THE GEOMETRY OF K3 SURFACES

LECTURES DELIVERED AT THE SCUOLA MATEMATICA INTERUNIVERSITARIA CORTONA, ITALY JULY 31—AUGUST 27, 1988

DAVID R. MORRISON

1. Introduction

This is a course about K3 surfaces and several related topics. I want to begin by working through an example which will illustrate some of the techniques and results we will encounter during the course. So consider the following problem.

Problem. Find an example of C ⊂ X ⊂ ℙ^{3}, where C is a smooth curve of genus 3 and degree 8 and X is a smooth surface of degree 4.

Of course, smooth surfaces of degree 4 are one type of K3 surface. (For those who don’t know, a K3 surface is a (smooth) surface X which is simply connected and has trivial canonical bundle. Such surfaces satisfy χ(O_{X}) = 1, and for every divisor D on X, D · D is an even integer.)

We first try a very straightforward approach to this problem. Let C be any smooth curve of genus 3, and let Z be any divisor on C of degree 8. (For example, we may take Z to be the sum of any 8 points on C.) I claim that the linear system |Z| defines an embedding of C. This follows from a more general fact, which I hope you have seen before.

Theorem. Let C be a smooth curve of genus g, and let Z be a divisor on C of degree d > 2g. Then the linear system |Z| is base-point-free, and defines an embedding of C.

Proof. First suppose that P is a base point of Z. We have an exact sequence

0 → O_{C}(Z − P) → O_{C}(Z) → O_{P} (Z) → 0

and the assumption that P is a base point implies H^{0}(O_{C}(Z − P)) ≅ H^{0}(O_{C}(Z)). But deg(Z−P) = d−1 > 2g−2 and deg(Z) = d > 2g−2 so that both of these divisors Z and Z − P are non-special. By Riemann-Roch, h^{0}(O_{C}(Z − P)) = d − 1 − g + 1 ≠ d − g + 1 = h^{0}(O_{C}(Z)), a contradiction.

Similarly, suppose that P and Q are not separated by the linear system |Z|. (We include the case P = Q, where we suppose that the maximal ideal of C at P is not embedded.) Then in the long exact cohomology sequence associated to

0 → O_{C}(Z − P − Q) → O_{C}(Z) → O_{P +Q}(Z) → 0

we must have that the map H^{0}(O_{C}(Z)) → H^{0}(O_{P +Q}(Z)) is not surjective. This implies that H^{1}(O_{C}(Z − P − Q)) ≠ (0). But

H^{1}(O_{C}(Z − P − Q)) ≅ H^{0}(O_{C}(K_{C} − Z + P + Q))^{∗}

and deg(K_{C} − Z + P + Q) = 2g − 2 − d + 2 < 0 so this divisor cannot be effective (again a contradiction). Q.E.D.

I have included this proof because later we will study the question: on a surface, when does a linear system have base points, and when does it give an embedding?

To return to our problem, we have a curve C of genus 3 and a divisor Z of degree 8. This divisor is non-special and gives an embedding (since 8 > 2 · 3). By Riemann-Roch,

h^{0}(O_{C}(Z)) = 8 − 3 + 1 = 6

so |Z| maps C into ℙ^{5}.

The theory of generic projections guarantees that we can project C into ℙ^{3} from ℙ^{5} in such a way as to still embed C. So we assume from now on: C ⊂ ℙ^{3} is a smooth curve of degree 8 and genus 3. (The linear system |O_{C}(1)| is not complete.)

We now consider the map H^{0}(O_{ℙ^{3}}(k)) → H^{0}(O_{C}(k)) for various degrees k. We have

h^{0}(O_{ℙ^{3}}(k)) = \frac{(k + 1)(k + 2)(k + 3)}{6}

h^{0}(O_{C}(k)) = 8k − 3 + 1 = 8k − 2

(by Riemann-Roch).

It is easy to see that for k ≤ 3 we have h^{0}(O_{ℙ^{3}}(k)) < h^{0}(O_{C}(k)) while for k ≥ 4 we have h^{0}(O_{ℙ^{3}}(k)) > h^{0}(O_{C}(k)). This means that for k ≤ 3, the linear system cut out on C by hypersurfaces of degree k is incomplete, while for k ≥ 4 there must be a hypersurface of degree k containing C. In fact, for k = 4 we have

h^{0}(O_{ℙ^{3}}(4)) = 35
h^{0}(O_{C}(4)) = 30

so that there is at least a ℙ^{4} of quartic hypersurfaces X containing C. By Bertini’s theorem, the generic X is smooth away from C (the base locus of this ℙ^{4}). So we have almost solved our problem: we have constructed C ⊂ X ⊂ ℙ^{3} with C smooth, but X is only known to be smooth away from C.

Where do we go from here? We could continue to pursue these methods of projective geometry in ℙ^{3}. For example, we might consider a generic pencil inside our ℙ^{4} of quartics: the base locus of this pencil is C ∪ C′ with C′ another curve of degree 8. (C′ ≠ C since the arithmetic genus would be wrong). We could try varying the pencil and showing that the induced family of divisors C′ ∩C on C has no base point. But the arguments are very intricate, and I’m not sure if they work! So we will try a different approach.

In this second approach, we construct X first instead of C. What we want is a K3 surface X together with two curves H and C on X. (H is the hyperplane section from the embedding X ⊂ ℙ^{3}). We want C to have genus 3, |H| to define an embedding X ⊂ ℙ^{3}, and C should have degree 8 under this embedding. To translate these properties into numerical properties of H and C on X, notice that for any curve D ⊂ X we have

deg(K_{D}) = deg((K_{X} + D)|_{D}) = deg(D|_{D}) = D · D

(using the adjunction formula, and the fact that K_{X} is trivial). Thus, g(D) = \frac{1}{2}(D · D) + 1.

The numerical versions of our properties are: C · C = 4 (so that g(C) = 3), H · H = 4 (so that we get a quartic X ⊂ ℙ^{3}) and H · C = 8 (so that C will have degree 8 in ℙ^{3}). In addition, we want C to be smooth and |H| to define an embedding in ℙ^{3}.

The first step is to check that the topology of a K3 surface permits curves with these numerical properties to exist. The topological properties I have in mind concern the intersection pairing on H^{2}(X, ℤ). Each curve D on X has a cohomology class [D] ∈ H^{2}(X, ℤ), and the intersection number for curves coincides with the cup product pairing

H^{2}(X, ℤ) ⊗ H^{2}(X, ℤ) → H^{4}(X, ℤ) ≅ ℤ.

Poincaré duality guarantees that this is a unimodular pairing (i. e. in a basis, the matrix for the pairing has determinant ±1). In addition, the signature of the pairing (the number of +1 and −1 eigenvalues) can be computed as (3,19) for a K3 surface. The pairing is also even: this means x · x ∈ 2ℤ for all x ∈ H^{2}(X, ℤ). These three properties together imply that the isomorphism type of this bilinear form over ℤ is (−E_{8})^{⊕2} ⊕ U^{⊕3} where E_{8} is the unimodular even positive definite form of rank 8, and U is the hyperbolic plane. Let Λ denote this bilinear form; Λ is called the K3 lattice. We will return to study these topological properties in more detail (and define the terms!) in section 11.

In our example, we need a submodule of H^{2}(X, ℤ) ≅ Λ generated by 2 elements h and c such that the matrix of the pairing on these 2 elements is

(\begin{matrix}4 & 8 \\ 8 & 4 \end{matrix}) .

The fact that such a submodule exists is a consequence of Theorem (James [12]). Given an even symmetric bilinear form L over the integers such that L has signature (1, r − 1) and the rank r of L is ≤ 10, there exists a submodule of Λ isomorphic to L.

(Later^{1} we will study refinements of this theorem in which the rank is allowed to be larger.)

So there is no topological obstruction in our case. Moreover, later^{2} in the course we will partially prove the following:

Fact. Given a submodule L of Λ on which the form has signature (1, r − 1) with r ≤ 20, there exists a (20−r)-dimensional family of K3 surfaces {X_{t}}, each equipped with an isomorphism H^{2}(X_{t}, ℤ) ≅ Λ in such a way that elements of L correspond to cohomology classes of line bundles on X_{t}. Moreover, for t generic, these are the only line bundles on X_{t}, that is, the Néron-Severi group NS(X_{t}) [together with its intersection form] is isomorphic to L.

I said we would partially prove this: what we will not prove (for lack of time) is the global Torelli theorem and the surjectivity of the period map for K3 surfaces. This fact depends on those theorems.^{3}

To return to our problem: we now have a K3 surface X and two line bundles O_{X}(H), O_{X}(C) with the correct numerical properties, which generate NS(X). For any line bundle L on X, we have H^{2}(L) ≅ H^{0}(L^{∗})^{∗} (since K_{X} is trivial), from which follows the Riemann-Roch inequality:

h^{0}(L) + h^{0}(L^{∗}) ≥ \frac{L · L}{2} + 2

(since χ(O_{X}) = 2). Thus, if L · L ≥ −2 either L or L^{∗} is effective.

In our situation we conclude: ±H is effective and ±C is effective. Replacing H by −H if necessary we may assume H is effective. (The choice of C is then determined by H · C = 8.)

To finish our construction, we need another fact which will be proved later.^{4}

Fact. Let H be an effective divisor on a K3 surface X with H^{2} ≥ 4. If |H| has base points or does not define an embedding, then there is a curve E on X with E^{2} = −2 or E^{2} = 0. Moreover, when it does define an embedding H^{1}(O_{X}(H)) = 0 and H^{2}(O_{X}(H)) = 0.

In our situation, we wish to rule out the existence of such an E. We have^{5} E ∼ mH + nC since H and C generate NS(X). Thus, E^{2} = 4m^{2} + 16mn + 4n^{2} is always divisible by 4, so E^{2} = −2 is impossible. Moreover, if a rank 2 quadratic form ( \begin{matrix}a & b \\ b & c \end{matrix}) represents 0 then − det ( \begin{matrix}a & b \\ b & c \end{matrix}) = b^{2} − ac is a square. In our case, − det ( \begin{matrix}4 & 8 \\ 8 & 4 \end{matrix} ) = 48 is not a square, so E^{2} = 0 is impossible.

1Need cross-reference.

2Need cross-reference.

3For statements of the theorems and further discussion, see section 12.

4Need cross-reference: section 5?

5The symbol ∼ denotes linear equivalence.

We conclude that |H| is very ample. Then H · C = 8 implies −C is not effective, so that C must be effective. Furthermore, |C| is then very ample, so this linear system contains a smooth curve (which we denote again by C). By Riemann-Roch, h^{0}(O_{X}(H)) = \frac{H·H}{2} + 2 = 4, so |H| maps X into ℙ^{3} as a smooth quartic surface, and we have C ⊂ X ⊂ ℙ^{3} as desired.

2. K3 surfaces and Fano threefolds

We will use in this course a definition of K3 surfaces which is slightly different from the standard one. Namely, for various technical reasons which will appear later, it is convenient to allow K3 surfaces to have some singular points called rational double points. These will be the subject of a seminar later on;^{6} if you are not familiar with them, I suggest that you ignore the singularities for the moment and concentrate on smooth K3 surfaces.

(We do not use the term “singular K3 surface” to refer to these surfaces, because that term has a different meaning in the literature: it refers to a smooth K3 surface with Picard number 20. Cf. [27].)

Here is a convenient definition of rational double points: a complex surface X has rational double points if the dualizing sheaf ω_{X} is locally free, and if there is a resolution of singularities π : X̃ → X such that π^{∗}ω_{X} = ω_{X} = O_{X}(K_{X}). For those unfamiliar with the dualizing sheaf, what this means is: for every P ∈ X there is a neighborhood U of P and a holomorphic 2-form α = α(z_{1}, z_{2})dz_{1} ∧ dz_{2} defined on U − {P } such that π^{∗}(α) extends to a nowhere-vanishing holomorphic form on π^{−1}(U).

The structure of rational double points (sometimes called simple singularities) is well-known: each such point must be analytically isomorphic to one of the following:

\begin{matrix}A_{n} (n ≥ 1): & x^{2} + y^{2} + z^{n+1} = 0 \\ D_{n} (n ≥ 4): & x^{2} + yz^{2} + z^{n−1} = 0 \\ E_{6}: & x^{2} + y^{3} + z^{4} = 0 \\ E_{7}: & x^{2} + y^{3} + yz^{3} = 0 \\ E_{8}: & x^{2} + y^{3} + z^{5} = 0 \end{matrix}

and the resolution X̃ → X replaces such a point with a collection of rational curves of self-intersection −2 in the following configuration:

A_{n} (n curves)

D_{n} (n curves)

E_{6}

E_{7} E_{8} To return to the definition of K3 surfaces: a K3 surface is a compact complex

6See Appendix ??.

analytic surface X with only rational double points such that h^{1}(O_{X}) = 0 and ω_{X} ≅ O_{X}. (If X is smooth, the dualizing sheaf ω_{X} is the line bundle associated to the canonical divisor K_{X}, so this last condition says that the canonical divisor is trivial.)

If X is a K3 surface and π : X̃ → X is the minimal resolution of singularities (i. e. the one which appeared in the definition of rational double point) then it turns out that π^{∗} establishes an isomorphism H^{1}(O_{X}) ≅ H^{1}(O_{X}), and also we have ω_{X} = π^{∗}ω_{X} = π^{∗}O_{X} = O_{X}. Thus, the smooth surface X̃ is also a K3 surface.

We will concentrate on smooth K3 surfaces for quite a while, and only return to singular ones in several weeks.^{7} I have included the singular case today so that we don’t have to change the definitions later.

Here are some of the basic facts about smooth K3 surfaces. Let X be a smooth K3 surface.

(1) χ(O_{X}) = 2, because h^{1}(O_{X}) = 0 and h^{2}(O_{X}) = h^{0}(O_{X}(K_{X}))^{∗} = h^{0}(O_{X}) = 1.

(2) c_{1}^{2}(X) = 0. [Remember that c_{1}^{2}(X) = K_{X} · K_{X}.]

(3) Therefore, using Noether’s formula

c_{1}^{2}(X) + c_{2}(X) = 12χ(O_{X})

we find that c_{2}(X) = 24. Since b_{1}(X) = 2h^{1}(O_{X}) or 2h^{1}(O_{X}) − 1, we see that b_{1}(X) = 0. So the Betti numbers must be: b_{0} = 1, b_{1} = 0, b_{2} = 22, b_{3} = 0, b_{4} = 1 giving 24 as the topological Euler characteristic.

[For those who haven’t studied compact complex surfaces, I remind you that in the case of algebraic surfaces we always have b_{1}(X) = 2h^{1}(O_{X}). Kodaira proved that for non-algebraic complex surfaces this equality can only fail by 1, i. e., b_{1} = 2h^{1}(O_{X}) or 2h^{1}(O_{X}) − 1.] (Cf. [BPV, Theorem II.6].^{8})

(4) For any line bundle L on X, the Riemann-Roch theorem

χ(L) = χ(O_{X}) + \frac{L · L − K_{X} · L}{2}

becomes:

h^{0}(L) − h^{1}(L) + h^{0}(L^{∗}) = 2 + \frac{L · L}{2}

since h^{2}(L) = h^{0}(L^{∗}(K_{X})) = h^{0}(L^{∗}).

In particular, if L · L ≥ −2 then h^{0}(L) + h^{0}(L^{∗}) > 0, i. e., either L or L^{∗} is effective.

(5) The intersection form on H^{2}(X, ℤ) has a very explicit structure mentioned in the introduction. We postpone discussion of this structure until section 11.

7Need cross-reference: section 8?

8Per A. G.

(6) Let D be an irreducible reduced effective divisor on X. Then the adjunction formula K_{D} = (K_{X} + D)|_{D} yields

deg(K_{D}) = deg(D|_{D}) = D · D.

In particular,

g(D) = \frac{1}{2}(D · D) + 1.

(7) h^{0}(D) = g(D) + 1.

But even more is true: if we consider the exact sequence (when D is smooth)

0 → O_{X} → O_{X}(D) → O_{D}(D) → 0

since h^{1}(O_{X}) = 0 we have that H^{0}(O_{X}(D)) → H^{0}(O_{D}(D)) is surjective. Thus since O_{D}(D) ≅ O_{D}(K_{D}), the global sections of the line bundle O_{X}(D) induce the canonical map on D. So when D is not hyperelliptic, it must be embedded by O_{X}(D) and this embedding coincides with the canonical embedding of D.

To say this another way, if X is embedded in ℙ^{g} by the complete linear system |O_{X}(D)|, then a hyperplane section D = X ∩ℙ^{g−1} is canonically embedded in ℙ^{g−1} ⊂ ℙ^{g}. In brief: “a hyperplane section of a K3 surface is a canonical curve”.

This property can be considered as motivating the definition of a K3 surface. That is, we require K_{X} ∼ 0 so that the normal bundle O_{D}(D) of a hyperplane section agrees with the canonical bundle O_{D}(K_{D}), and we require h^{1}(O_{X}) = 0 so that the rational map H^{0}(O_{X}(D)) → H^{0}(O_{D}(D)) is surjective.

We now ask: What kind of threefold has a K3 surface as its hyperplane section? If Y is such a threefold, we must have h^{1}(O_{Y}) = 0 to guarantee that H^{0}(O_{Y} (X)) → H^{0}(O_{X}(X)) is surjective, and by adjunction

K_{X} = (K_{Y} + X)|_{X}

so that we need K_{Y} = −X. (Note that by the Lefschetz hyperplane theorem (cf. [9]), we have h^{1}(O_{X}) = 0 so X is a K3 surface.) That is, Y is embedded by its anti-canonical linear system |O_{Y} (−K_{Y})|. This is called a Fano threefold.

We can ask the same question for higher dimension, of course.

Definition. A Fano variety is a complex projective variety Y with O_{Y} (−K_{Y}) ample. A Fano variety has index r if r is the maximum integer such that −K_{Y} ∼ rH for some ample H on Y. The coindex of Y is defined to be c = dim(Y) − r + 1.

The linear system |H| is called the fundamental system of the Fano variety.

Lemma. If |H| is very ample and we choose r general hyperplanes H_{1}, …, H_{r}, then Y ∩ H_{1} ∩ ⋯ ∩ H_{r} is a variety of dimension c − 1 with trivial canonical bundle.

(The proof is easy: use the adjunction formula.)

Thus, the Fano varieties related to elliptic curves are the ones of coindex 2; the ones related to K3 surfaces (and canonical curves) are the ones of coindex 3. To see that Fano varieties of coindex 3 are in fact related to K3 surfaces, rather than to some other surface with trivial canonical bundle, we need to recall the Kodaira vanishing theorem.

Theorem (Kodaira). Let L be an ample divisor on Y. Then H^{i}(O_{Y} (K_{Y} + L)) = 0 for i > 0.

Corollary. Let Y be a Fano variety. Then H^{i}(O_{Y}) = 0 for i > 0.

Proof. Take L = −K_{Y}, which is ample, so that O_{Y} (K_{Y} + L) = O_{Y}. then H^{i}(O_{Y}) = (H^{i}(O_{Y} (K_{Y} + L)) = 0 for i > 0. Q.E.D.

Corollary. Let Y be a Fano variety of coindex c, and let X = Y ∩ H_{1} ∩ ⋯ ∩ H_{r} be a linear section of dimension c − 1 (which has trivial canonical bundle). Then H^{i}(O_{X}) = 0 for 0 < i < dim X = c − 1.

[This justifies the statement made earlier that Fano varieties of coindex 3 have linear surface sections which are K3 surfaces.]

Proof. Let Z = Y ∩H_{1}∩⋯∩H_{r−1} be a linear section of dimension c so that X = Z∩H; then O_{Z}(K_{Z}) = O_{Z}(−H) and Z is again a Fano variety. We apply Kodaira vanishing (in its dual form) to conclude that H^{i}(O_{Z}(−H)) = 0 for 0 ≤ i < c = dim Z. Thus, in the long exact cohomology sequence associated to

0 → O_{Z}(−H) → O_{Z} → O_{X} → 0

we find that H^{i}(O_{Z}) ≅ H^{i}(O_{X}) for i + 1 < c. The statement now follows from the previous corollary. Q.E.D.

