of codimension 2 and 3 when d = 4, 5.

For instance, when d = 4, the fibre consists of ‘4 points’, or, more precisely, a zero-dimensional subscheme of length 4 which is arithmetically Gorenstein, and is therefore the complete intersection of two conics. When d = 5, we obtain ‘5 points’ which are the Pfaffian locus of an antisymmetric 5 × 5 matrix of linear forms.

The nice result is that everything globalizes in terms of vector bundles on X.

Theorem 7.1 (C-E). The datum of a degree 4 Gorenstein cover f : Z ⟶ X is equivalent to the datum of a rank 3 bundle E and a rank 2 bundle V on X, and of a section s of V^{∨} ⊗ Sym^{2}(E^{∨}).

One can also achieve, by twisting E by a line bundle L and V by L^{2}, that det(V) ≅ det(E).

Theorem 7.2 (Ca). The datum of a degree 5 Gorenstein cover f : Z ⟶ X is equivalent to the datum of a rank 4 bundle E and a rank 5 bundle F on X, such that det(E)^{−2} ≅ detF, and of a section s of E^{∨} ⊗ (detE) ⊗ Λ^{2}(F). Z is the Pfaffian locus of s.

Recently D. Hahn and R. Miranda ([Hahn],[H-M]) were able to show that, for degree d = 4, an entirely analogous result holds also in the non Gorenstein case, simply replacing the section s we have in the Gorenstein case with the exterior product Λ^{2}(s) = η, a section with values in the Grassmanian locus inside Λ^{2}(Sym^{2}(E^{∨})) ⊗ (detE) (η is said to be a totally decomposable section, since locally it can be written as a single wedge product of two local sections of Sym^{2}(E^{∨})). The advantage of doing so is that one allows η to vanish at some points of X, exactly the points where the fibre is the fat point ℂ[x, y]/(x, y, z)^{2}.

Theorem 7.3 (Hahn, Miranda). The datum of a degree 4 cover f : Z ⟶ X is equivalent to the datum of a rank 3 bundle E on X and a totally decomposable section of Λ^{2}(Sym^{2}(E^{∨})) ⊗ (detE).

Also here the proof goes via representation theory, since again the kernel of the contraction mapping Sym^{2}(E)^{∨} ⊗ E ≅ Sym^{2}(E^{∨}) ⊗ E ⟶ E^{∨} is precisely Λ^{2}(Sym^{2}(E^{∨})) ⊗ (detE). Moreover, the embedding of Λ^{2}(Sym^{2}(E^{∨})) ⊗ (detE) inside Sym^{2}(E)^{∨} ⊗ E can be written quite explicitly ([Hahn], page 16), as the map sending

ab⊗(c∧d∧e) ↦ ac∧bd⊗e−ad∧bc⊗e+ad∧be⊗c−ae∧bd⊗c+ae∧bc⊗d−ac∧be⊗d.

Conversations with Rick Miranda make us hope to be able to extend the Casnati structure theorem to the non Gorenstein case. We refer to the cited papers, especially [C-E] and [Ca] for several examples and applications.

The simplest but highly non trivial case is the one of surfaces with K^{2} = 5, q = 0 and p_{g} = 3 such that the canonical system |K| is base point free. Then the canonical map yields the finite Gorenstein covering of degree 5 of the projective plane ℙ^{2}.

In this case the condition q = 0 implies easily, by the main corollary to Horrock’s theorem, that E is a direct sum of line bundles, and the corresponding component of the moduli space is thus shown to be unirational.

It would be interesting to consider also examples where q ≠ 0.

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AUTHORS’ ADDRESS F. Catanese, Mathematisches Instiut der Georg-August-Universität Göttingen, Bunsenstr. 3 - 5, 37073 Göttingen (Germany). e-mail : catanese@uni-math.gwdg.de