3. Examples of canonical curves, K3 surfaces, and Fano varieties of coindex 3

Before beginning the examples, let us recall the theory of the Hirzebruch surfaces 𝔽_{n} (or Σ_{n}). These are defined as 𝔽_{n} = ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(n)), and are ℙ^{1}-bundles over ℙ^{1}. 𝔽_{n} has a distinguished section of its ℙ^{1}-bundle structure (when n > 0), given by σ_{∞} = ℙ(O_{ℙ^{1}}(n)) ⊂ ℙ(O_{ℙ^{1}} ⊕O_{ℙ^{1}}(n)). [When n = 0, the section exists but is no longer unique.] σ_{∞} satisfies: σ_{∞}^{2} = −n, σ_{∞} · f = 1 (where f is a general fiber) and also f^{2} = 0; moreover, σ_{∞} and f generate Pic(𝔽_{n}). There are other sections ℙ(O_{ℙ^{1}}) ⊂ 𝔽_{n} which are linearly equivalent to σ_{∞} + nf.

We need to know the canonical bundle formula for 𝔽_{n}, and it is derived as follows. Write

K_{𝔽_{n}} = aσ_{∞} + bf

and use the fact that σ_{∞} and f are smooth rational curves:

−2 = deg(K_{σ_{∞}}) = deg((a + 1)σ_{∞} + bf)|_{σ_{∞}} = −n(a + 1) + b
−2 = deg(K_{f}) = deg(aσ_{∞} + (b + 1)f)|_{f} = a.

This implies a = −2, b = −n − 2 so K_{𝔽_{n}} = −2σ_{∞} − (n + 2)f.

Example 1. Write −2K_{𝔽_{4}} = σ_{∞} + (3σ_{∞} + 12f) and choose a smooth divisor D ∈ |3σ_{∞} + 12f|. (We will see in a moment^{9} that this can be done.) Let X be the double cover branched on D + σ_{∞} ∈ | − 2K_{𝔽_{4}}|. Then h^{1}(O_{X}) = h^{1}(O_{𝔽_{4}}) = 0 and

K_{X} = π^{∗}(K_{𝔽_{4}} + \frac{1}{2}(D + σ_{∞})) = π^{∗}(K_{𝔽_{4}} − K_{𝔽_{4}}) ∼ 0

so that X is a K3 surface.^{10} We have the following picture of the branch locus:

In fact, the double cover of f is branched in 4 points (because −2K_{F_{4}} · f = 4), and so is a smooth elliptic curve for general f by the Hurwitz formula. Thus, X is a K3 surface with a pencil of curves of genus 1.

To see that D exists and to describe this all more explicitly, consider the complement of σ_{∞} in 𝔽_{4}. This is a ℂ-bundle over ℙ^{1}, isomorphic to the total space of the (bundle associated to the) sheaf O_{ℙ^{1}}(4). If we restrict further to ℂ ⊂ ℙ^{1}, we can choose coordinates: x in the fiber direction and t in the base.

There will be another set of coordinates: s = \frac{1}{t} and y in the fiber direction. To see how these are related, let e be a nonvanishing section of O_{ℙ^{1}}(4)|_{t-chart} and let f be one of O_{ℙ^{1}}(4)|_{s-chart}. There is a global section of O_{ℙ^{1}}(4) with a zero of order 4 at t = 0 and no other zero: this must be given by t^{4}e and 1 ·f when restricted to the two charts. So we have t^{4}e = 1 · f which determines the transition e = t^{−4}f. Since arbitrary sections are to be represented by xe or yf we have yf = xe = xt^{−4}f so that xt^{−4} = y . The 2 boxed equations are the transition functions.

Now on this space we have the line bundle L = O(σ_{∞} + 4f) which has a section vanishing at (x = 0) ∪ (y = 0), and D is the zero-locus of a section of L^{⊗3}. If we let ε and φ be trivializing sections for L in the (x, t) and (y, s) charts respectively, then xε = yφ so that the transition is ε = \frac{y}{x}φ = t^{−4}φ.

9I. e., below.

10See also the appendix on double covers.

Let us write the section of L^{⊗3} whose zero-locus is D in the form f(x, t)ε^{⊗3} = g(y, s)φ^{⊗3}. Then f(x, t)t^{−12} = g(y, s). A monomial x^{i}t^{j} can appear in f only if

x^{i}t^{j−12} = (ys^{−4})^{i}(s^{−1})^{j−12} = y^{i}s^{12−4i−j}

is holomorphic, i. e., 12 − ri − j ≥ 0. This implies that f has the form

f(x, t) = kx^{3} + a_{4}(t)x^{2} + b_{8}(t)x + c_{12}(t).

We take k ≠ 0 so that there are truly three points of intersection with the fiber; then by a coordinate change we may assume k = 1.

Finally we may describe the double cover: it has the form

z^{2} = x^{3} + a_{4}(t)x^{2} + b_{8}(t)x + c_{12}(t)

in one chart, and

w^{2} = y^{3} + (s^{4}a_{4}(\frac{1}{s}))y^{2} + (s^{8}b_{8}(\frac{1}{s}))y + (s^{12}c_{12}(\frac{1}{s}))

in the other chart. As is well-known, this compactifies nicely when x → ∞ or y → ∞, and branches at ∞ in the fiber direction, giving a family of elliptic curves with a section.

What remains to be checked is that D is smooth when a, b, c are chosen generically. We leave this as an exercise.

Example 2. Let X → ℙ^{2} be the double cover of ℙ^{2} branched in a smooth curve C of degree 6. Then^{11}

h^{1}(O_{X}) = h^{1}(O_{ℙ^{2}}) = 0

K_{X} = π^{∗}(K_{ℙ^{2}} + \frac{1}{2}C) = π^{∗}(−3H + \frac{1}{2}(6H)) ∼ 0

so that X is a K3 surface. π expresses the inverse image of a general line in ℙ^{2} as a double cover of the line branched in 6 points, so the inverse image is a curve of genus 2. Thus, X is a K3 surface with a curve of genus 2.

Examples 3, 4, 5. Let us find all complete intersections in projective space ℙ^{n+k} which are either a canonical curve, a K3 surface, or a Fano variety of coindex 3. Let X be the variety in question, and notice that in all three cases we want

K_{X} = (2 − n)H|_{X}

where n = dim(X). [In the Fano case, r = n − 2 so c = n − (n − 2) + 1 = 3.]

Now if X is the intersection of hypersurfaces V_{1}, …, V_{k} in ℙ^{n+k} of degrees d_{1}, …, d_{k}, we may assume d_{i} ≥ 2. An easy induction with the adjunction formula gives

K_{X} = (K_{ℙ^{n+k}} + V_{1} + ⋯ + V_{k})|_{X}
= ((−n − k − 1) + d_{1} + ⋯ + d_{k})H|_{X}.

11See the appendix on double covers.

So we need

−n − k − 1 + d_{1} + ⋯ + d_{k} = 2 − n

or

∑_{i=1}^{k}(d_{i} − 1) = n + 1 + 2 − n = 3.

The solutions are 3 = 3, 3 = 2 + 1 and 3 = 1 + 1 + 1 corresponding to

Example 3. A quartic hypersurface in ℙ^{n+1} (this is a K3 with curve section of genus 3 when n = 2);

Example 4. The intersection of a quadric and a cubic in ℙ^{n+2} (this is a K3 with curve section of genus 4 when n = 2); and

Example 5. The intersection of three quadrics in ℙ^{n+3} (this is a K3 with curve section of genus 5 when n = 2).

Example 6. The canonical bundle of the Grassmannian Gr(2, 5) satisfies K_{Gr(2,5)} = −5Σ, where Σ is the Schubert cycle of codimension 1. The linear system |Σ| induces the Plucker embedding of the Grassmannian Gr(2, 5) → ℙ(Λ^{2}ℂ^{5}) = ℙ^{9}.

Let Y be the double cover of Gr(2, 5) branched along a divisor D ∈ |2Σ|. Then K_{Y} = π^{∗}(K_{Gr(2,5)} + \frac{1}{2}D) = π^{∗}(−5Σ + Σ) = −4π^{∗}(Σ) so that Y has index 4. Since dim(Y) = 6, the coindex is 6 − 4 + 1 = 3.

The fundamental system |π^{∗}(Σ)| satisfies:

π^{∗}(Σ)^{6} = 2 · Σ^{6} = 2 · 5 = 10

and so has degree 10. Moreover, since H^{0}(π^{∗}Σ) ≅ H^{0}(Σ) ⊕ H^{0}(Σ − \frac{1}{2}D) and H^{0}(Σ − \frac{1}{2}D) = H^{0}(O_{Gr(2,5)}) ≅ ℂ, the fundamental system maps Y to ℙH^{0}(π^{∗}Σ) ≅ ℙ^{10}. The genus of a linear curve section is 6, since 2g − 2 = 10.

Examples 7, 8, 9, 10. Mukai has investigated the question: which compact complex homogeneous spaces are Fano varieties of coindex 3? (Note that ℙ^{n} and the quadric Q^{n} ⊂ ℙ^{n+1}, which are the Fano varieties of coindex 0 and 1 respectively, are complex homogeneous spaces). The answers Mukai found (cf. [21]) are the following (we give no details about his methods):

Variety  Dimension Degree ; of |H| Ambient space \begin{matrix}Genus \\ of C \end{matrix}
7)  S0(10)/U(5)  10  12  ℙ^{15}  7
8) U(6)/U(2) × U(4) = Gr(2, 6)  8  14  ℙ^{14}  8
9)  Sp(3)/U(3)  6  16  ℙ^{13}  9
10)  G_{2}/P  5  18  ℙ^{13}  10

where G_{2} is the exceptional Lie group of that name, and P is the maximal parabolic associated to the long root in the Dynkin diagram • > ≡ •.

[In the table, H represents the ample generator of the Picard group (which is isomorphic to ℤ), the ambient space refers to the embedding φ_{|H|} (it turns out that |H| is in fact very ample), and C is a linear curve section of the embedded variety.]

In the examples we have given with linear curve section of genus g, 2 ≤ g ≤ 10, it will turn out later^{12} that the general K3 surface with a (primitive) curve of that genus belongs to the family we have described. It is also possible to give such a description for g = 12 (we give the example below), but not for g = 11 or g ≥ 13. We will give several examples of Fano varieties of genus 11 and coindex 3, but all have Picard number ≥ 2 and so only give proper subsets of the set of all K3 surfaces of genus 11.

Example 11a. ℙ^{3} × ℙ^{3}.

Pic(ℙ^{3} × ℙ^{3}) is generated by H_{1} = ℙ^{2} × ℙ^{3} and H_{2} = ℙ^{3} × ℙ^{2}, with K_{ℙ^{3}×ℙ^{3}} = −4H_{1} − 4H_{2}. Thus, r = 4, c = 6 − 4 + 1 = 3 and of course the dimension is 6. The fundamental system is H = H_{1} + H_{2}, and H^{6} = (_{3}^{6})H_{1}^{3}H_{2}^{3} = (_{3}^{6}) = 20. The mapping φ_{|H|} is the Segre embedding ℙ^{3} × ℙ^{3} ↪ ℙ^{15}. Since 2g − 2 = 20 we have g = 11.

Example 11b. ℙ^{2} × Q^{3}, where Q^{3} ⊂ ℙ^{4} is a quadric.

Pic(ℙ^{2} × Q^{3}) is generated by H_{1} = ℙ^{1} × Q^{3} and H_{2} = ℙ^{2} × (ℙ^{3} ∩ Q^{3}), with K_{ℙ^{2}×Q^{3}} = −3H_{1} − 3H_{2}. Thus, r = 3, c = 5 − 3 + 1 = 3 and the dimension is 5. The fundamental system is H = H_{1} + H_{2} of degree H^{5} = (_{2}^{5})H_{1}^{2}H_{2}^{3} = (_{2}^{5}) · 2 = 20. The mapping φ_{|H|} is induced by the Segre embedding of ℙ^{2} × ℙ^{4}:

ℙ^{2} × Q^{3} ⊂ ℙ^{2} × ℙ^{4} ↪ ℙ^{14}.

Since 2g − 2 = 20 we have g = 11.

Example 11c. ℙ^{1} × V_{5}^{3}, where V_{5}^{3} is a 3-dimensional linear section of Gr(2, 5) ⊂ ℙ^{9} (and thus has degree 5 in ℙ^{6}).

Pic(ℙ^{1} × V_{5}^{3}) is generated by H_{1} = ℙ^{0} × V_{5}^{3} and H_{2} = ℙ^{1} × (V_{5}^{3} ∩ ℙ^{5}) with K_{ℙ^{1}×V_{5}^{3}} = −2H_{1} −2H_{2}. (V_{5}^{3} is a Fano 3-fold of index 2.) Thus, r = 2, c = 4−2+1 = 3 and the dimension is 4. The fundamental system is H = H_{1} + H_{2} of degree H^{4} = (_{1}^{4})H_{1} · H_{2}^{3} = (_{1}^{4}) · 1 · 5 = 20. The mapping φ_{|H|} is induced by the Segre embedding of ℙ^{1} × ℙ^{6}:

ℙ^{1} × V_{5}^{3} ⊂ ℙ^{1} × ℙ^{6} ↪ ℙ^{13}.

Since 2g − 2 = 20, we once again have g = 11.

Example 11d. We give no details on this, but Mori and Mukai have found that if we take a smooth conic C ⊂ ℙ^{2} and choose a degree 5 map C → ℙ^{1} so that the induced curve C ⊂ ℙ^{1} ×ℙ^{2} has bidegree (5,2), then the blowup of ℙ^{1} ×ℙ^{2} with center C is a Fano 3-fold of index 1 with degree of the fundamental system = 20 and so g = 11. This is an example with Picard number 3.

12Need cross-reference.

Example 12. Let F_{1}, F_{2}, F_{3} be general skew-symmetric bilinear forms on ℂ^{7}. Let

Y = {w ∈ Gr(3, 7) | F_{1}(w, w) = F_{2}(w, w) = F_{3}(w, w) = 0}.

Mukai has shown that this Y is a smooth Fano 3-fold of degree 22 (and g = 12). In fact, the map φ_{|H|} associated to the fundamental system factors through the inclusion Y ⊂ Gr(3, 7) and we have

Y  ⟶^{φ_{|H|}} ℙ^{13} = span of Y
⋂  ⋂ [linear subspace]
Gr(3, 7) ⟶^{Plucker} ℙ^{34}

Example Km. Our final example of K3 surfaces will include some non-algebraic ones. Let T = ℂ^{2}/Γ be a complex torus of complex dimension 2. (Thus, Γ ⊂ ℂ^{2} is an additive subgroup such that there is an isomorphism of ℝ-vector spaces Γ⊗ℝ ≅_{ℝ} ℂ^{2}. In particular, Γ is a free ℤ-module of rank 4.) Let (z, w) be coordinates on ℂ^{2}, and define i(z, w) = (−z, −w). Since Γ is a subgroup under addition, i(Γ) = Γ. Thus, i descends to an automorphism ĩ: T → T.

What are the fixed points of ĩ? To find them, we need to know the solutions to i(z, w) ≡ (z, w) mod Γ. These solutions are {(z, w) | (2z, 2w) ∈ Γ} and so ĩ has as fixed points \frac{1}{2}Γ/Γ. (There are 16 of these.)

Let X = T/ĩ; X is called a Kummer surface. This surface has 16 singular points at the images of the fixed points of ĩ. To see the structure of these singular points, consider the action of i on a small neighborhood U of (0,0) in ℂ^{2}. Then U/i is isomorphic to a neighborhood of a singular point of X.

To describe U/i, we note that the invariant functions on U are generated by z^{2}, zw, and w^{2}. Thus, if we let r = z^{2}, s = zw and t = w^{2} we can write

U/i ≅ {(r, s, t) near (0, 0, 0) | rt = s^{2}}.

This is a rational double point of type A_{1}.

dz ∧ dw is a global holomorphic 2-form on ℂ^{2}, invariant under the action of Γ, and so descends to a form on T. It is also invariant under the action of i (since d(−z) ∧ d(−w) = dz ∧ dw), so we get a form dz ∧ dw on X − {singular points}. In local coordinates, dr = 2zdz, dt = 2wdw so that

dz ∧ dw = \frac{dr ∧ dt}{4zw} = \frac{dr ∧ dt}{4s}.

It is easy to check that this form induces a global nowhere vanishing holomorphic 2-form on the minimal resolution X̃ of X.

To finish checking that X is a K3 surface, we use the fact that H^{1}(O_{X}) ≅ H^{1}(O_{X}) ≅ {elements of H^{1}(O_{T}) invariant under ĩ}. Now H^{1}(O_{T}) ≅ H^{0,1}(T), the space of global differential forms of type (0,1). This space is generated by dz̄ and dw̄; since ĩ^{∗}(dz̄) = −dz̄ and ĩ^{∗}(dw̄) = −dw̄ there are no invariants. It follows that H^{1}(O_{X}) ≅ H^{1}(O_{X}) = (0), and that X and X̃ are both K3 surfaces. Notice that when T (or equivalently Γ) is chosen generally, then T is not algebraic, nor are X or X̃.

3.1. Addendum to section 3. There is a beautiful 3-fold, which I believe was first constructed by Segre, which shows that projective K3 surfaces (specifically quartics in ℙ^{3}) can have 15 or 16 singularities of type A_{1}.

The threefold is defined in ℙ^{5} by two equations

x_{1}^{2}y_{1}^{2} + x_{2}^{2}y_{2}^{2} + x_{3}^{2}y_{3}^{2} − 2x_{1}y_{1}x_{2}y_{2} − 2x_{1}y_{1}x_{3}y_{3} − 2x_{2}y_{2}x_{3}y_{3} = 0
x_{1} + x_{2} + x_{3} + y_{1} + y_{2} + y_{3} = 0.

(Of course this is really in ℙ^{4}, but the equations are more symmetric this way.) This is a quartic 3-fold which has 15 singular lines:

(a) 8 lines of the form ℓ_{1} = ℓ_{2} = ℓ_{3} = ℓ′_{1} + ℓ′_{2} + ℓ′_{3} = 0 where ℓ_{α} ∈ {x_{α}, y_{α}} and ℓ′_{α} is different from ℓ_{α} for each α (e.g. x_{1} = x_{2} = x_{3} = y_{1} + y_{2} + y_{3} = 0)

(b) 6 lines of the form x_{i} = y_{i} = ℓ_{j} + ℓ_{k} = ℓ′_{j} + ℓ′_{k} = 0 where ℓ_{α} ∈ {x_{α}, y_{α}}, ℓ′_{α} is different from ℓ_{α} and (i, j, k) is a permutation of (1,2,3).

(c) the line x_{1} = y_{1}, x_{2} = y_{2}, x_{3} = y_{3}, x_{1} + x_{2} + x_{3} = 0.

A general hyperplane section of this 3-fold is a quartic surface with 15 A_{1} singularities. On the other hand, for the general point P of this variety Y, if T_{P} (Y) is the (projective) tangent plane to Y at P then T_{P} (Y) ∩ Y is a quartic surface with 16 A_{1} singularities: one at P and 15 at the intersection with the lines.

Note.

(i) For the general point P, the rank of Pic({ \atop T_{P} (Y) ∩ Y} is 17, where { \atop T_{P} (Y) ∩ Y} is the minimal resolution of T_{P} (Y) ∩ Y.

(ii) T_{P} (Y) ∩ Y is a (quartic) Kummer surface (see Nikulin’s theorem in section 8). (iii) Every Kummer surface coming from a principally polarized abelian surface

can be represented as such a T_{P} (Y) ∩ Y. This has been “sort of” proved by Van der Geer.^{13}

3.2. Appendix: Double covers. A double cover is constructed from a variety X, a line bundle L, and a section s ∈ H^{0}(L^{⊗2}) whose zero-locus is D. (This is called the double cover of X branched along D.) For simplicity we assume X and D smooth.

To describe the construction, we need an open cover {U_{i}} of X such that L|_{U_{i}} is trivial. Let t⃗_{i} denote coordinates on U_{i}, and choose a nowhere zero section e_{i} ∈ H^{0}(U_{i}, L|_{U_{i}}) to trivialize L|_{U_{i}}. Then every section of L|_{U_{i}} may be written in the form s_{i}(t⃗_{i})e_{i} for some functions s_{i} on U_{i}.

13Need a literature reference, and an explanation of “sort of.”

The sections e_{i} are related by the transition functions: e_{i} = λ_{ij}e_{j}. Thus, if s = {s_{i}(t⃗_{i})e_{i}} is a global section, we must have

s_{i}(t⃗_{i})e_{i} = s_{i}(t⃗_{i})λ_{ij}e_{j} = s_{j}(t⃗_{i})e_{j}

so that: s_{i}(t⃗_{i})λ_{ij} = s_{j}(t⃗_{j}).

In the case of the double cover construction, we have s = {s_{i}(t⃗_{i})e_{i}^{⊗2}}, a section of L^{⊗2} and so

s_{i}(t⃗_{i})λ_{ij}^{2} = s_{j}(t⃗_{j})

where {λ_{ij}} are transition functions for L. Define

V_{i} = {(t⃗_{i}, x_{i}) ∈ U_{i} × ℂ | x_{i}^{2} = s_{i}(t⃗_{i})}.

The double cover is Y = ⋃ V_{i}, with projection map π : Y → X given by π(t⃗_{i}, x_{i}) = t⃗_{i}. The coordinate charts V_{i} are to be patched by:

(t⃗_{j}, x_{j}) = (t⃗_{j}(t_{i}), x_{i}λ_{ij})

[so that x_{i}^{2}λ_{ij}^{2} = s_{i}λ_{ij}^{2} = s_{j} = x_{j}^{2}.]

Lemma.

(1) K_{Y} = π^{∗}(K_{X} + L), i. e., O_{Y} (K_{Y}) = π^{∗}(O_{X}(K_{X}) ⊗ L).

(2) If M is another line bundle on X, then

H^{0}(π^{∗}M) ≅ H^{0}(M) ⊕ H^{0}(M ⊗ L^{−1}).

[Property #1 is often written: K_{Y} = π^{∗}(K_{X} + \frac{1}{2}D), which is a bit sloppy.]

Proof of (1). Let dt⃗_{i} denote the differential form dt_{i}^{1} ∧ ⋯ ∧ dt_{i}^{n} on U_{i}. dt⃗_{i} is a section of O_{X}(K_{X})|_{U_{i}} which trivializes that bundle; since

dt⃗_{i} = |\frac{∂(t⃗_{i})}{∂(t⃗_{j})}| dt⃗_{j}

[the notation means the “Jacobian determinant”], we see that w_{ij} = |\frac{∂(t⃗_{i})}{∂(t⃗_{j})} | give

transition functions for O_{X}(K_{X}).

Since X and D are smooth, we may assume (after shrinking the U_{i}’s) that s_{i}(t⃗_{i}) = t_{i}^{1} (the first coordinate). Then x_{i}^{2} = t_{i}^{1} so that (x_{i}, t_{i}^{2}, …, t_{i}^{n}) form coordinates on V_{i}. Moreover,

dt⃗_{i} = dt_{i}^{1} ∧ dt_{i}^{2} ∧ ⋯ ∧ dt_{i}^{n}
= 2x_{i}dx_{i} ∧ dt_{i}^{2} ∧ ⋯ ∧ dt_{i}^{n}.

Thus, if e_{i} is a trivializing section of O_{X}(K_{X})|_{U_{i}} and ẽ_{i} is a trivializing section of O_{Y} (K_{Y})|_{V_{i}} we have π^{∗}(e_{i}) = 2x_{i}ẽ_{i}. It follows that

ẽ_{i} = \frac{1}{2x_{i}} π^{∗}(e_{i}) = \frac{1}{2} \frac{λ_{ij}}{x_{j}} π^{∗}(w_{ij})π^{∗}(e_{j})
= π^{∗}(λ_{ij}w_{ij})ẽ_{j}

(because π^{∗}(λ_{ij}) = λ_{ij} due to this function being independent of x_{i}). So the transition functions for O_{Y} (K_{Y}) and π^{∗}O_{X}(K_{X}) ⊗ L are the same. Q.E.D.

Proof of (2). Let μ_{ij} be transition functions for M, and let ε_{i} be a trivializing section of M|_{U_{i}}. A global section of π^{∗}(M) is given by {f_{i}(t⃗_{i}, x_{i})ε_{i}} with

f_{i}(t⃗_{i}, x_{i})μ_{ij} = f_{j}(t⃗_{j}, x_{j}).

Now the maps (t⃗_{i}, x_{i}) ↦ (t⃗_{i}, −x_{i}) are compatible and give an automorphism of Y whose quotient is X. We let this automorphism act on H^{0}(π^{∗}M) and write

f_{i}(t⃗_{i}, x_{i}) = f_{i}^{+}(t⃗_{i}, x_{i}) + f_{i}^{−}(t⃗_{i}, x_{i})

where we have decomposed according to +1 and −1 eigenspaces. {f_{i}^{+}} and {f_{i}^{−}} give global sections [since the automorphism was global].

Now f_{i}^{+}(t⃗_{i}, x_{i}) involves only even powers of x_{i}: we may write

f_{i}^{+}(t⃗_{i}, x_{i}) = g_{i}(t⃗_{i}, x_{i}^{2}) = g_{i}(t⃗_{i}, s_{i}(t⃗_{i}))

and so we get a section of M.

Similarly, f_{i}^{−}(t⃗_{i}, x_{i}) involves only odd powers of x_{i}: if we write

f_{i}^{−}(t⃗_{i}, x_{i}) = x_{i}h_{i}(t⃗_{i}, x_{i}^{2}) = x_{i}h_{i}(t⃗_{i}, s_{i}(t⃗_{i}))

then {h_{i}} give a section of M ⊗ L^{−1} since

h_{j} = \frac{f_{j}^{−}}{x_{j}} = \frac{f_{i}^{−}μ_{ij}}{x_{i}λ_{ij}} = h_{i}(\frac{μ_{ij}}{λ_{ij}}).

Q.E.D.

An explicit example of all of this is given in Example C1 in the next section.

4. Elliptic K3 surfaces

I will give a very brief^{14} sketch of the following fact: if X is a K3 surface with a nonsingular connected elliptic curve E and a smooth rational curve C such that C · E = 1, then X is constructed from a Weierstrass equation as in example 1. Note that h^{0}(E) = 2, by basic fact (7) from section 2.

14For more details, see Hartshorne II.3.2, Deligne’s “Formulaire”, or Tate.

Consider X as an elliptic curve E over the function field k(Γ), where Γ is the base of the elliptic fibration X →^{f} Γ, one of whose fibers is E. The curve C (which is a section of f) can be considered as a point P ∈ E.

Now O_{E}(P) is a line bundle of degree 1 on E. By Riemann-Roch, we have h^{0}(nP) = n. Thus, we have sections

1  generating H^{0}(O_{E}(P))
1^{2}, x  generating H^{0}(O_{E}(2P))
1^{3}, 1 · x, y  generating H^{0}(O_{E}(3P))
1^{4}, 1^{2}x, 1 · y, x^{2}  generating H^{0}(O_{E}(4P))
1^{5}, 1^{3}x, 1^{2}y, 1x^{2}, xy  generating H^{0}(O_{E}(5P))
and 1^{6}, 1^{4}x, 1^{3}y, 1^{2}x^{2}, 1xy, x^{3}y^{2} contained in H^{0}(O_{E}(6P)).

It follows that there is an equation relating all of these sections of O_{E}(6P), of the form

c_{1}y^{2} + a_{1}xy + a_{3}y = c_{2}x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Some standard linear algebra reduces this to an equation of the form

y^{2} = x^{3} + b_{4}x + b_{6}

in Weierstrass form.

To complete the analysis, we must see how to do all of this in affine charts, and how the different equations relate in the overlap; one eventually gets that X is the double cover of ℙ(O_{Γ} ⊕ L) for a line bundle L which is divisible by 2, and that b_{4}, b_{6} are sections of L^{⊗2} and L^{⊗3} respectively. Computing the canonical bundle forces the construction to be that of example 1.

If we want X to have only rational double points, we must take the branch locus to have only simple singularities.^{15}

Every elliptic surface which is a K3 surface must be of the type described above. In particular, the parameter curve for the elliptic fibration must be isomorphic to ℙ^{1}. In fact, the canonical bundle formula for an elliptic fibration π : X → C implies that κ(X) = 1 either if g(C) ≥ 2 or if g(C) = 1 and π is not a trivial fibration. (Here, κ denotes the Kodaira dimension.^{16}) Moreover, if g(C) = 1 and π is trivial, then h^{1}(O_{X}) = 2, which prevents X from being a K3 surface. Thus, when X is a K3 surface C must have genus 0.

4.1. Elliptic K3 surfaces, continued. We now use elliptic K3 surfaces to produce some examples of badly-behaved linear systems on K3 surfaces. First, we recall the Kodaira-Ramanujan vanishing theorem (to be proved in section 5).

15Need references to literature.

16This is occurring for the first time.

Theorem (Kodaira-Ramanujan). Let X be a smooth projective surface, let L be a divisor on X which is nef and big. (That is, L · C ≥ 0 for all curves C on X, and L^{2} > 0.) Then H^{i}(O_{X}(K_{X} + L)) = 0 for i > 0.

In section 1, we saw three important properties which many linear systems |K_{X} +L| have: (a) the higher cohomology may vanish, i. e., H^{i}(O_{X}(K_{X} +L)) = 0 for i > 0, (b) |K_{X} + L| may have no base points, and (c) φ_{|K_{X}+L|} may give an embedding. We now give some examples of linear systems on K3 surfaces for which these properties fail. The first example shows that the hypothesis “L^{2} > 0” in the Kodaira-Ramanujan theorem cannot be relaxed.

Example A. Let X be an elliptic K3 surface, and let E be a fiber of the elliptic pencil. Consider L = kE. The map φ_{|L|} factors through the elliptic pencil f : E → Γ ≅ ℙ^{1} and in fact O_{X}(L) = f^{∗}O_{ℙ^{1}}(k). Thus, h^{0}(O_{X}(L)) = h^{0}(O_{ℙ^{1}}(k)) = k + 1. Moreover, since −kE is not effective, h^{2}(O_{X}(L)) = 0. But then

h^{0}(O_{X}(L)) − h^{1}(O_{X}(L)) = \frac{L · L}{2} + 2 = 2

which implies that h^{1}(O_{X}(L)) = k − 1, which is nonzero for k ≥ 2. Note that L^{2} = 0 in this example, and L is nef.

Example B. Let f : X → Γ be an elliptic K3 surface with fiber E, and let C be a section of f. (In particular, X has a Weierstrass form, but we shall not need it for this example.) Consider L = C + kE; note that L^{2} = 2k − 2.

Let us check that L is nef for k ≥ 2. If D is any irreducible curve on X, then D ·(C +kE) < 0 implies that D is a component of C +kE, that is, D = C or D = E. But

C · (C + kE) = k − 2 ≥ 0
E · (C + kE) = 1

so that D · (C + kE) ≥ 0. Hence L = C + kE is nef.

We may then use the Kodaira-Ramanujan vanishing theorem and Riemann-Roch to compute:

h^{0}(O_{X}(L)) = \frac{L · L}{2} + 2 = k + 1.

On the other hand, h^{0}(O_{X}(kE)) = k + 1 as well by Example A. Since the exact sequence

0 → O_{X}(kE) → O_{X}(C + kE) → O_{C}(C + kE) → 0

induces an injection

0 → H^{0}(O_{X}(kE)) → H^{0}(O_{X}(C + kE)),

we see that H^{0}(O_{X}(kE)) ≅ H^{0}(O_{X}(L)) and that C is a fixed component of |L|. In particular, every point of C is a base point of |L|. Note that in this example L is nef, L^{2} > 0 and there exists an E with E^{2} = 0 and L · E = 1.

Example C1. Here are two examples of linear systems without base points which do not give embeddings. Let X be the double cover of ℙ^{2} branched along a curve of degree 6. In the notation of the double cover appendix, we have L = O_{ℙ^{2}}(3) and K_{ℙ^{2}} = −3H so the canonical bundle of the double cover is trivial. If we take M = O_{ℙ^{2}}(n), then H^{0}(π^{∗}M) contains 2 pieces: H^{0}(M), coming from the pullback from ℙ^{2}, and H^{0}(M ⊗ L^{−1}) = H^{0}(O_{ℙ^{2}}(n − 3)). We conclude: all sections of π^{∗}O(1) and π^{∗}O(2) come from ℙ^{2}, so the maps φ_{|π^{∗}O(1)|} and φ_{|π^{∗}O(2)|} factor through the projection to ℙ^{2}. That is, if X is the double cover of ℙ^{2} branched along a smooth curve of degree 6, then |π^{∗}O_{ℙ^{2}}(1)| and |π^{∗}O_{ℙ^{2}}(2)| both factor through the map π : X → ℙ^{2}. In particular, these linear systems do not give embeddings. It is only with |π^{∗}O(3)| that we can embed the K3 surface.

Example C2. To see another example of this phenomenon, let D be a smooth divisor in | − 2K_{𝔽_{0}}|. (Note that 𝔽_{0} ≅ ℙ^{1} × ℙ^{1} ≅ a smooth quadric in ℙ^{3}, and that D is a curve of “type (4,4)”: the complete intersection of 𝔽_{0} with a quartic surface.) Let π : X → 𝔽_{0} be the double cover branched on D. Then X is a K3 surface. If we let C be the graph in ℙ^{1} × ℙ^{1} of a degree k map ℙ^{1} → ℙ^{1}, then C is a smooth rational curve of “type (1, k)”, and C · D = 4k + 4. Thus, π^{−1}(C) is a hyperelliptic curve of genus 2k + 1 by the Hurwitz formula. In particular, if L = π^{−1}(C) then the map φ_{|L|} induces the canonical map |K_{π^{−1}(C)}| on π^{−1}(C). Since that map has degree 2, φ_{|L|} cannot be an embedding.

Note that in this case L is nef, L^{2} > 0, and there is a curve E = π^{−1}(F) (where F is a “type (0,1) fibre” of 𝔽_{0}) which is elliptic with E^{2} = 0 and L · E = 2.

Example C3. The genus of the curves in the previous example was always odd; to get an even genus case, start with 𝔽_{1} ≅ blowup of ℙ^{2} at a point P. Let D̄ be a curve in ℙ^{2} of degree 6 with a node at P so that the proper transform D of D̄ is a smooth curve in | − 2K_{𝔽_{1}}|. Let C̄ be an irreducible curve in ℙ^{2} of degree k with a point of multiplicity (k − 1) at P. [For example, if P = [1, 0, 0], take an equation for C which is the general linear combination of monomials x_{0}^{a}x_{1}^{b}x_{2}^{c} with a + b + c = k, a ≤ 1.] The proper transform C of C̄ is a smooth rational curve. Since the local intersection multiplicity of C̄ and D̄ at P is 2(k − 1), we get C · D = 4k − 2(k − 1) = 4k + 2.

Now we repeat the construction of example C2: π : X → 𝔽_{1} branched on D is a K3 surface with a hyperelliptic curve π^{−1}(C) of genus 2k. The linear system |L| = |π^{−1}(C)| cannot embed X, and we have: L is nef, L^{2} > 0, and there is an elliptic E = π^{−1}(fiber on 𝔽_{1}) with E^{2} = 0 and L · E = 2.

We will see in section 6 that all examples with φ_{|L|} not birational are of these types.

5. Reider’s method

Reider’s method, only a few years old, is now one of the most important tools for studying linear systems on algebraic surfaces. It has almost completely supplanted the older method of studying “d-connectedness” of divisors, although it is closely related to that method. We present Reider’s method here for arbitrary surfaces, and then give a refinement for K3 surfaces in the next section.

We begin with a version of the Hodge index theorem for surfaces.

Theorem. Let L, D be divisors on a smooth projective surface X with L^{2} > 0. Then either

(a) L^{2}D^{2} < (L · D)^{2}, or

(b) L^{2}D^{2} = (L · D)^{2} and D ∼ ∼ ∼ sL for some s ∈ ℚ. (Here, ∼ ∼ ∼ denotes numerical equivalence.)

“Proof”: (Based on the version of the Hodge index theorem given in Hartshorne). Consider the intersection form on (Pic(X)/∼ ∼ ∼)⊗ℚ. Hartshorne’s version of the index theorem says: if H is ample and H · Δ = 0 then either Δ^{2} < 0 or Δ ∼ ∼ ∼ 0. If we choose a basis H, Δ_{1}, …, Δ_{r−1} of (Pic(X)/∼ ∼ ∼) ⊗ ℚ with H · Δ_{i} = 0 for all i, the index theorem says that the intersection form on (Pic(X)/∼ ∼ ∼) ⊗ ℚ has signature (1, r − 1). [It must be negative definite on the span of Δ_{1}, …, Δ_{r−1}.]

Now consider the span of D and L inside (Pic(X)/∼ ∼ ∼) ⊗ ℚ. If this span has dimension 2, then since L^{2} > 0 it must have signature (1,1). [There can be at most one positive eigenvalue.] This is true if and only if

det (\begin{matrix}L^{2} & L · D \\ L · D & D^{2} \end{matrix}) < 0,

giving case (a).

If the span has dimension 1, then D ∼ ∼ ∼ sL for some s ∈ ℚ and an easy computation shows L^{2}D^{2} = (L · D)^{2}. Q.E.D.

We will apply this to prove a very technical-looking lemma, which contains the key computations for the Reider method.

Let us say that 2 divisors L and D on X satisfy condition (∗)_{d} if:

\{ \begin{matrix}L · D ≥ 0 \\ L · (L − 2D) ≥ 0 \\ D · (L − D) ≤ d. \end{matrix} (∗)_{d}

Technical Lemma. Let X be a smooth projective surface and let L and D be divisors such that L^{2} > 0, D ≁ ∼ ∼ 0, and L and D satisfy condition (∗)_{d}. Then either

(i) 0 < L · D ≤ min\{2d, \frac{1}{2}L^{2}\}, max\{0, −d + L · D\} ≤ D^{2} ≤ \frac{(L·D)^{2}}{L^{2}} , or (ii) 0 ≤ L · D ≤ min\{d − 1, \frac{1}{2}L^{2}\}, −d + L · D ≤ D^{2} < 0.

Moreover, if D^{2} = \frac{(L·D)^{2}}{L^{2}} in case (i), then L ∼ ∼ ∼ sD where s = \frac{L^{2}}{L·D}.

Proof. We have

L · D ≤ \frac{1}{2}L^{2} (1)

− d + L · D ≤ D^{2} (2)

which account for 2 of the inequalities in either case.

Case 1. L · D = 0.

Here, we must be in case (ii) and what must be shown is L · D < d and D^{2} < 0. The first is a consequence of the second, in light of (2).

If D^{2} = 0, then L^{2}D^{2} = (L · D)^{2} = 0 so by Hodge index, D ∼ ∼ ∼ sL. This means D ∼ ∼ ∼ 0, contrary to hypothesis.

In any case, D^{2} ≤ (L · D)^{2}/L^{2} = 0, proving this case.

Case 2. L · D > 0, D^{2} ≥ 0.

Multiply eq. (2) by (L · D): (−d + L · D)(L · D) ≤ D^{2}(L · D)

Multiply eq. (1) by D^{2}: D^{2}(L · D) ≤ D^{2}(\frac{1}{2}L^{2})

Use Hodge index: \frac{1}{2}L^{2}D^{2} ≤ \frac{1}{2}(L · D)^{2}.

Thus, since L · D > 0 we get

−d + L · D ≤ \frac{1}{2}(L · D)

or L · D ≤ 2d.

The remaining inequalities are clear.

Case 3. L · D > 0, D^{2} < 0.

All inequalities are clear in this case.

Q.E.D.

Corollary. Under the hypotheses L^{2} > 0, D ≁ ∼ ∼ 0 there are the following possible solutions for (∗)_{0}, (∗)_{1}, (∗)_{2}:

Solution for (∗)_{0}: None.

Solution for (∗)_{1}:

L · D = 2, D^{2} = 1, L^{2} = 4, L ∼ ∼ ∼ 2D

L · D = 1, D^{2} = 0
L · D = 0, D^{2} = −1

Solution for (∗)_{2}: all solutions for (∗)_{1}, and: L · D = 4, D^{2} = 2, L^{2} = 8, L ∼ ∼ ∼ 2D

L · D = 3, D^{2} = 1, 6 ≤ L^{2} ≤ 9, (if L^{2} = 9 then L ∼ ∼ ∼ 3D)
L · D = 2, D^{2} = 0
L · D = 1, D^{2} = −1

L · D = 0, D^{2} = −2.

(The proof consists of enumerating cases in the conclusion of the “technical lemma”.)

For the next step in the Reider method, we need to construct some vector bundles on the surface X. The construction proceeds by means of extensions of sheaves, which are now review.

Suppose that 0 → B → E → A → 0 is a short exact sequence of sheaves of O_{X}-modules, and consider the functor Hom(−, B). This is a half-exact contravariant functor, and leads to a long exact sequence which begins

0 → Hom(A, B) → Hom(E, B) → Hom(B, B) → Ext^{1}(A, B) →

where Ext^{1}(−, B) is the first derived functor. The extension class of the sequence 0 → B → E → A → 0 is the image of the identity map 1_{B} ∈ Hom(B, B) in Ext^{1}(A, B). [Conversely, any element of Ext^{1}(A, B) is the extension class of some extension.] Notice that 1_{B} maps to zero in Ext^{1}(A, B) if and only if there is some map φ : E → B such that the composite B ⊂ E →^{φ} B is the identity on B. That is, the extension class is 0 if and only if the sequence is split, so that E = A ⊕ B.

The key to the bundle constructions we need is to use Serre duality (for sheaves which may not be locally free) to interpret an H^{1} cohomology group as the dual of an Ext^{1} group, and then build an extension. So recall this form of Serre duality (proved in Hartshorne’s book), which we state only for surfaces.

Theorem (Serre Duality). Let X be a smooth projective surface and let F be a sheaf of O_{X}-modules. Then

H^{1}(F ⊗ ω_{X}) ≅ Ext^{1}(F, O_{X})^{∗}.

We use these techniques to build two kinds of vector bundles. First, if L is a line bundle on X with H^{1}(O_{X}(K_{X} + L)) ≠ 0, and e ∈ H^{1}(O_{X}(K_{X} + L))^{∗} is a nonzero element, define E_{e,L} to be the extension

0 → O_{X} → E_{e,L} → O_{X}(L) → 0

with extension class e ∈ H^{1}(O_{X}(K_{X} + L))^{∗} = Ext^{1}(O_{X}(L), O_{X}).

Second, if L is a line bundle on X and Z is a zero-cycle with H^{1}(I_{Z}(K_{X} + L)) ≠ 0, we define a “universal” extension as follows. For any complex vector space V,

Ext^{1}(I_{Z}(L), O_{X} ⊗ V) ≅ Ext^{1}(I_{Z}(L), O_{X}) ⊗ V.

In particular, this holds for V = H^{1}(I_{Z}(K_{X} + L)) ≅ Ext^{1}(I_{Z}(L), O_{X})^{∗}. We may regard the identity mapping on Ext^{1}(I_{Z}(L), O_{X}) as an element

id ∈ Ext^{1}(I_{Z}(L), O_{X}) ⊗ Ext^{1}(I_{Z}(L), O_{X})^{∗} ≅ Ext^{1}(I_{Z}(L), O_{X} ⊗ H^{1}(I_{Z}(K_{X} + L))

and thus get an extension

0 → O_{X} ⊗ H^{1}(I_{Z}(K_{X} + L)) → E(Z, L) → I_{Z}(L) → 0.

Before proceeding, let’s pause and show why this particular cohomology group H^{1}(I_{Z}(K_{X}+L)) is so interesting. Suppose we have a line bundle L with H^{1}(O_{X}(K_{X}+ L)) = 0, and we consider the exact sequence

0 → I_{Z}(K_{X} + L) → O_{X}(K_{X} + L) → O_{Z}(K_{X} + L) → 0.

The long exact cohomology sequence has 4 interesting terms:

\begin{aligned}0 & → & H^{0}(I_{Z}(K_{X} + L)) & → & H^{0}(O_{X}(K_{X} + L)) & → & H^{0}(O_{Z}(K_{X} + L)) \\ & → & H^{1}(I_{Z}(K_{X} + L)) & → & 0. \end{aligned}

We say that Z fails to impose independent conditions on |K_{X}+L| if H^{1}(I_{Z}(K_{X}+L)) ≠ 0. The key cases for the Reider method will be (1) Z = P is a point (in which case this condition means that P is a base point) or (2) Z = P + Q is a pair of points, possibly infinitely near (in which case this condition means that either P or Q is a base point, or the map φ_{|K_{X}+L|} fails to separate P and Q [in the infinitely near case: fails to have injective differential at P]). We will use the sheaf E(Z, L) to extract information about these situations.

We need the following lemma, whose proof we do not give here.

Lemma. Let X be a smooth projective surface, L be a nef and big divisor (i) If e is a general element of H^{1}(O_{X}(K_{X} + L))^{∗} then E_{e,L} is locally free. (ii) If for every Z′ ⫋ Z we have h^{1}(I_{Z′}(K_{X} + L)) < h^{1}(I_{Z′}(K_{X} + L)) then E(Z, L)

is locally free.

In these cases, we let E_{e,L} and E(Z, L) denote the corresponding vector bundles, and write

E_{e,L} = O_{X}(E_{e,L});
E(Z, L) = O_{X}(E(Z, L)).

We will be primarily interested in the case of rank 2 bundles; notice that in this case we have a sequence

0 → O_{X} → O_{X}(E) → I_{Z}(L) → 0

where E = E_{e,L} or E(Z, L), and Z = ∅ in the first case (assuming rank E = 2).

Definition. Let E be a rank 2 vector bundle on a smooth projective surface X.

(i) We say that E has the strong Bogomolov property if there are a zero-cycle A, line bundles M, N ∈ Pic(X) and an exact sequence

0 → M → O_{X}(E) → I_{A} ⊗ N → 0

such that h^{0}((M ⊗ N^{−1})^{⊗k}) > 0 for some k > 0.

(ii) If L = c_{1}(E) is nef, we say that E has the weak Bogomolov property if there are a zero-cycle A, line bundles M, N ∈ Pic(X) and an exact sequence

0 → M → O_{X}(E) → I_{A} ⊗ N → 0

such that L · (M ⊗ N^{−1}) ≥ 0.

[Clearly, when c_{1}(E) is nef the strong Bogomolov property implies the weak one. Notice also: O_{X}(L) = M ⊗ N.]

The reason for making this somewhat strange looking definition is

Bogomolov’s Theorem. If E is a rank 2 vector bundle on a smooth projective surface X with c_{1}^{2}(X) > 4c_{2}(X) then E has the strong Bogomolov property.

The proof of this theorem is far beyond the scope of this course; it is essential for doing Reider’s method on arbitrary surfaces, but as we will see in the next section, for K3 surfaces we get better results by using a slightly different method (which avoids Bogomolov’s theorem).

The final ingredient in Reider’s method is the following proposition:

Proposition. Let E be a rank 2 vector bundle on a smooth projective surface X such that L = c_{1}(E) is nef and big. Suppose that there is a section s : O_{X} → O_{X}(E) whose zero-locus Z has dimension 0. [This is the case if and only if we have an exact sequence of sheaves

0 → O_{X} →^{s} O_{X}(E) → I_{Z}(L) → 0.] (*)

If E has the weak Bogomolov property with associated sequence

0 → M → O_{X}(E) → I_{A} ⊗ N → 0, (**)

then there is an effective divisor D containing Z such that N = O_{X}(D). Moreover, if D = 0 then Z = ∅ and the sequence (*) splits.

Proof. We assemble (*) and (**) into a diagram and consider the induced map α :

O_{X} → I_{A} ⊗ N shown below.

0

↓

O_{X}

↓s ↘^{α}
0 ⟶ M ⟶ O_{X}(E) ⟶ I_{A} ⊗ N ⟶ 0

↓

I_{Z}(L)

↓

0

Suppose first that α is identically 0. Then the image of s lies in M, and there is an induced map O_{X} → M. For x ∉ Z ∪ A, the maps on fibers O_{X,x} → E_{x} and M_{x} → E_{x} are injective; thus, O_{X,x} → M_{x} is an isomorphism for such values of x. But a map between line bundles which is an isomorphism away from a codimension 2 set like Z ∪ A must be an isomorphism everywhere; it follows that M ≅ O_{X}.

But now O_{X}(L) = M ⊗ N = N so that M ⊗ N^{−1} = O_{X}(−L). The weak Bogomolov property then implies that L · (−L) ≥ 0, contradicting the assumption “L is big”.

Thus, α is not identically zero, so it defines a non-trivial section of I_{A} ⊗ N; composing with the inclusion I_{A} ⊗ N ⊂ N we get a section of N. The zero-locus D of this section contains Z (since the map s, through which our section O_{X} → N factors, vanishes on Z), and satisfies N = O_{X}(D).

It remains to prove the last statement. Suppose that D = 0. Since D is by definition the subset of X where the composite map O_{X} →^{α} I_{A} ⊗ N ⊂ N fails to be surjective on fibers, we see that this map is an isomorphism. (In particular, I_{A} = O_{X} and A = ∅.) Now in the diagram

0

↓

O_{X}

↓  ↘^{α}
O_{X}(E) ⟶^{β} N ⟶ 0

↓

I_{Z}(L)

↓

0

the map α^{−1} ∘ β : O_{X}(E) → O_{X} gives a splitting of the sequence. This implies that O_{X}(E) = O_{X} ⊕ I_{Z}(L); since E is locally free, Z must be empty as well. Q.E.D.

We can now give Mumford’s proof of the Kodaira-Ramanujan vanishing theorem, which is the “0^{th} case” of Reider’s method. (In retrospect, this proof is a special case of Reider’s method, but in fact it preceded Reider’s work by about 10 years.)

Mumford’s proof of Kodaira-Ramanujan vanishing. Let L be nef and big, and suppose H^{1}(O_{X}(K_{X} + L)) ≠ 0. Choose e ≠ 0 to be a general element of H^{1}(O_{X}(K_{X} + L))^{∗} and consider the vector bundle E_{e,L}, which has a defining sequence

0 → O_{X} → O_{X}(E_{e,L}) → O_{X}(L) → 0.

Since e ≠ 0, this sequence is not split.

Now c_{1}(E_{e,L}) = L and c_{2}(E_{e,L}) = 0. Thus, c_{1}^{2}(E_{e,L}) = L^{2} > 0 = 4c_{2}(E_{e,L}) so that Bogomolov’s theorem applies, and we have the strong (and the weak) Bogomolov property:

0 → M → O_{X}(E_{e,L}) → I_{A} ⊗ N → 0.

By the proposition, N = O(D) for some effective divisor D; moreover, since the defining sequence is not split, D ≠ 0. This implies that D ≁ ∼ ∼ 0, since D is effective.

Now L · D ≥ 0 since L is nef and D is effective; L · ((L − D) − D) ≥ 0 since M = O_{X}(L − D), N = O_{X}(D); and

0 = c_{2}(E_{e,L}) = c_{2}(M) + c_{2}(I_{A} ⊗ N) + c_{1}(M) · c_{1}(I_{A} ⊗ N)
= deg A + (L − D) · D ≥ (L · D) · D

so that L and D satisfy (∗)_{0}. But by the technical lemma, there are no solutions to (∗)_{0} with D ≁ ∼ ∼ 0 and L^{2} > 0, a contradiction.

Hence, H^{1}(O_{X}(K_{X} + L)) = 0. Q.E.D.

We assemble all of our pieces of the Reider method for E(Z, L) in the rank 2 case into the following theorem.

Theorem (Reider’s method). Let X be a smooth projective surface, let L be a nef and big line bundle on X, and let Z be a zero-cycle of degree d > 0 such that h^{1}(I_{Z}(K_{X} + L)) = 1 but for every Z′ ⫋ Z, h^{1}(I_{Z′}(K_{X} + L)) = 0. Suppose that the vector bundle E(Z, L) satisfies the weak Bogomolov property. Then there is an effective divisor D containing Z such that L and D satisfy (∗)_{d} [with L^{2} > 0 and D ≁ ∼ ∼ 0]. In particular, L and D satisfy the conclusion of the “technical lemma”: either

(i) 0 < L · D ≤ min\{2d, \frac{1}{2}L^{2}\}, max\{0, −d + L · D\} ≤ D^{2} ≤ \frac{(L·D)^{2}}{L^{2}} , or (ii) 0 ≤ L · D ≤ min\{d − 1, \frac{1}{2}L^{2}\}, −d + L · D ≤ D^{2} < 0.

Proof. Since h^{1}(I_{Z}(K_{X} +L)) = 1 and h^{1}(I_{Z′}(K_{X} +L)) = 0 for all Z′ ⫋ Z, the vector bundle E(Z, L) exists and has rank 2. Its defining sequence has the form

0 → O_{X} → O_{X}(E(Z, L)) → I_{Z}(L) → 0.

Since E(Z, L) satisfies the weak Bogomolov property, there is another sequence

0 → M → O_{X}(E(Z, L)) → I_{A} ⊗ N → 0

with L · (M ⊗ N^{−1}) ≥ 0. By the proposition, N = O_{X}(D) for an effective divisor D containing Z. Since Z ≠ ∅, D ≠ 0 so D ≁ ∼ ∼ 0.

We compute Chern classes of E = E(Z, L):

L = c_{1}(E) = c_{1}(M) + c_{1}(I_{A} ⊗ N) = c_{1}(M) + D

which implies c_{1}(M) = L − D, and

d = deg Z = c_{2}(E) = c_{2}(M) + c_{2}(I_{A} ⊗ N) + c_{1}(M) · c_{1}(I_{A} ⊗ N)
= deg(A) + c_{1}(M) · c_{1}(N).

which implies d = deg(A) + (L − D) · D.

It remains to verify (∗)_{d}. Since L is nef and D is effective, L · D ≥ 0. By the weak Bogomolov property, since M ⊗ N^{−1} = O_{X}((L − D) − D) we have L · (L − 2D) ≥ 0. And finally, by the computation of c_{2},

d = deg A + (L − D) · D ≥ (L − D) · D.

Q.E.D.

As an application, we prove Reider’s original theorem.

Reider’s Theorem. Let X be a smooth projective surface, and let L be a nef line bundle on X.

(I) If P is a base point of |K_{X} + L| and L^{2} ≥ 5 then there is an effective divisor D containing P such that either

(a) L · D = 0, D^{2} = −1

or

(b) L · D = 1, D^{2} = 0. (II) If P and Q are not base points of |K_{X} + L|, and P and Q are not separated by

the map φ_{|K_{X}+L|} (including the infinitely mear case in which the differential of φ_{|K_{X}+L|} at P has a kernel in the direction corresponding to the infinitely near point Q), and if L^{2} ≥ 9, then there is an effective divisor D containing P + Q such that either

(a) L · D = 0, D^{2} = −1 or −2

(b) L · D = 1, D^{2} = 0 or −1

(c) L · D = 2, D^{2} = 0 or

(d) L · D = 3, D^{2} = 1, L^{2} = 9, L ∼ ∼ ∼ 3D.

Proof. (I) Let Z = P and note that H^{0}(O_{Z}(K_{X} + L)) ≅ ℂ. In view of the exact sequence

0 → H^{0}(I_{Z}(K_{X} + L)) → H^{0}(O_{X}(K_{X} + L)) → ℂ → H^{1}(I_{Z}(K_{X} + L)) → 0

we have h^{1}(I_{Z}(K_{X} + L)) = 1 if and only if P is a base point of |K_{X} + L|. Moreover,

c_{1}^{2}(E(Z, L)) = L^{2} > 4 deg(Z) = 4c_{2}(E(Z, L)),

so by Bogomolov’s theorem, E(Z, L) satisfies the Bogomolov property. The statement now follows from the previous theorem together with the list of solutions to (∗)_{1}. (We omitted all cases with L^{2} ≤ 4.)

(II) This time, if Z = P + Q we have H^{0}(O_{Z}(K_{X} + L)) ≅ ℂ^{2}. If neither P nor Q is a base point, the image of the map H^{0}(O_{X}(K_{X} + L)) → H^{0}(O_{X}(K_{X} + L)) has dimension at least 1, and it has dimension exactly 1 if and only if P and Q are not separated by φ_{|K_{X}+L|}. In this case, h^{1}(I_{Z}(K_{X} + L)) = 1 while for Z′ ⫋ Z (i. e. Z′ = P or Z′ = Q), h^{1}(I_{Z′}(K_{X} + L)) = 0 since neither is a base point.

In this case, c_{1}^{2}(E(Z, L)) = L^{2} > 4 deg(Z) = 4c_{2}(E(Z, L)) so the statement again follows from Bogomolov’s theorem, the previous theorem, and the list of solutions to (∗)_{2} (with L^{2} ≥ 9). Q.E.D.

Corollary 1. Let L be a nef line bundle on a smooth projective surface X such that L^{2} ≥ 10. If φ_{|K_{X}+L|} is not birational, then there is a (possibly irrational) pencil {D_{t}} such that D_{t}^{2} = 0 and L · D_{t} = 1 or 2.

Proof. Assume first that the base locus of |K_{X} + L| is a proper subvariety of X. If φ_{|K_{X}+L|} is not birational, then there is some Zariski-open subset U ⊂ X such that no point of U is a base point of |K_{X} + L|, but for every P ∈ U there is some Q ≠ P, Q ∈ U which is not separated from P by the map φ_{|K_{X}+L|}. Thus, every point P ∈ U must be contained in some curve D from cases (a), (b), or (c) of part (II) of Reider’s theorem. Since each such D contains a 1-parameter family of points but U has dimension 2, there must be at least a 1-parameter family of such curves D. If we taken an irreducible component of this family of dimension ≥ 1, it cannot consist of curves D with D^{2} < 0, since such curves do not move in algebraic families. Thus, there is a family of curves {D_{t}} with parameter space of dimension ≥ 1 such that D_{t}^{2} = 0 and L · D_{t} = 1 or 2.

A similar argument in the case that the base locus of |K_{X} + L| is all of X shows the existence of a family {D_{t}} with D_{t}^{2} = 0 and L · D_{t} = 1 (from part (I) of Reider’s theorem). Q.E.D.

Corollary 2.

(i) If −K_{X} is nef and big then |−3K_{X}| is birational. (Del Pezzo surfaces)

(ii) if K_{X} is nef and big then |5K_{X}| is birational. (Surfaces of general type)

Proof. Take L = ∓4K_{X}. Then L^{2} = 16K_{X}^{2} ≥ 16. Furthermore, L · D is always divisible by 4, so the cases D^{2} = 0, L · D = 1 or 2 cannot occur. By Corollary 1, |K_{X} + L| is birational. Q.E.D.

Remark. If we take L = ∓3K_{X} then L^{2} = 9K_{X}^{2} ≥ 9. A similar argument shows: if in addition K_{X}^{2} > 1, then |−2K_{X}| resp. |4K_{X}| is birational.

5.1. Addendum to section 5. There is some further information on generalized Del Pezzo surfaces which can easily be obtained from Reider’s theorem.

Proposition. Let X be a generalized Del Pezzo surface, that is, a surface for which −K_{X} is nef and big, and let m ≥ 1.

(1) If |−mK_{X}| has a base point, then m = 1 and K_{X}^{2} = 1.

(2) If there are two points P and Q which are not separated by φ_{|−mK_{X}|} and which do not lie on smooth rational curves with self-intersection −2, then either m = 1, K_{X}^{2} ≤ 2 or m = 2, K_{X}^{2} = 1.

Proof. Let L = −(m + 1)K_{X}, so that |K_{X} + L| = |−mK_{X}|.

(1) L^{2} ≥ 5 if and only if m = 1, K_{X}^{2} ≥ 2 or m ≥ 2. In this case, if there is a base point then there is an effective divisor D with either L · D = 0, D^{2} = −1 or L · D = 1, D^{2} = 0. In the first case, K_{X} · D + D^{2} = −1 while in the second case, K_{X} · D + D^{2} = −\frac{1}{m+1}. In neither case can K_{X} · D + D^{2} be an even integer, so such a D cannot exist.

2) L^{2} ≥ 10 if and only if m = 1, K_{X}^{2} ≥ 3 or m = 2, K_{X}^{2} ≥ 2 or m ≥ 3. In this case, if there exist such points P and Q which are not base points, then there is an effective divisor D with either one of the properties in (1) [which is impossible] or L · D = 1, D^{2} = −1 or L·D = 2, D^{2} = 0. We compute again: K_{X} ·D+D^{2} = \frac{−1}{m+1} −1 in the first case, and = \frac{−2}{m+1} in the second case; again, neither can be an even integer. Q.E.D.

Before leaving the topic of Reider’s method in general, we give a bit more information about the condition h^{1}(I_{Z′}(K_{X} + L)) < h^{1}(I_{Z}(K_{X} + L)) for Z′ ⫋ Z in a slightly special case.

Suppose that X is regular (so that H^{1}(O_{X}(K_{X})) ≅ H^{1}(O_{X})^{∗} = 0) and that there is a smooth curve C ∈ |L| containing Z. Then we can make a big diagram of exact sheaf sequences:

0  0
↓  ↓
I_{C/X}(K_{X} + L) = I_{C/X}(K_{X} + L)
↓  ↓
0 → I_{Z/X}(K_{X} + L) → O_{X}(K_{X} + L) → O_{Z}(K_{X} + L) → 0
↓  ↓  ||
0 → I_{Z/C}(K_{X} + L) → O_{C}(K_{X} + L) → O_{Z}(K_{X} + L) → 0
↓  ↓
0  0

Using the standard isomorphism I_{C/X}(K_{X} + L) ≅ O_{X}(K_{X}), I_{Z/C}(K_{X} + L) ≅ O_{C}(K_{C} − Z), and O_{C}(K_{X} + L) ≅ O_{C}(K_{C}), this becomes:

0  0
↓  ↓
O_{X}(K_{X})  = O_{X}(K_{X})
↓  ↓
0 → I_{Z/X}(K_{X} + L) → O_{X}(K_{X} + L) → O_{Z}(K_{X} + L) → 0
↓  ↓  ||
0 → O_{C}(K_{C} − Z) → O_{C}(K_{C}) → O_{Z}(K_{C}) → 0
↓  ↓
0  0

This has the following interpretation: H^{1}(I_{Z/X}(K_{X} + L)) measures the failure of the points Z to impose independent conditions on the linear system |K_{X} + L|. Now |K_{X} + L| induces the canonical linear system |K_{C}| on C, so we are measuring the linear dependence relations among the points Z in the canonical space ℙH^{0}(O_{C}(K_{C})). The “geometric version of Riemann-Roch” (see Griffiths-Harris), relates this to H^{0}(O_{C}(Z)). In our case, looking at the long exact sequence associated to the left vertical sequence we find (since H^{1}(O_{X}(K_{X})) = 0, H^{2}(O_{X}(K_{X})) ≅ H^{0}(O_{X})^{∗} ≅ ℂ, and H^{2}(I_{Z/X}(K_{X} + L)) ≅ H^{2}(O_{X}(K_{X} + L)) ≅ H^{0}(O_{X}(−L))^{∗} = (0)):

0 → H^{1}(I_{Z/X}(K_{X} + L)) → H^{1}(O_{C}(K_{C} − Z)) → ℂ → 0
||≀
H^{0}(O_{C}(Z))^{∗}

In particular, if r = h^{1}(I_{Z/X}(K_{X} + L)) then h^{0}(O_{C}(Z) = r + 1 and E(Z, L) has rank r + 1.

The condition h^{1}(I_{Z′/X}(K_{X} + L)) < h^{1}(I_{Z/X}(K_{X} + L)) for all Z′ ⫋ Z can now be interpreted in the following way: h^{0}(O_{Z′}(C)) < h^{0}(O_{Z}(C)). In other words, this condition means that the linear system |Z| on X has no base points.

5.2. Second addendum to section 5. When the bundle E(Z, L) comes from a base-point-free linear system |Z| on a smooth curve C ∈ |L|, there is an easy way to guarantee that E(Z, L) is generated by its global sections (when X is regular). Namely, Z ⊂ C ⊂ X gives rise to the exact sequence

0 → I_{C/X}(L) → I_{Z/X}(L) → I_{Z/C}(L) → 0

which can also be written as

0 → O_{X} → I_{Z/X}(L) → O_{C}(K_{C} − Z) → 0. (*)

Now the map H^{0}(O_{X}(E(Z, L)) → H^{0}(I_{Z/X}(L)) is surjective since X is regular; thus, (*) induces an extra section of E(Z, L) and there is an exact sequence

0 → (O_{X})^{⊕r} ⊕ O_{X} → O_{X}(E(Z, L)) → O_{C}(K_{C} − Z) → 0.

This sequence is exact on global sections; thus, E(Z, L) is generated by global sections if O_{C}(K_{C} −Z) is. The latter happens whenever |K_{C} −Z| has no base points. Thus, if both |Z| and |K_{C} −Z| have no base points, O_{C}(E(Z, L)) is locally free and generated by global sections.

6. Linear systems on K3 surfaces

We will use Reider’s method to investigate linear systems on K3 surfaces, but with one difference: instead of Bogomolov’s theorem, we will find another technique for ensuring that the vector bundles E(Z, L) have the weak Bogomolov property. To begin, we need to give the computation of χ(E ⊗ E^{∗}) due to Mukai and Lazarsfeld.

Computing Euler characteristics for vector bundles requires the Hirzebruch-Rie-mann-Roch theorem, which we now review. Let E be a vector bundle of rank m on a smooth projective variety X of dimension n, and let

c_{t}(E) = 1 + c_{1}(E)t + ⋯ + c_{n}(E)t^{n}

be the Chern polynomial. There is a “splitting principle” for calculating Chern classes which states that any formula which can be proved under the assumption that E is a direct sum of line bundles in fact holds in general. If we pretend that E = L_{1} ⊕ ⋯ ⊕ L_{m} and write c_{t}(L_{j}) = 1 + λ_{j}t, then by the multiplicativity of c_{t}(E) this splitting principle corresponds to a formal factorization

c_{t}(E) = ∏_{j=1}^{m}(1 + λ_{j}t) mod t^{n+1}

and so c_{j}(E) is the j^{th} elementary symmetric function in {λ_{1}, …, λ_{m}}.

Let us work now in the graded ring ⨁ H^{2i}(X, ℤ), in which we denote an element by [a_{0}, a_{1}, …] and the i^{th} component by [a_{0}, a_{1}, …]_{i} = a_{i}. (The grading is given by deg H^{2i}(X, ℤ) = i.) We may regard λ_{j} as an element of H^{2}(X, ℤ), and e^{λ_{j}} as an element of our ring (via truncated power series): [1, λ_{j}, λ_{j}^{2}/2!, …]. With these conventions, define the Chern character of E to be

ch(E) = ∑_{j=1}^{m} e^{λ_{j}} ∈ ⨁ H^{2i}(X, ℤ).

This definition is to be interpreted in the non-split case as follows: it is symmetric in {λ_{1}, …, λ_{m}}, and so can be written in terms of the elementary symmetric functions, and hence in terms of the Chern classes c_{i}(E).

The first few terms in the Chern character are

ch(E) = [rank(E), c_{1}(E), \frac{c_{1}^{2}(E) − 2c_{2}(E)}{2}, \frac{c_{1}^{3}(E) − 3c_{1}(E)c_{2}(E) + 3c_{3}(E)}{6}, …].

Here are some properties of ch(E) which can easily be verified from the definition (and the splitting principle):

(1) ch(E ⊕F) = ch(E)+ch(F) (More generally, ch is additive in exact sequences)

(2) ch(E ⊗ F) = ch(E) · ch(F)

(3) ch(E^{∗})_{i} = (−1)^{i} ch(E)_{i}.

The Hirzebruch-Riemann-Roch theorem also involves the Todd class of X, another element of ⨁ H^{2i}(X, ℤ). This is a bit complicated to define, but it begins as

td(X) = [1, \frac{c_{1}(X)}{2}, \frac{c_{1}^{2}(X) + c_{2}(X)}{12}, \frac{c_{1}(X)c_{2}(X)}{24}, …]

where c_{i}(X) = c_{i}(T_{X}) are the Chern classes of the tangent bundle. The Hirzebruch-Riemann-Roch theorem says:

χ(E) = (ch(E) · td(X))_{n}.

As an example, consider a line bundle L on a surface X. We have c_{t}(L) = 1 + c_{1}(L)t and so ch(L) = [1, L, \frac{1}{2}L^{2}]. In addition, for a surface we have td(X) = [1, −\frac{1}{2}K_{X}, X(O_{X})]. Thus,

χ(L) = χ(O_{X}) + −\frac{1}{2}K_{X} · L + \frac{1}{2}L^{2}

giving the familiar formula.

We return to the case of a K3 surface X, and consider the vector bundle E = E(Z, L) on X. We define r + 1 = rank E, d = deg Z, and 2g − 2 = L^{2} = c_{1}^{2}(E). In particular, in the special case in which there is a smooth curve C ∈ |L| containing Z, the linear system |Z| on C satisfies: genus(C) = g, deg(O_{C}(Z)) = d, h^{0}(O_{C}(Z))−1 = r. (It is also base-point-free, to get the vector bundle E(Z, L).)

Now comes the miraculous computation of Mukai and Lazarsfeld:

ch(E) = [r + 1, c_{1}(E), \frac{c_{1}^{2}(E) − 2c_{2}(E)}{2}]
ch(E^{∗}) = [r + 1, −c_{1}(E), \frac{c_{1}^{2}(E) − 2c_{2}(E)}{2}]
ch(E ⊗ C^{∗}) = [(r + 1)^{2}, 0, rc_{1}^{2}(E) − (2r + 2)c_{2}(E)]
td(X) = [1, 0, 2]

since X is a K3 surface. Thus

χ(E ⊗ E^{∗}) = 2(r + 1)^{2} + rc_{1}^{2}(E) − (2r + 2)c_{2}(E)
= 2(r + 1)^{2} + r(2g − 2) − (2r + 2)d
= 2 − 2[−(r + 1)^{2} − (r + 1)(g − 1) + g + (r + 1)d]
= 2 − 2[g − (r + 1)((r + 1) + (g − 1) + d)]
= 2 − 2(g − (r + 1)(r − d + g)).

The miracle is this: the number

ρ(g, r, d) = g − (r + 1)(r − d + g)

is called the Brill-Noether number, and is very important in the theory of special linear systems on curves. As an example, we have:

Part of the Brill-Noether Theorem. If C is a general curve of genus g, then every g_{d}^{r} in C (that is, a linear system with h^{0} = r + 1 and degree d) satisfies ρ(g, r, d) ≥ 0.

So the Mukai-Lazarsfeld computation says: χ(E ⊗ E^{∗}) = 2 − 2ρ(g, r, d), where g = \frac{1}{2}c_{1}^{2}(E) + 1, r = rank(E) − 1 and d = c_{2}(E). But even more is true: by Serre duality, since K_{X} = 0 we have

H^{2}(E ⊗ E^{∗}) ≅ H^{0}((E ⊗ E^{∗})^{∗} ⊗ K_{X})^{∗}
≅ H^{0}(E^{∗} ⊗ E)^{∗}

so that h^{2}(E ⊗ E^{∗}) = h^{0}(E ⊗ E^{∗}). In particular

2h^{0}(E ⊗ E^{∗}) − h^{1}(E ⊗ E^{∗}) = 2 − 2ρ.

The conclusion is: if ρ < 0 then h^{0}(E ⊗ E^{∗}) > 1. (To restate in the case Z ⊂ C: if the linear system O_{C}(Z) has negative Brill-Noether number, then the vector bundle E(Z, C) has an extra endomorphism.) It is this extra endomorphism which gives us the weak Bogomolov property.

Proposition. Let E be a vector bundle of rank 2 on a smooth projective surface X. Suppose that L = c_{1}(E) is nef, and that h^{0}(E ⊗ E^{∗}) > 1. Then E has the weak Bogomolov property.

Proof. H^{0}(E ⊗ E^{∗}) ≅ Hom(E, E); let φ : E → E be a (sheaf) homomorphism which is not a scalar multiple of the identity 1_{E}. (This exists since dim Hom(E, E) > 1.)

If we choose a point x ∈ X, and choose an eigenvalue λ of the fiber map φ_{x} : E_{x} → E_{x}, then φ̃ = φ−λ·1_{E} has the property that rank(φ̃) is less than 2 somewhere, while φ̃ ≢ 0. We replace φ by φ̃, and assume this about φ.

Now Ker(φ) is a subsheaf of the locally free sheaf O_{X}(E), and so is torsion-free. Thus, it has a rank, and away from a set of codimension 2 on X it is locally free of that rank. [In fact, since E/ Ker(φ) ≅ Im(φ) ⊂ E is also torsion-free, Ker(φ) is reflexive and hence locally free.] Since

0 ⫋ Ker(φ) ⫋ E

we see that rank(Ker(φ)) = 1; this implies that Ker(φ) is a line bundle.

We have naturally Ker(φ) ⊂ Ker(φ^{2}) ⊂ E, and Ker(φ^{2}) is locally free. Thus, either it has rank 1 and Ker(φ) = Ker(φ^{2}), or it has rank 2 and Ker(φ^{2}) = E.

If Ker(φ) = Ker(φ^{2}), then I claim that φ : Im(φ) → E is injective. For if x ∈ Im(φ) with φ(x) = 0 then x = φ(y) for some y and thus φ^{2}(y) = 0. But then φ(y) = 0 so that x = 0, proving the claim. Thus, φ gives a splitting of the sequence

0 → Ker(φ) → E →^{\underleftarrow{φ}} Im(φ) → 0

and we have E = Ker(φ) ⊕ Im(φ). If L · c_{1}(Ker φ) ≥ L · c_{1}(Im(φ)) then

0 → Ker(φ) → E → Im(φ) → 0

provides a Bogomolov sequence; if L · c_{1}(Ker φ) ≤ L · c_{1}(Im φ) then

0 → Im(φ) → E → Ker(φ) → 0

is the desired sequence.

On the other hand, if Ker(φ^{2}) = E then Im φ ⊂ Ker φ so there is a non-trivial section O → O(Ker φ) ⊗ O(Im φ)^{∗}. Since L is nef, this implies L · (O(Ker φ) ⊗ O(Im φ)^{∗}) ≥ 0. But then

0 → Ker φ → E → Im φ → 0

has the weak Bogomolov property. Q.E.D.

We now come to the first main theorem about linear systems on K3 surfaces.

Theorem. Let X be a smooth projective K3 surface, and let L be a nef line bundle on X.

(I) If P is a base point of |L| and L^{2} ≥ 2 then there is an effective divisor D

containing P such that L · D = 1, D^{2} = 0. (II) If P and Q are not base points of |L|, and P and Q are not separated by the

map φ_{|L|} (including the infinitely near case in which the differential of φ_{|L|} at P has a kernel in the direction corresponding to the infinitely near point Q), and if L^{2} ≥ 4 then there is an effective divisor D containing P + Q such that either

(a) L · D = 0, D^{2} = −2,

(b) L · D = 1 or 2, D^{2} = 0, or

(c) L · D = 4, D^{2} = 2, L^{2} = 8, L ∼ ∼ ∼ 2D.

Proof. As in the proof of Reider’s theorem, the key step is to show that bundles E(Z, L) satisfy the weak Bogomolov property when deg Z = 1 (for part (I)), and deg Z = 2 (for part (II)). (We have Z = P and Z = P + Q, respectively.) We have r + 1 = 2 and for deg Z = d:

ρ = g − 2(1 − d + g) = 2d − 2 − g.

Thus, when g > 0 in the case d = 1 (i. e. L^{2} > 0 to have nef and big) or when g > 2 in case d = 2 (i. e. L^{2} ≥ 4) we have ρ < 0; by the “miraculous computation” of χ(E ⊗ E^{∗}) and the previous proposition, E satisfies the weak Bogomolov property. Thus, by the “Reider’s method theorem”, there is an effective divisor D containing Z satisfying (∗)_{1} resp. (∗)_{2}. Noting that D^{2} ∈ 2ℤ for a K3 surface, we get the solutions listed in the theorem. Q.E.D.

To complete the story of linear systems on K3 surfaces we need some converse statements.

Proposition 1. Let X be a smooth projective K3 surface, let L be a nef and big line bundle on X, and suppose there is an effective divisor D such that L·D = 1, D^{2} = 0. Then |L| has a fixed component.

Proof. Consider the divisor L − gD, where L^{2} = 2g − 2. We use the following:

Standard Trick. On a K3 surface, if Γ is a divisor with Γ^{2} ≥ −2 then Γ or −Γ is effective. If Γ^{2} ≥ −2 and L · Γ > 0 for some nef divisor L, then it is Γ which is effective.

Proof of the Trick.

H^{2}(O_{X}(Γ)) ≅ H^{0}(O_{X}(K_{X} − Γ))^{∗}
≅ H^{0}(O_{X}(−Γ))^{∗}.

Thus, h^{0}(Γ) + h^{0}(−Γ) ≥ χ(Γ) = \frac{1}{2}(Γ^{2}) + 2 ≥ 1 so either Γ or −Γ is effective. If L · Γ > 0 and L is nef then −Γ cannot be effective. Q.E.D.

Resuming the proof of the proposition, we have

(L − gD)^{2} = 2g − 2 − 2g = −2
L · (L − gD) = 2g − 2 − g = g − 2.

If g ≥ 3 then L − gD is effective; if g = 2 then either L − 2D or 2D − L is effective. In the latter case let D̃ = L − D. Then L · D̃ = 1, (D̃)^{2} = 0 and L − 2D̃ = 2D − L is effective. Thus, replacing D by D̃ if necessary, we may assume that L − gD is effective.

But now h^{0}(D) ≥ 2 implies h^{0}(gD) ≥ g + 1 while h^{0}(L) = g + 1. Thus, L − gD is a fixed component of |L|. Q.E.D.

Proposition 2. Let |L| be a nef and big linear system on a smooth K3 surface without base points. Suppose that there is an effective divisor D on X such that L · D = 2 and D^{2} = 0. Then φ_{|L|} has degree 2; moreover, any smooth C ∈ |L| is hyperelliptic.

Proof. By Bertini’s theorem, the general C ∈ |L| is smooth. Now φ_{|L|}|_{C} induces the canonical map φ_{|K_{C}|} on C. On the other hand, the linear system O_{C}(D) has degree 2 and dimension 1, so that C is hyperelliptic. Thus, φ_{|K_{C}|} = φ_{|L|}|_{C} has degree 2; since this is true for the general hyperplane section φ_{|L|} itself has degree 2. Q.E.D.

In the course of proving the next proposition, we will need some facts about rational double points, whose proofs we omit.

Facts about rational double points. Let C_{1}, …, C_{n} be a collection of smooth rational curves on a smooth surface X such that C_{i}^{2} = −2 and ∪C_{i} is connected, and suppose that the intersection matrix (C_{i} ·C_{j}) is negative definite. Then there is a linear combination C = ∑ n_{i}C_{i} with n_{i} ∈ ℤ, n_{i} > 0 such that −C · C_{i} ≥ 0 for all i, and C^{2} = −2. Moreover, there is a contraction map π : X → X̄ such that π(∪C_{i}) = P is a point, and π|_{X−∪C_{i}} : X − ∪C_{i} → X̄ − P is an isomorphism. P ∈ X̄ is a rational double point.

Proposition 3. Let X be a smooth K3 surface, let |L| be a nef and big base-point-free linear system on X, and suppose there is an effective curve D such that L · D = 0, D^{2} = −2. Then every irreducible component D_{i} of D satisfies L · D_{i} = 0, D_{i}^{2} = −2.

Moreover, if C_{1}, …, C_{n} is a maximal connected set of irreducible curves such that L · D_{i} = 0, C_{i}^{2} = −2, then there is a contraction π : X → X̄of ∪ C_{i} to a rational double point, and the map φ_{|L|} factors through π.

Proof. If we write D = ∑ n_{i}D_{i} with n_{i} > 0, then 0 = L · D = ∑ n_{i}L · D_{i} and each L·D_{i} ≥ 0 implies L·D_{i} = 0 for all i (since L is nef). By Hodge index, since L·D_{i} = 0 we have D_{i}^{2} < 0. But then D_{i}^{2} = −2 and D_{i} is a smooth rational curve. Since Hodge index implies that these curves have a negative-definite intersection matrix, the maximal connected components can be contracted to rational double points.

Now suppose that C_{1}, …, C_{n} is a maximal connected set of such curves. Also suppose L^{2} ≥ 4. Suppose that C = ∑ n_{i}C_{i} satisfies C · C_{i} ≤ 0 for all i, and C^{2} = −2. Consider the linear system |L − C|. Note that (L − C)^{2} = L^{2} − 2 ≥ 0 and L · (L − C) = 2g − 2, so L − C is effective.

Suppose that L − C is not nef, and let Γ be an irreducible curve such that (L − C) · Γ < 0. If Γ^{2} ≥ 0 then |Γ| moves, so that (L − C) · Γ cannot be negative since L − C is effective. Thus, Γ^{2} = −2. We have C · Γ > L · Γ ≥ 0, so that Γ cannot be a component of C, and must be connected to Supp(C). By our assumption about the maximality of C, it follows that L · Γ > 0. If we let x = L · Γ, y = C · Γ and 2g − 2 = L^{2} (so that g ≥ 3) then 0 < x < y and the intersection matrix for L, C, Γ is:

[ \begin{matrix}2g − 2 & 0 & x \\ 0 & −2 & y \\ x & y & −2 \end{matrix} ]

By Hodge index, this must have determinant ≥ 0. Thus, (using also the relation 0 < x < y):

0 ≤ 2x^{2} + (4 − y^{2})(2g − 2) < 2y^{2} + (4 − g^{2})(2g − 2)

which implies (since g > 2):

y^{2} < 4 (\frac{g − 1}{g − 2}) = 4 (\frac{1}{g − 2}) ≤ 8.

It follows that y = 2 and x = 1. But then (C + Γ)^{2} = 0 and L · (C + Γ) = 1 so that |L| has a fixed component, contrary to hypothesis.

Thus, |L − C| is nef. But now by Riemann-Roch,

h^{0}(L − C) = \frac{1}{2}(L − C)^{2} + 2 = \frac{1}{2}L^{2} + 2 − 1 = h^{0}(L) − 1.

Since C imposes only 1 condition on H^{0}(L), φ_{|L|}(C) must be a point, so φ_{|L|} factors through the contraction π : X → X̄ of C.

In the case L^{2} = 2, h^{0}(L − C) ≥ \frac{1}{2}(L − C)^{2} + 2 = 2. Since C is not a fixed component of L, we have

2 ≤ h^{0}(L − C) < h^{0}(L) = 3

so that h^{0}(L − C) = h^{0}(L) − 1 and the result follows as before. Q.E.D.

In sum, we have almost finished the proof of:

Theorem. ^{17} Let L be a nef and big linear system on a K3 surface.

17Reference for this theorem?

(1) |L| has base points if and only if there is a divisor D such that L · D = 1, D^{2} = 0.

(2) In the case of no base points:

(a) If π : X → X̄ denotes the contraction of all effective curves C with L · C = 0, C^{2} = −2 to rational double points, then φ_{|L|} factors through π.

(b) The induced map φ̄ : X̄ → ℙ^{g} has degree 2 if and only if either L^{2} = 2, or L ∼ ∼ ∼ 2D for a divisor D with D^{2} = 2, or there is a divisor D with L · D = 2, D^{2} = 0. Otherwise, φ̄ : X̄ → ℙ^{g} is an embedding.

Notice that in the degree 2 case, the image surface φ̄(X̄)_{red} has degree \frac{1}{2}(2g − 2) = g − 1 in ℙ^{g}. It is thus a surface of minimal degree, and so is a scroll or the Veronese. (The Veronese case corresponds to L ∼ ∼ ∼ 2D.) Proof: The statement does not say “D is effective”, but that follows in each case since L · D > 0, D^{2} ≥ −2.

The only remaining thing to prove is that φ̄ is an embedding when the degree is not 2. We omit the proof for infinitely near points (which is complicated for the rational double points), and simply show that φ̄ separates distinct points.

Let P̄, Q̄ ∈ X̄ with P̄ ≠ Q̄ and suppose they are not separated by φ̄. Choose P ∈ π^{−1}(P̄), Q ∈ π^{−1}(Q̄); then P and Q are not separated by φ_{|L|}. If φ̄ (and φ) do not have degree 2, there is an effective divisor D containing P and Q with L · D = 0, D^{2} = −2. But since P and Q do not belong to the same maximal connected set of irreducible curves C_{i} with L · C_{i} = 0, C_{i}^{2} = −2, this is impossible: any such D would have D^{2} ≤ −4 (being supported on 2 different connected components). (This is another fact about rational double points.) “Q.E.D.”

6.1. Addendum to section 6.

Corollary. If |L| is a nef and big linear system on a K3 surface, then |3L| induces an embedding of X into projective space.

Proof. If not, either 2 = (3L)^{2} = 9L^{2} (impossible), or 3L ∼ ∼ ∼ 2D with D^{2} = 2 so that L^{2} = \frac{4}{9}D^{2} (impossible), or there is some D with D^{2} = 0 and 3L · D = 1 or 2 (also impossible). Q.E.D.

7. The geometry of canonical curves and K3 surfaces

We now wish to discuss some of the connections between the geometry of K3 surfaces in projective space ℙ^{g}, and the geometry of their hyperplane sections, which are canonical curves. We begin by mentioning a theorem of Lazarsfeld, which can be partially proved with the tools we have developed.

Theorem (Lazarsfeld [13]). Let X be a smooth K3 surface, and let L be a nef and big line bundle which generates the Picard group Pic(X). (It is easy to see that this implies that |L| is base-point-free.) Then every smooth C ∈ |L| satisfies the Brill-Noether property in the following form: for each line bundle L on C, ρ(L) ≥ 0.

Partial Proof. We treat the case h^{0}(L) = 2: suppose that L satisfies h^{0}(L) = 2 and ρ(L) < 0 and write L = O_{C}(Z) for some effective divisor Z on C. The bundle E(Z, L) has rank 2, and since ρ(E(Z, L)) = ρ(L) < 0 it satisfies the weak Bogomolov property. Thus, there is an effective divisor D containing Z such that L and D satisfy (∗)_{d}, where d = deg L. (Note that Z ≠ ∅, so that D ≠ 0.) By the assumption on Pic(X), D ∼ kL for some k ∈ ℤ. But now L · D ≥ 0 implies k ≥ 0 while L · (L − 2D) ≥ 0 implies k ≤ \frac{1}{2}. We find that k = 0 and so D = 0, a contradiction. Q.E.D.

Thus, the general principle is that the existence of a g_{d}^{1} on C with ρ < 0 forces the existence of some divisor D on X in addition to C. We make this more explicit in the low degree cases.

Proposition. Let L be a nef and big linear system without base points on a K3 surface.

(1) If there is a smooth C ∈ |L| which is hyperelliptic, then φ_{|L|} has degree 2. Conversely, if φ_{|L|} has degree 2 then every smooth C ∈ |L| is hyperelliptic.

(2) Suppose L^{2} = 8 or L^{2} ≥ 12. If there is a smooth C ∈ |L| which is trigonal (that is, which has a g_{3}^{1}) but not hyperelliptic, then there is an effective divisor D with L · D = 3, D^{2} = 0. Conversely, when such a divisor exists, every smooth C ∈ |L| is trigonal.

(3) Suppose L^{2} = 12, 14 or 18. If there is a smooth C ∈ |L| which is tetragonal (that is, which has a g_{4}^{1}) but not hyperelliptic or trigonal, then there is an effective divisor D such that either L · D = 4, D^{2} = 0 or L · D = 6, D^{2} = 2. If L^{2} = 12 or 14, the converse also holds: if there is a divisor of either kind, then every smooth C ∈ |L| is tetragonal.

(We will return to the case of L^{2} = 18 and a g_{4}^{1} on C a bit later.)

Proof. In each case, we are given a g_{d}^{1} on C, call it |Z|, with d the minimum possible value for such systems on C. Because it is the minimum, |Z| has no base points. We may thus choose Z ∈ |Z| consisting of distinct points.

Consider the bundle E(Z, L): this has rank 2, and ρ = g −2(1−d+g) = 2d−2−g. Thus, for g > 2d − 2 (which is equivalent to L^{2} = 2g − 2 > 4d − 6) we have ρ < 0. For d = 2, 3, 4 this is implied by L^{2} > 0, L^{2} ≥ 8, L^{2} ≥ 12 respectively.

By the Reider method, there is an effective divisor D containing Z such that L and D satisfy (∗)_{d}. Notice that C ∈ |L| is irreducible and is not a component of D (for D − C effective would imply

0 ≤ C · (D − C) = C · D − C^{2} ≤ −C · D ≤ 0

(using C^{2} ≥ 2C · D) and hence C · D = C^{2} = 0, a contradiction). Thus, C · D ≥ # Supp(C ∩ D) ≥ # Supp(Z) = d.

By induction on d, we can see that L and D do not satisfy (∗)_{d−1}: this follows from the “converse statement” for d − 1 in each case. (The “converse statement” for d − 1 = 1 is the statement that |L| has a fixed component when there is a D satisfying (∗)_{1}.) Thus, we need solutions to (∗)_{d} under the additional conditions: D · (L − D) = d, L · D ≥ d, D^{2} ∈ 2ℤ. We have done the case d = 2 before: the d = 3 and d = 4 cases are:

Solution  Restrictions
L · D = 5, D^{2} = 2 10 ≤ L^{2} ≤ \frac{25}{2}
d = 3
L · D = 3, D^{2} = 0  6 ≤ L^{2}

L · D = 8, D^{2} = 4 16 ≤ L^{2} ≤ 16
d = 4 L · D = 6, D^{2} = 2 12 ≤ L^{2} ≤ \frac{36}{2}
L · D = 4, D^{2} = 0  8 ≤ L^{2}

To match the numerical statements given in to proposition, we need a lemma:

Lemma. If |L| is a base-point-free linear system on a K3 surface with L^{2} = 12 and if there is a divisor D such that L · D = 5, D^{2} = 2 then smooth curves C ∈ |L| are hyperelliptic.

The proof of this is easy: (L − 2D)^{2} = 0 and L · (L − 2D) = 2 so φ_{|L|} must have degree 2.

To finish the proof of the proposition, notice that the existence of a curve D with D^{2} = 0, L · D = d implies that each smooth C ∈ |L| has a g_{d}^{1}, namely O_{C}(D). The only remaining thing to check is the converse statement when d = 4, D^{2} = 2, L · D = 6. In that case, each smooth C has a g_{6}^{2}. But since L^{2} ≠ 18 (i. e. g ≠ 10), the image of C under this g_{6}^{2} cannot be a smooth plane sextic. The pencil residual to a singular point of the image is a g_{4}^{1} (or smaller—but if there were something smaller, every C ∈ |L| would be hyperelliptic or trigonal). Q.E.D.

In order to further study the trigonal case, we recall the Enriques-Babbage-Petri theorem.

Theorem (Enriques-Babbage-Petri; cf. ACGH). A (non-hyperelliptic) canonical curve C ⊂ ℙ^{g−1} is cut out by the quadrics containing it if and only if C is not trigonal and not a smooth plane quintic. In the case in which C is not cut out by quadrics, the quadrics containing C cut out a surface of minimal degree; in the trigonal case, this is a scroll ruled by the trisecant lines of C spanned by the g_{3}^{1}.

The following theorem was originally proved by Saint-Donat using different techniques, and including the case g = 6 (in which the statement must be modified to include the smooth plane quintic hyperplane section case).

Theorem (Saint-Donat). If a K3 surface X ⊂ ℙ^{g}, g ≥ 5, is not cut out by quadrics and g ≠ 6, then every smooth hyperplane section of X is trigonal, and there is a family of curves D of degree 3 with D^{2} = 0 which cut out the trigonal series on the hyperplane sections. Moreover, the ℙ^{2}’s spanned by the plane cubics D sweep out a threefold scroll (of minimal degree), the base locus of the quadrics through X.

Proof. Let P ∉ X be a point contained in all quadrics through X. A general hyperplane H through P meets X in a smooth curve C (by Bertini’s theorem).

Since H^{1}(O_{ℙ^{g}}(1)) = 0 and the natural map H^{0}(O_{ℙ^{g}}(1)) → H^{0}(O_{X}(1)) is an isomorphism, it follows from the exact sequence

0 → I_{X}(1) → O_{ℙ^{g}}(1) → O_{X}(1) → 0

that H^{0}(I_{X}(1)) = H^{1}(I_{X}(1)) = 0. Now from the exact sequence

0 → I_{X}(1) → I_{X}(2) → I_{C}(2) → 0

it follows that H^{0}(I_{X}(2)) ≅ H^{0}(I_{C}(2)). In particular, P is contained in all quadrics through C.

Thus, by the Enriques-Babbage-Petri theorem C is trigonal. By the previous proposition, there is a D with L · D = 3, D^{2} = 0 where O_{X}(L) = O_{X}(1); every smooth C ∈ |L| is then trigonal.

We sketch a proof of the “moreover” statement. The base locus of the quadrics through C is the intersection of H with the base locus of the quadrics through X. Since the former is a scroll swept by the trisecant lines, the latter is a scroll swept by the linear span of the D’s (which cut C in the trisecant lines)—i. e., by the ℙ^{2}’s containing the plane cubics D. Q.E.D.

To ensure that you are not left with the wrong impression about special linear systems on hyperplane sections of K3 surfaces, I return to the example of L^{2} = 18 (g = 10) and g_{4}^{1}’s. I need a lemma.

Lemma. A smooth plane sextic curve has no g_{4}^{1}.

Proof. We first show: any set of k ≤ 4 distinct points P_{1}, …, P_{k} in ℙ^{2} impose independent conditions on cubics. This is a fairly straightforward fact, but it is amusing to prove it as an application of Reider’s method. If it were false, after re-ordering the points if necessary, upon blowing up P_{1}, …, P_{k−1} with a map π : X → ℙ^{2} the anti-canonical series |−K_{X}| would have a base point at P_{k}. It is easy to check that −K_{X} is nef (the worst case is three collinear points which prevents −K_{X} from being ample). But since K_{X}^{2} ≥ 6, the application of Reider’s theorem to generalized Del Pezzo surfaces given in section 5 implies that |−K_{X}| can have no base point.

Now given a smooth plane sextic with a g_{k}^{1}, k ≤ 4, base-point-free, we would have k distinct points not imposing independent conditions on |K_{C}|. (Choosing an element in the g_{k}^{1} with distinct points.) Since |K_{C}| coincides with O_{ℙ^{2}}(3) restricted to C, any such set of points must impose independent conditions, a contradiction. Q.E.D.

Example (due to Donagi and the author). Consider again the example we gave in the appendix to section 3 (on double covers): the double cover of ℙ^{2} branched along a curve of degree 6 with map π : X → ℙ^{2}. Define L by O_{X}(L) = π^{∗}O_{ℙ^{2}}(3). We have L · π^{∗}O_{ℙ^{2}}(1) = 6 so that for a smooth C ∈ |L|, either π(C) is a curve of degree 6 or π maps C onto a cubic curve and deg(π|_{C}) = 2.

The computation we made earlier shows that both cases occur: on a linear space of codimension 1 in |L|, deg(π|C) = 2 but the generic curve C ∈ |L| is not the pullback of a curve in ℙ^{2}. Now the curves which are double covers of elliptic curves all carry a g_{4}^{1}, which they inherit from a g_{2}^{1} on the elliptic curve. (In fact, there is a 1-parameter family of g_{4}^{1}’s.) On the other hand, for general C ∈ |L|, π(C) is a smooth plane sextic and so carries no g_{4}^{1}. We conclude from our proposition that there is a curve D such that L · D = 6, D^{2} = 2, so that every curve carries a g_{6}^{2}; something which is obvious from the geometry. (In fact, Hodge index implies L ∼ ∼ ∼ 3D so that the g_{6}^{2}’s produced by the Reider method are exactly those induced by the map π : X → ℙ^{2}.)

Quite a bit more is known about the connections between special linear systems on hyperplane sections of a fixed K3 surface than we have covered here. To briefly indicate what else is known, consider our basic setup: a linear system |Z| on C with ρ < 0, the bundle E(Z, L) (where C ∈ |L|) and the corresponding divisor D on X. Suppose for simplicity that H^{1}(O_{X}(D − L)) = 0 so that the natural map H^{0}(O_{X}(D)) → H^{0}(O_{C}(D)) is an isomorphism. (It is easy to see that D − L cannot be effective.) In this case, the basic inequality D · (C − D) ≤ d which forms a part of (∗)_{d} can be interpreted as follows: h^{0}(O_{C}(D)) = \frac{1}{2}D^{2} + 2 and deg(O_{C}(D)) = D · C so that

d ≥ D · (C − D) = deg(O_{C}(D)) − 2h^{0}(O_{C}(D)) + 4.

If we define for any line bundle L on C the Clifford index

ν(L) = deg(L) − 2(h^{0}(L) − 1)

(which is “d − 2r” for a g_{d}^{r}) then this says:

ν(O_{C}(Z)) ≥ ν(O_{C}(D)).

The Clifford index of C is defined to be:

ν(C) = min{ν(L) | L ∈ Pic(C), h^{0}(L) ≥ 2, h^{1}(L) ≥ 2}.

The theorems are then:

Theorem (Donagi-Morrison). If C is a smooth curve on smooth K3 surface X with C^{2} > 0 and |Z| is a base-point-free g_{d}^{1} on C with ρ < 0 then there is a divisor D on X containing Z such that ν(O_{C}(D)) ≤ ν(O_{C}(Z)) and the function C′ ↦ ν(O_{C′}(D)) is constant for smooth C′ ∈ |C|.

Theorem (Green-Lazarsfeld). If C is a smooth curve on a smooth K3 surface X with C^{2} > 0 then ν(C′) = ν(C) for every smooth C′ ∈ |C|. Moreover if ν(C) < [\frac{g−1}{2}] (which implies that linear systems at the minimum have ρ < 0) then there is a divisor D on X such that ν(C) = ν(O_{C}(D)).

For the proofs (which use techniques related to the ones we have discussed here). I refer you to the original papers (J. Diff. Geo. 1988 and Inventiones Math. 1987, resp.).

We now apply our investigations of the geometry of canonical curves and K3 surfaces to give characterizations of K3 surfaces of low degree, showing that they almost always coincide with the examples we have constructed.

Theorem. Let L be a nef line bundle on a smooth K3 surface X. Let π : X → X̄ be the contraction of all irreducible curves C_{i} on X with C_{i}^{2} = −2, C_{i} · L = 0 to rational double points.

(1) If L^{2} = 2 and there does not exist a divisor D with D^{2} = 0, L · D = 1 then φ_{|L|} induces a map X̄ → ℙ^{2} of degree 2, expressing X̄ as the double cover of ℙ^{2} branched on a curve with only simple singularities.

(2) If L^{2} = 4 and there does not exist a divisor D with D^{2} = 0, L · D = 1 or 2 then φ_{|L|} embeds X̄ as a quartic surface in ℙ^{3} with only rational double points.

(3) If L^{2} = 6 and there does not exist a divisor D with D^{2} = 0, L · D = 1 or 2 then φ_{|L|} embeds X̄ as a generically transverse intersection of a quadric and a cubic in ℙ^{4} with only rational double points.

(4) If L^{2} = 8 and there does not exist a divisor D with D^{2} = 0, L · D = 1, 2, or 3 then either φ_{|L|} embeds X̄ as a generically transverse intersection of three quadrics in ℙ^{5} with only rational double points, or L ∼ ∼ ∼ 2D for some divisor D and φ_{|L|} induces a map X̄ → V ⊂ ℙ^{5} of degree 2 from X̄ to the Veronese V.

Proof. First notice that in all cases we have assumed there is no D with L · D = 1, D^{2} = 0 so that |L| has no base points. In addition, our previous results about when φ_{|L|} has degree 2 agree with the statements made here. In fact, the only thing left to prove for (1) is the statement that the branch curve has simple singularities. But these are exactly the singularities producing only rational double points on the double cover X̄.

In the case L^{2} = 4, since φ_{|L|} is an embedding (X is not hyperelliptic), it embeds X̄ as a hypersurface in ℙ^{3}, which has degree L^{2} = 4.

In the case L^{2} = 6, consider the sequence

0 → H^{0}(IX̄(2)) → H^{0}(O_{ℙ^{4}}(2)) → H^{0}(OX̄(2)).

Since h^{0}(O_{ℙ^{4}}(2)) = 15 and h^{0}(OX̄(2)) = \frac{1}{2}(2L)^{2} + 2 = 14, we have h^{0}(IX̄(2)) ≥ 1. Let q ∈ H^{0}(IX̄(2)) and let Q be the corresponding quadric, so that φ_{|L|}(X̄) ⊂ Q. Note that Q is irreducible since φ_{|L|}(X̄) is not contained in any hyperplane of ℙ^{4}.

If x_{0}, …, x_{4} denote coordinates in ℙ^{4}, then x_{i}q ∈ H^{0}(IX̄(3)) for i = 0, …, 4. On the other hand, since h^{0}(O_{ℙ^{4}}(3)) = 35 and h^{0}(OX̄(3)) = \frac{1}{2}(3L)^{2} + L = 29 we have h^{0}(IX̄(3)) ≥ 6. Thus, there is some section r ∈ H^{0}(IX̄(3)) whose associated cubic R does not contain Q. Then R ∩ Q is a surface of degree 6 containing φ_{|L|}(X̄); since φ_{|L|}(X̄) also has degree L^{2} = 6 it follows that φ_{|L|}(X̄) = R ∩ Q is a generically transverse intersection.

Finally, if L^{2} = 8 and φ_{|L|} is an embedding we have h^{0}(O_{ℙ^{5}}(2)) = 21 while h^{0}(OX̄(2)) = \frac{1}{2}(2L)^{2} +L = 18 so that h^{0}(IX̄(2)) ≥ 3. By our assumptions (that there is no D with L·D = 3, D^{2} = 0), φ_{|L|}(X̄) is cut out by the quadrics containing it. Thus, there are 3 elements Q_{1}, Q_{2}, Q_{3} in the linear system |IX̄(2)| whose intersection is generically transverse so that dim(Q_{1}∩Q_{2}∩Q_{3}) = 2. Since φ_{|L|}(X̄) ⊂ Q_{1}∩Q_{2}∩Q_{3} and deg φ_{|L|}(X̄) = 8 = deg(Q_{1} ∩Q_{2} ∩Q_{3}) it follows that φ_{|L|}(X̄) = Q_{1} ∩Q_{2} ∩Q_{3}. Q.E.D.

Mukai has given some further characterizations using vector bundle techniques; we will describe his method, and do the case of g = 8 in detail.

A vector bundle E is simple if it has no endomorphisms other than scalar multiplies of the identity, that is, if h^{0}(E ⊗ E^{∗}) = 1. Mukai originally made the computation of X(E ⊗ E^{∗}) for bundles on a K3 surface X because of the fact that the tangent space to the moduli space of vector bundles on X at the point [E] can be naturally identified with H^{1}(E ⊗ E^{∗}). Thus, in the simple case

X(E ⊗ E^{∗}) = 2 − h^{1}(E ⊗ E^{∗}) = 2 − 2ρ(g, r, d)

when E = E(Z, L) for a g_{d}^{r} (namely Z) on a smooth C ∈ |L| of genus g. It follows that

h^{1}(E ⊗ E^{∗}) = 2ρ(g, r, d).

We have used this computation previously to see that E(Z, L) cannot be simple when ρ < 0. Now, however we consider the case of ρ = 0. The bundles in this case will be rigid (that is, will have no local deformations), and so form some kind of discrete invariants of the K3 surface X. It is natural to expect that the embeddings into such bundles will yield information about the geometry of X.

To apply this idea, we need to recall:

Another Part of the Brill-Noether Theorem (cf. ACGH). Suppose that

ρ(g, r, d) = g − (r + 1)(r − d + g) ≥ 0

and that r − d + g ≥ 0, r ≥ 0. Then every smooth curve of genus g has a line bundle L with deg L = d and h^{0}(L) ≥ r + 1.

Corollary. Every smooth curve of genus 8 has a complete base-point-free g_{d}^{1} for some d ≤ 5.

Proof. For g = 8, r = 1, d = 5 we have ρ = 0; thus, there is a line bundle L = O_{C}(Z) with deg Z = 5, h^{0}(O_{C}(Z)) ≥ 2.

Suppose that h^{0}(O_{C}(Z)) = k+2 with k ≥ 0. Pick points P_{1}, …, P_{k} on C such that P_{i} is not a base point of |Z − P_{1} − ⋯ − P_{i−1}|. Then h^{0}(O_{C}(Z − P_{1} − ⋯ − P_{k})) = 2 and degree (z −P_{1} −⋯−P_{k}) = 5−k. Now if |Z −P_{1} −⋯−P_{k}| has as its base points Q_{1} + ⋯ + Q_{ℓ}, we have h^{0}(O_{C}(Z − ∑ P_{i} − ∑ Q_{j})) = 2 and deg(Z − ∑ P_{i} − ∑ Q_{j}) = 5 − k − ℓ ≤ 5. Q.E.D.

Theorem (A more precise version of a theorem of Mukai). Let L be a nef line bundle on a smooth K3 surface X, and let π : X → X̄ be the contraction of all irreducible curves C_{i} on X with C_{i}^{2} = −2, C_{i} · L = 0 to rational double points. Suppose that L^{2} = 14 and there does not exist a divisor D with D^{2} = 0, L · D = 1, 2, 3 or 4 or D^{2} = 2, L · D = 6. Then φ_{|L|} embeds X̄ in ℙ^{8} ∩ Gr(2, 6) of the Grassmannian Gr(2, 6) in its Plucker embedding Gr(2, 6) ⊂ ℙ^{14}.

Proof. ^{18} Since there is no D with D^{2} = 0, L · D = 1 or 2, the linear system |L| has no base points, and φ_{|L|} defines an embedding of X̄ into ℙ^{8}.

Let C ∈ |L| be a smooth curve. By the corollary above, C has a linear system |Z| which is a complete base-point-free g_{d}^{1} for some d ≤ 5. Our assumptions about non-existence of divisors D imply that d cannot be less than 5, so d = 5. Thus, we get a vector bundle E(Z, L) for which ρ = 0.

I claim that the linear system |K_{C} − Z| on C has no base points. For if P were a base point of this system, then |Z + P | would be a g_{6}^{2} on C. Since C does not have genus 10, φ_{|Z+P |} cannot be an embedding of C. But this means for some Q ∈ C, |Z+P −2Q| is a g_{4}^{1}. We do not have g_{4}^{1}’s on C by our assumptions on the non-existence of divisors D.

It follows that E(Z, L) is generated by its global sections. To see how many sections, we prove a

Lemma. If |Z| is a g_{d}^{r} on C ∈ |L| of genus g, then

h^{0}(O_{X}(E(Z, L))) = g + 1 − ν(O_{C}(Z))
= g + 1 − d + 2r.

18Mukai’s theorem covers the case of a generic K3 surface, and his proof is quite different.

Proof. The sequence

0 → H^{0}((O_{X})^{⊕r}) → H^{0}(O_{X}(E(Z, L))) → H^{0}(I_{Z}(L)) → 0

is exact since X is regular. Now

h^{0}(I_{Z}(L)) = h^{0}(O_{X}(L)) − h^{0}(O_{Z}(L)) + h^{1}(I_{Z}(L))
= g + 1 − d + r.

So h^{0}(O_{X}(E(Z, L))) = r + h^{0}(I_{Z}(L)) = g + 1 − d + 2r. Q.E.D.

In the case of the theorem we are proving, g = 8, r = 1 and d = 5 so that h^{0}(E) = 6.

Since E = E(Z, L) is generated by its global sections, there is a regular map π : X → Gr(2, H^{0}(E)^{∗}) = Gr(2, 6) defined by

x ↦ {φ ∈ H^{0}(E)^{∗} | if v ∈ H^{0}(E) vanishes at x then φ(v) = 0}

(cf. Griffiths and Harris, p. 207).

Moreover, since O_{X}((Λ^{2}E)^{∗∗}) = O_{X}(L), we have

X  ⟶^{φ_{|L|}} ℙ(H^{0}(O_{X}(L))^{∗}) = ℙ(H^{0}(⋀^{2} E)^{∗})
↓  ↓
Gr(2, H^{0}(E)^{∗}) ⟶^{Pℓ}  ℙ(⋀^{2} H^{0}(E)^{∗})

where Pℓ is the Plucker embedding.

(The map between projective spaces is given by the dual of the natural map

⋀^{2} H^{0}(E) → H^{0}(⋀^{2} E).

In non-intrinsic terms, the diagram becomes

X  ⟶^{φ_{|L|}} ℙ^{8}
φ_{E} ↓  ↓
Gr(2, 6) ⟶^{Pℓ} ℙ^{14}

where the map ℙ^{8} → ℙ^{14} is the one referred to above.)

Now dim Gr(2, 6) = 8 and Pℓ maps it to ℙ^{14}; intersecting with the linear ℙ^{8} = ℙ(H^{0}(O_{X}(L))^{∗}) produces ℙ^{8} ∩ Gr(2, 6) of codimension 6 in ℙ^{8}. That is, ℙ^{8} ∩ Gr(2, 6) is a surface containing φ_{|L|}(X̄). Since deg(ℙ^{8} ∩ Gr(2, 6)) = deg Gr(2, 6) = 14 and deg(φ_{|L|}(X̄)) = L^{2} = 14 it follows that φ_{|L|}(X̄) = ℙ^{8} ∩ Gr(2, 6). Q.E.D.

8. Kummer surfaces

In this section we will prove a theorem of Nikulin which says that a K3 surface is a Kummer surface if and only if it has 16 singular points of type A_{1}. The proof is rather combinatorial in nature, but the combinatorial analysis has a nice byproduct: with it, we will be able to prove a Torelli-type theorem for Kummer surfaces, which is an important step in proving the Torelli theorem for all K3 surfaces.

We start with a K3 surface X which has A_{1} singularities at points P_{1}, …, P_{k} and is smooth elsewhere. Let π : X̃ → X be the minimal desingularization, let E_{i} = π^{−1}(P_{i}), and let e_{i} be the class of E_{i} in H^{2}(X̃, ℤ). The combinatorial analysis is devoted to the following problem: for which subsets J ⊂ {1, …, k} is ∑_{i∈J} E_{i} divisible by 2 in Pic(X̃)? (By the Lefschetz (1,1) theorem, this is equivalent to asking when \frac{1}{2} ∑_{i∈J} e_{i} ∈ H^{2}(X̃, ℤ).) The final step in showing that X is a Kummer surface when k = 16 will be to use the divisibility by 2 of ∑_{i=1}^{16} E_{i} to construct a double cover of X̃ and thus recover the complex torus out of which the Kummer surface is constructed; our goal must therefore be to show that ∑_{i=1}^{16} e_{i} is in fact divisible by 2.

Definition. Let L be a free ℤ-module of finite rank equipped with a symmetric bilinear form L × L → ℤ (which we denote by (x, y) ↦ x · y). We define

L^{#} = {x ∈ L ⊗ ℚ | x · y ∈ ℤ for all y ∈ L}

and note the natural map L^{#} → Hom(L, ℤ) which sends x to the function y ↦ x · y. The form on L is nondegenerate if this map is an isomorphism, and in that case the cokernel L^{#}/L is called the discriminant-group of the form. (This is necessarily a finite group, since L ⊂ L^{#} and they become isomorphic after tensoring with ℚ.)

As an example, let L be the ℤ-span of e_{1}, …, e_{k} in H^{2}(X̃, ℤ), for our K3 surface X as above. Then e_{i} · e_{j} = −2δ_{ij} so that L^{#} is generated by \frac{1}{2}e_{1}, …, \frac{1}{2}e_{k}. It follows that L^{#}/L ≅ (ℤ/2ℤ)^{k}, generated by \{\frac{1}{2}e_{i}\}.

In our example, if we augment L by including the elements \frac{1}{2} ∑_{i∈J} e_{i} which are contained in H^{2}(X̃, ℤ) we get an inclusion L ⊂ M of free ℤ-modules of the same rank. This leads to:

Property 1 of Discriminant-Groups. Let L ⊂ M be an inclusion of free ℤ-mod-ules of the same (finite) rank, let M be equipped with a nondegenerate symmetric bilinear form, and consider L ⊂ M ⊂ M^{#} ⊂ L^{#}. Then

|L^{#}/L| = |M^{#}/M| · [M:L]^{2},

where |G| denotes the order of a finite group and [G:H] denotes the index of H in G.

Proof. We have

|L^{#}/L| = [L^{#}:M^{#}][M^{#}:M][M:L]

so it suffices to show that [L^{#}:M^{#}] = [M:L]. Now

L^{#}/M^{#} ≅ Hom(L, ℤ)/ Hom(M, ℤ),

so we must compute the order of this latter group.

There is an exact sequence

0 → Hom(M, ℤ) → Hom(L, ℤ) →^{δ} Hom(M/L, ℚ/ℤ) → 0

constructed as follows: pick an integer n such that nM ⊂ L, and define

δ(φ)(x) = \frac{1}{n}φ(nx) mod ℤ

for φ ∈ Hom(L, ℤ), x ∈ M mod L. If x ∈ L then \frac{1}{n}δ(nx) = δ(x) ∈ ℤ, so that this is well-defined. It is easy to see that Ker(δ) = Hom(M, ℤ): if φ is in the kernel, then \frac{1}{n}φ(nx) = φ(x) ∈ ℤ for all x ∈ M, and conversely. To see that δ is surjective, pick a basis e_{1}, …, e_{r} of M and for ψ ∈ Hom(M/L, ℚ/ℤ) pick g_{i} ∈ ℚ such that g_{i} ≡ ψ(e_{i}) mod ℤ for all i. Then defining φ ∈ Hom(M, ℚ) by φ(e_{i}) = g_{i}, we see that φ is integer-valued on L, so φ|_{L} ∈ Hom(L, ℤ), and that δ(φ|_{L}) = ψ.

Finally, to compute the order of Hom(M/L, ℚ/ℤ): it is well known that this has the same order as M/L itself. To see this, it suffices to check it for a cyclic group ℤ/dℤ; the homomorphisms ℤ/dℤ → ℚ/ℤ are classified by the image of 1, which must go to some element of \frac{1}{d}ℤ/ℤ ⊂ ℚ/ℤ. Q.E.D.

Consider now an inclusion L ⊂ Λ of free ℤ-modules with symmetric bilinear forms which do not necessarily have the same rank. (We assume that the form on Λ restricts to the form on L.) The saturation of L in Λ is defined to be M = (L ⊗ ℚ) ∩ Λ; this has the same rank as L. If M is saturated, and the form on M is nondegenerate, then N := M^{⊥} is also saturated and M ⊕ N ⊂ Λ is an inclusion of modules of the same rank.

A form Λ is unimodular if Λ = Λ^{#}, or equivalently, if the map Λ → Hom(Λ, ℤ) is an isomorphism.

Property 2 of Discriminant-Groups. Let Λ be a free ℤ-module with a unimodular form, let M be a saturated submodule on which the form is non-degenerate, and let N = M^{⊥}. Then there is a natural isomorphism M^{#}/M →^{≅} N^{#}/N.

Proof. For a given x ∈ M^{#}, there is an associated φ_{x} ∈ Hom(M, ℤ) defined by φ_{x}(y) = x · y. Now M is saturated means that Λ/M is torsion-free, so every homomorphism M → ℤ can be extended to a homorphism Λ → ℤ. Pick such an extension φ ∈ Hom(Λ, ℤ); since Λ is unimodular there is some λ ∈ Λ corresponding to φ: we have x · y = λ · y for all y ∈ M.

Now λ − x ∈ N ⊗ ℚ, and for all z ∈ N we have (λ − x) · z = λ · z ∈ ℤ. Sending x ↦ λ − x defines the homomorphism M^{#}/M → N^{#}/N. It is easy to check that it is well-defined (i. e. does not depend on the choice of extension φ). To see that it is injective, suppose that λ − x ∈ N. Then x ∈ Λ ∩ (M ⊗ ℚ) and so (since M is saturated) x ∈ M.

Finally, since we have M^{#}/M ↪ N^{#}/N, if we reverse the roles of M and N we get N^{#}/N ↪ M^{#}. This implies that the groups have the same order, and that the inclusions are isomorphisms. Q.E.D.

To return to our example of a K3 surface with A_{1} singularities, the minimal resolution X̃ has a cohomology group H^{2}(X̃, ℤ) with a symmetric bilinear form, which is unimodular by Poincaré duality. We have the submodule L of Λ = H^{2}(X̃, ℤ) generated by e_{1}, …, e_{k}; its saturation M = (L ⊗ ℚ) ∩ Λ; and the orthogonal complement N = M^{⊥}. L^{#}/L ≅ (ℤ/2ℤ)^{k} is naturally a vector space over 𝔽_{2}, as is any subgroup of it.

Lemma. Let α = dim_{𝔽_{2}}(M/L). Then

k − 2α ≤ 22 − k.

Proof. Since |M/L| = 2^{α} and |L^{#}/L| = 2^{k}, it follows from property 1 that |M^{#}/M| = 2^{k−2α}. Since M^{#}/M is a sub-quotient of L^{#}/L ≅ (ℤ/2ℤ)^{k}, it follows that M^{#}/M ≅ (ℤ/2ℤ)^{k−2α}.

Now by property 2, N^{#}/N ≅ (ℤ/2ℤ)^{k−2α}. On the other hand, the second Betti number of a smooth K3 surface is 22, which implies that N has rank 22 − k (since L and M have rank k). Thus, N^{#} and N^{#}/N can both be generated by 22 − k (or fewer) elements; since (ℤ/2ℤ)^{k−2α} requires at least k − 2α elements to generate it, k − 2α ≤ 22 − k. Q.E.D.

Corollary. If the K3 surface X has 16 singular points of type A_{1}, there is at least a five-dimensional 𝔽_{2}-vector space of linear combinations ∑_{i∈J} E_{i} which are divisible by 2 in the Picard group.

(A nice way to think about this result is: the topology of the K3 surface, as represented by H^{2}(X̃, ℤ), forces the existence of certain double covers of X̃ [and X], corresponding to the divisors divisible by 2.)

To proceed further, we need to investigate the kinds of double covers which can occur. One kind we already know: the double cover of a Kummer surface by a complex torus. To investigate the double covers in general, consider the following set-up: given a subset J ⊂ \{1, …, k\} such that \frac{1}{2}_{i} ∑_{i∈J} e_{i} ∈ M, let X_{J} → X be the resolution of the singular points not in J. If we form the double cover Ỹ of X̃ branched on ∑_{i∈J} E_{i}, we get a diagram

Ỹ
↓η
X̃ → X_{J} → X.

Let D_{j} = \frac{1}{2}η^{∗}(E_{j}) for j ∈ J; then

D_{j}^{2} = \frac{1}{4}η^{∗}(E_{j})^{2} = \frac{1}{4} · 2(E_{j})^{2} = −1

so that D_{j} is an exceptional curve of the first kind. Blowing down all the D_{j}’s for j ∈ J with a map α : Ỹ → Y, it is easy to see that the induced rational map Y −−→ X_{J} is in fact regular. So we get a bigger diagram:

Ỹ →^{α} Y
↓η  ↓
X̃ → X_{J} → X.

The inverse image of P_{j} in Y is the point Q_{j} = α(D_{j}), and the map Y → X_{J} is unramified away from the points P_{j} and Q_{j} (j ∈ J).

We need to compute some invariants of the surface Y. First, we have

K_{Y} = η^{∗}(K_{X} + \frac{1}{2} ∑_{i∈J} E_{i}) = ∑_{i∈J} D_{i}

and

K_{Y} = α^{∗}(K_{Y}) + ∑_{i∈J} D_{i}

which implies that K_{Y} = 0. Second, we can compute the topological Euler characteristic as follows (using the fact that for an unramified cover, the topological Euler characteristic multiplies by degree)

χ_{top}(Y) = χ_{top}(Y − ⋃_{i∈J} Q_{i}) + #(J)
= 2χ_{top}(X_{J} − ⋃_{i∈J} P_{i}) + #(J)
= 2χ_{top}(X̃ − ⋃_{i∈J} E_{i}) + #(J)
= 2(24 − 2 · #(J)) + #(J)
= 48 − 3 · #(J),

since χ_{top}(X̃) = 24 (always the case for a smooth K3 surface).

Lemma (from classification of surfaces). If J ≠ ∅, then #(J) = 8 or 16, with 16 points if and only if X is a Kummer surface and X = X_{J}.

Proof: Suppose first that Ỹ is Kähler (and so Y is Kähler). The only connected Kähler surfaces with K_{Y} = 0 are K3 surfaces (with χ_{top} = 24) and complex tori (with χ_{top} = 0). #(J) ≠ 0 implies that Y is connected, and the 2 cases correspond to #(J) = 8 and #(J) = 16 respectively. In the latter case, we must have X = X_{J} because the exceptional curves of X_{J} → X lift to smooth rational curves on Y and there are no such curves on a complex torus.

To handle the non-Kähler case, there are several options. One can assume in the definition that X is Kähler in an appropriate sense; this has the disadvantage that the tools we are developing are used in the proof that every K3 surface is Kähler, so it is not a good idea to assume that fact here. Alternatively, there are arguments using deformation theory, or the analysis of the behavior of the signature under ramified covers, which can be used to eliminate the non-Kähler case. Both arguments are too far from our topics here, so we omit them. “Q.E.D.”

We can construct an example of a cover with #(J) = 8 in the following way. Let T = ℂ^{2}/Γ be a complex torus and let Y = T/ĩ be its Kummer surface. (We retain the notation of Example Km from section 3.) Pick a point t ∈ \frac{1}{2}Γ and let Γ̄ be the group generated by Γ and t. Translation by t acts naturally on T, and the quotient by that action is another torus T̄ = ℂ^{2}/Γ̄, which has a Kummer surface X.

\begin{matrix}T & → & T̄ \\ ↓ & & ↓ \\ Y & & X \end{matrix}

The translation by t descends to an automorphism of Y, because z ↦ −z ↦ −z +t and z ↦ z + t ↦ −z − t differ by an element 2t ∈ Γ. Since X is the quotient of T by the group generated by ĩ and our translation, we get an induced map Y → X.

Let us compute the fixed points of the action of the translation on Y. If the image of z is fixed, then either z + t ≡ z mod Γ or z + t ≡ −z mod Γ. The first is clearly impossible, so we must have the second case: for some γ ∈ Γ, z = \frac{1}{2}(−t + γ). Since −z = \frac{1}{2}(−t + (2t − γ)) has the same form, this set of 16 points on T descends to a set of 8 points on Y. The map Y → X is unramified away from those 8 points.

The set J on X over which Y → X ramifies is the image of \{\frac{1}{2}(−t + γ) mod (Γ, ĩ)\} on X. Each such point satisfies 2(\frac{1}{2}(−t + γ)) = −t + γ ∈ Γ̄, so that the points all have order 2. There are 8 of them, and they form a subgroup which can naturally be identified with \frac{1}{2}Γ/Γ + t ⊂ \frac{1}{2} Γ̄/Γ̄. This is the set J ⊂ \{ points of order 2 on T̄\}.

Conversely, if we had started with a hyperplane J ⊂ \frac{1}{2} Γ̄/Γ̄ we could construct the double cover as follows: T̄ → T̄/J → T̄/T̄_{2} ≅ T̄ where T̄_{2} is the subgroup of points of order 2. The map T̄/J → T̄ is the quotient by the involution which is the non-trivial element in T̄_{2}/J ≅ ℤ/2ℤ and repeating the construction above (with T̄/J in place of T) produces the cover Y → X branched on the points of J.

In order to efficiently use the fact that #(J) can only be 0, 8 or 16 we change our notation a bit and phrase our constructions in terms of binary linear codes.

Definition. Let I be a finite set, and let 𝔽_{2}^{I} denote the 𝔽_{2}-vector space of maps from I to 𝔽_{2}. A binary linear code is an 𝔽_{2}-subspace V ⊂ 𝔽_{2}^{I}.

In our situation of a K3 surface X with A_{1} singularities at P_{1}, …, P_{k}, we let I = \{1, …, k\} and use the isomorphism L^{#}/L ≅ 𝔽_{2}^{I} induced by sending \frac{1}{2}e_{i} to the map φ_{i} such that φ_{i}(j) = δ_{ij} to produce the associated code V ⊂ 𝔽_{2}^{I} corresponding to M/L ⊂ L^{#}/L. Explicitly, φ ∈ V if and only if the set J_{φ} = {i | φ(i) = 1} is one of our distinguished subsets: \frac{1}{2} ∑_{i∈J_{φ}} E_{i} ∈ Pic(X̃).

Another example of a binary linear code is the universal binary linear code of dimension α: for an 𝔽_{2}-vector space W of dimension α, the space of linear maps W^{∗} = Hom(W, 𝔽_{2}) gives a code W^{∗} ⊂ 𝔽_{2}^{W}. A slight variant on this comes from noticing that φ(0) = 0 for any linear map, so that restricting linear maps to W − {0} gives a code W^{∗} ⊂ 𝔽_{2}^{W −{0}}.

The code W^{∗} ⊂ 𝔽_{2}^{W} is called “universal” because of the following construction. Let V ⊂ 𝔽_{2}^{I} be any code with dim V = α, let W = Hom(V, 𝔽_{2}) and define a tautological map τ : I → W by

τ(i)(φ) = φ(i)

for φ ∈ V. Then τ^{∗}(W^{∗}) = V; the only thing further which must be specified in order to describe V ⊂ 𝔽_{2}^{I} completely is the fibers of the map τ.

Theorem. Let V ⊂ 𝔽_{2}^{I} be a binary linear code with α = dim_{𝔽_{2}} V and k = #(I). Suppose that for every nonzero φ ∈ V, we have #{i | φ(i) = 1} = d, a number independent of φ. Then 2^{α} | 2d, k ≥ \frac{d}{2^{α−1}} (2^{α}−1), and for all nonzero w ∈ Hom(V, 𝔽_{2}) we have #\{i | τ(i) = w\} = \frac{d}{2^{α−1}} , where τ : I → Hom(V, 𝔽_{2}) is the tautological map.

Before giving the proof, we point out 2 applications to our situation.

Corollary 1. A K3 surface with 16 A_{1}-singularities is a Kummer surface.

Corollary 2. The code of a K3 surface with 15 A_{1}-singularities is isomorphic to W^{∗} ⊂ 𝔽_{2}^{W −{0}} for an 𝔽_{2}-vector space W of dimension 4.

Proof of Corollaries. Suppose that the K3 surface X with k singularities of type A_{1} is not a Kummer surface. Then for each non-empty distinguished subset J we have #(J) = 8. This means that the theorem applies with d = 8, and we find: α ≤ 4 and k ≥ 2^{4−α}(2^{α} − 1).

On the other hand, we know that k − 2α ≤ 22 − k, i. e., that α ≥ k − 11. Since α ≤ 4 we conclude that k ≤ 15. (Hence if there are 16 points, X must be a Kummer surface.) If k = 15, we have \frac{d}{2^{α−1}} = 1 so that each τ^{−1}(w) has cardinality 1 for w ≠ 0; since W − {0} contains only 15 points, the tautological map induces an isomorphism between the code of X and the code W^{∗} ⊂ 𝔽_{2}^{W −{0}}. Q.E.D.

Proof of the Theorem. For each w ∈ W = Hom(V, 𝔽_{2}), define

a_{w} = #{i | τ(i) = w}.

Now for each φ ∈ V we can write

{i | φ(i) = 1} = ⋃_{w | w(φ)=1}{i | τ(i) = w}.

Thus, if φ ≠ 0 we have

∑_{w | w(φ)=1} a_{w} = d

which implies

∑_{w | w(φ)=0} a_{w} = k − d.

while if φ = 0 then

∑_{w | w(φ)=1} a_{w} = 0

and

∑_{w | w(φ)=0} a_{w} = k.

We can combine these formulas as

∑_{w∈W}(−1)^{w(φ)}a_{w} = \{ \begin{matrix}k − 2d & if & φ ≠ 0 \\ k & if & φ = 0. \end{matrix} (*)

Define a matrix A = (A_{wφ})_{{w∈W \atop φ∈V}} by A_{wφ} = (−1)^{w(φ)}. (This is a 2^{α} × 2^{α} matrix). A is a Hadamard matrix, that is, an N × N matrix whose entries are all ±1 such that AA^{T} = diag(N, …, N). To see this, we compute

(AA^{T})_{wu} = ∑_{φ∈V} A_{wφ}A_{uφ}
= ∑_{φ∈V} (−1)^{w(φ)}(−1)^{u(φ)}
= ∑_{φ∈V} (−1)^{(w+u)(φ)}
= 2^{α}δ_{wu}

(since w + u is identically 0 if and only if w = u; otherwise, w + u takes the values 0 and 1 equally often.)

Now we compute using (*): on the one hand,

∑_{φ∈V} ∑_{w∈W}(−1)^{w(φ)}a_{w}A_{uφ} = ∑_{w∈W} a_{w}(AA^{T})_{wu} = 2^{α}a_{u}

while on the other hand,

∑_{φ∈V} ∑_{w∈W}(−1)^{w(φ)}a_{w}A_{uφ} = k(−1)^{u(0)} + ∑_{φ∈V,φ≠0}(k − 2d)(−1)^{u(φ)}
= 2d(−1)^{u(0)} + (k − 2d) ∑_{φ∈V} (−1)^{u(φ)}

= \{ \begin{matrix}2d & if & u ≠ 0 \\ 2d + 2^{α}(k − 2d) & if & u = 0. \end{matrix}

Thus, a_{u} = \frac{2d}{2^{α}} ∈ ℤ for u ≠ 0 while a_{0} = k − \frac{d}{2^{α−1}} (2^{α} − 1) ≥ 0 for u = 0, from which the theorem follows. Q.E.D.

We can derive some further structure in the case of Kummer surfaces on the code by using this theorem.

Proposition. Let V ⊂ 𝔽_{2}^{I} be the code associated to a K3 surface with 16 A_{1} singularities. Then dim V = 5, and the set I has a natural structure of an affine space of dimension 4 over 𝔽_{2} determined by: J ⊂ I is an affine hyperplane if and only if J = {i | φ(i) = 1} for some φ ∈ V with φ ≢ 0, φ ≢ 1.

Proof. Pick a point i_{0} ∈ I, and define

V_{0} = {φ ∈ V | φ(i_{0}) = 0}.

Then V_{0} has codimension (at most) 1 in V, and V_{0} ⊂ 𝔽_{2}^{I−{i_{0}}} is a code satisfying the hypotheses of the theorem with d = 8, k = 15. It follows that dim V_{0} = 4 and so that dim V = 5 (since dim V ≥ 5).

Now if W = Hom(V_{0}, 𝔽_{2}), then the code V_{0} ⊂ 𝔽_{2}^{I−{i_{0}}} is isomorphic to the code W^{∗} ⊂ 𝔽_{2}^{I−{i_{0}}}. If we extend the isomorphism I − {i_{0}} ≅ W − {0} (from corollary 2 above) to an isomorphism I ≅ W, then for each φ ∈ V_{0}, φ ≢ 0 we have that {i | φ(i) = 1} is the complement of a linear hyperplane in W (and so is an affine hyperplane). Moreover, every element in V − V_{0} can be written in the form φ + φ_{1} with φ ∈ V_{0}, where φ_{1}(i) = 1 for all i. For such elements, if φ ≢ 0 (i. e. φ + φ_{1} ≢ 1) we have that {i | (φ + φ_{1})(i) = 1} is a linear hyperplane in W (and so also an affine hyperplane). Thus, I ≅ W has the desired structure. Q.E.D.

An automorphism of a code V ∈ 𝔽_{2}^{I} is an isomorphism σ : I → I such that σ^{∗}(V) = V.

Proposition. If V ⊂ 𝔽_{2}^{I} is the code associated to a K3 surface with 16 A_{1} singularities, then Aut(V ⊂ 𝔽_{2}^{I}) ≅ AGL(4, 𝔽_{2}), the affine general linear group (which is generated by the general linear group, and by translations). In particular, the affine 𝔽_{2}-space structure on I is uniquely determined.

Proof. We identify I with W = Hom(V_{0}, 𝔽_{2}) as in the previous proposition. Consider σ : W → W defined by σ(w) = w + w_{0} (for some fixed w_{0} ∈ W). Then for φ ∈ V, σ^{∗}(φ)(w) = φ(w + w_{0}) = φ(w) + φ(w_{0}), so that

σ^{∗}(φ) = \begin{cases}φ & if φ(w_{0}) = 0, \\ φ + φ_{1} & if φ(w_{0}) = 1. \end{cases}

In particular, σ^{∗}(V) = V.

So the translations lie in Aut(V ⊂ 𝔽_{2}^{I}). Given an arbitrary σ ∈ Aut(V ⊂ 𝔽_{2}^{I}), by composing with a translation we may assume σ(i_{0}) = i_{0} (i. e. φ(0) = 0 under the identification I ≅ W). But then σ preserves V_{0}, and so σ ∈ Aut(V_{0} ⊂ 𝔽_{2}^{I−{i_{0}}}) = Aut(W^{∗} ⊂ 𝔽_{2}^{W −{0}}). Now σ induces a linear automorphism σ^{∗} : V_{0} → V_{0}, and thus a linear automorphism (σ^{∗})^{∗} : Hom(V_{0}, 𝔽_{2}) → Hom(V_{0}, 𝔽_{2}); since this latter space is ≅ W, we get that σ = σ^{∗∗} is a linear map on W. Conversely, any linear map σ ∈ GL(W) preserves the subspace W^{∗} ⊂ 𝔽_{2}^{W} and so acts on the code. Q.E.D.

The description of the affine space structure we have given above is an abstract one, based solely on the code. Using however the fact that the K3 surface in question is actually a Kummer surface, we can give an alternate description of this structure.

Lemma. let T = ℂ^{2}/Γ, let T_{2} = \frac{1}{2}Γ/Γ, and let X be the Kummer surface of T. If we fix an origin on T, then T_{2} has a natural 𝔽_{2}-vector space structure; forgetting the choice of origin leads to an affine 𝔽_{2}-space structure. Under the map T_{2} ≅ Sing(X) induced by the quotient map ξ : T → X, this coincides with the structure determined by the code.

Proof. What needs to be checked is that the subsets J ⊂ I such that \frac{1}{2} ∑_{i∈J} E_{i} ∈ Pic(X̃) and |J| = 8 exactly correspond to the affine hyperplanes of T_{2}. But we already checked that the affine hyperplanes are among the subsets J (in our construction of an example of the cover branched on 8 points); since the number of such subsets is 30 (= the number of hyperplanes), these must be all of the subsets. Q.E.D.

As the final step in our analysis of Kummer surfaces, we consider again the basic diagram

\begin{matrix}T̃ & →^{ρ} & T \\ ↓η \\ X̃ & → & X \end{matrix}

which relates the surfaces, and consider the map

r = η_{∗}ρ^{∗} : H^{2}(T, ℤ) → H^{2}(X̃, ℤ).

If x and y are cohomology classes in H^{2}(T, ℤ), then by the projection formula, η^{∗}(r(x)) · η^{∗}(r(y)) = 2r(x) · r(y). Thus,

r(x) · r(y) = \frac{1}{2}η^{∗}(r(x)) · η^{∗}(r(y))
= \frac{1}{2}η^{∗}η_{∗}(ρ^{∗}(x)) · η^{∗}η_{∗}(ρ^{∗}(y))
= \frac{1}{2}(2ρ^{∗}(x)) · (2ρ^{∗}(y))
= 2ρ^{∗}(x) · ρ^{∗}(y)
= 2x · y.

In other words, the map r induces an inclusion H^{2}(T, ℤ) ↪ H^{2}(X̃, ℤ) which multiplies the intersection form by 2. Note that Im r ⊂ M^{⊥}, and rank M^{⊥} = 22 − 16 = 6 = rank Im r.

If x_{1}, …, x_{6} is a basis for H^{2}(T, ℤ), since H^{2}(T, ℤ) is unimodular (i. e., is isomorphic to H^{2}(T, ℤ)^{#}) it is easy to see that (Im r)^{#} is generated by \frac{1}{2}r(x_{1}), …, \frac{1}{2}r(x_{6}) and so that

(Im r)^{#}/(Im r) ≅ (ℤ/2ℤ)^{6}.

Lemma. M^{⊥} = (Im r). (That is, in our previous notation, N = Im r.) In particular, Im r is saturated.

Proof. Since N is the saturation of Im r, we have

2^{6} = |(Im r)^{#}/ Im r| = |N^{#}/N|[N: Im r]^{2}.

On the other hand, N^{#}/N ≅ M^{#}/M ≅ (ℤ/2ℤ)^{k−2α} and for a Kummer surface k = 16, α = 5. Thus, |N^{#}/N| = 2^{6} so that [N: Im r] = 1, i. e., N = Im r. Q.E.D.

As a consequence of this lemma, r induces an isomorphism (which we also denote by r):

r : H^{2}(T, 𝔽_{2}) →^{≅} N^{#}/N.

There are several other groups isomorphic to these—we introduce names for the isomorphisms. First, let T_{2} = \frac{1}{2}Γ/Γ be the set of points of order 2 on T (which we identify with I ≅ Sing(X) in the natural way). There is then a isomorphism

s : Hom(Λ^{2}T_{2}, 𝔽_{2}) →^{≅} H^{2}(T, 𝔽_{2}).

Second, property 2 of discriminant-groups gives us an isomorphism

q : N^{#}/N →^{≅} M^{#}/M

(since M and N are saturated). Finally, the identification L^{#}/L ≅ 𝔽_{2}^{T_{2}} which sends M/L to the code V ⊂ 𝔽_{2}^{T_{2}} induces an inclusion

p : M^{#}/M → 𝔽_{2}^{T_{2}}/V.

The image of p is U/V, where U is the subspace of 𝔽_{2}^{T_{2}} corresponding to M^{#}/L ⊂ L^{#}/L.

We let t = pqrs : Hom(Λ^{2}T_{2}, 𝔽_{2}) → 𝔽_{2}^{T_{2}}/V be the composite map.

Proposition. Let φ, ψ ∈ Hom(T_{2}, 𝔽_{2}) be nonzero linear functions, and let χ_{φψ} ∈ 𝔽_{2}^{T_{2}} be the (nonlinear) function defined by

χ_{φψ}(i) = \begin{cases}1 & if φ(i) = ψ(i) = 0, \\ 0 & otherwise. \end{cases}

(χ_{φψ} is the characteristic function of the intersection of hyperplanes φ = ψ = 0.) Then

t(φ ∧ ψ) ≡ χ_{φψ} mod V.

Proof. By definition of p,

p (−\frac{1}{2} ∑_{i | φ(i)=χ(i)=0} e_{i}) = χ_{φψ} mod V.

Thus, the definition of q : N^{#}/N →^{≅} M^{#}/M shows that it suffices to prove that

rs(φ ∧ ψ) + −\frac{1}{2} ∑_{i | φ(i)=χ(i)=0} e_{i} ∈ H^{2}(X̃, ℤ)

i. e., is an integral class; in other words, that

2rs(φ ∧ ψ) − ∑_{i | φ(i)=χ(i)=0} e_{i}

is divisible by 2 in H^{2}(X̃, ℤ). (Note that rs(φ ∧ ψ) ∈ H^{2}(X̃, ℚ)!)

Now φ, ψ ∈ Hom(T_{2}, 𝔽_{2}) are induced by some homomorphisms φ̃, ψ̃ ∈ Hom(Γ, ℤ) (using T_{2} = \frac{1}{2}Γ/Γ ≅ Γ/2Γ), and the cohomology classes in H^{1}(T, ℤ) are Poincaré dual to the (real) hypersurfaces φ̃ = 0 and ψ̃ = 0 respectively. Thus, the class of φ̃ ∧ ψ̃ is dual to \{φ̃ = ψ̃ = 0\}; let x denote this class. Note that rs(φ ∧ ψ) = \frac{1}{2}r(x). We have that x passes through the points {i | φ(i) = ψ(i) = 0} ⊂ T_{2} (and only those points of T_{2}), so that the proper transform of x on T̃ is the integral class

ρ^{∗}(x) − ∑_{i | φ(i)=χ(i)=0} D_{i} ∈ H^{2}(T̃, ℤ).

But this proper transform is invariant under the involution on T̃, so applying η_{∗} yields a cohomology class which is divisible by 2 in H^{2}(X̃, ℤ). That is, 2 divides (in H^{2}(X̃, ℤ)) the class

η_{∗}(ρ^{∗}(x) − ∑_{i | φ(i)=χ(i)=0} D_{i}) = η_{∗}ρ^{∗}(x) − ∑_{i | φ(i)=χ(i)=0} e_{i}
= r(x) − ∑_{i | φ(i)=χ(i)=0} e_{i}
= 2rs(φ ∧ ψ) − ∑_{i | φ(i)=χ(i)=0} e_{i},

as required. Q.E.D.

As a concluding remark about the combinatorics of Kummer surfaces, consider the case of a projective Kummer surface, say a quartic Kummer surface in ℙ^{3}. There is then a nef line bundle with a class λ ∈ M^{⊥}, λ^{2} = 4. Let M̃ = M ⊕ ℤ(λ); then M̃^{#}/M̃ ≅ (ℤ/2ℤ)^{6} × ℤ/4ℤ while M̃^{⊥} has rank 5! There must thus be an additional class in (M̃ ⊗ ℚ) ∩ Λ.

In fact, this class has the form \frac{1}{2}(λ − ∑_{i=1}^{6} e_{i}) where e_{1}, …, e_{6} have been chosen so that if e_{6} is the origin of a vector space structure on T_{2} then e_{1}, …, e_{4} form a basis and e_{5}“ = ”e_{1} + ⋯ + e_{4} in this basis. There are 16 such classes, and each leads to a hyperplane in ℙ^{3} passing through 6 of the nodes of the quartic, which is everywhere tangent to the quartic along a plane conic. Moreover, each of the 16 singular points lies in 6 of these planes, and each of the 16 planes contains 6 of the singular points. This famous “16-6 configuration” can be viewed in the photograph from Hudson’s book.

9. The Torelli theorem for Kummer surfaces

A Hodge structure of weight n is a free ℤ-module of finite rank H_{ℤ} together with a direct sum decomposition H_{ℤ}⊗ℂ = ⨁_{p=0}^{n} H^{p,n−p} such that \overline{H^{p,n−p}} = H^{n−p,p}. The n^{th} cohomology group of an algebraic variety (or a Kähler manifold) has a natural Hodge structure of weight n, and the Torelli problem asks whether the Hodge structure determines the isomorphism type of the variety. Generally in order for the Torelli problem to have a positive answer, some extra structure (like a polarization of the Hodge structure) must be added.

The easiest case of a Torelli theorem is that of complex tori (weight 1 Hodge structures). If T = ℂ^{n}/Γ is a complex torus, then we can identify H_{1}(T, ℤ) ≅ Hom(H^{1}(T, ℤ), ℤ) and recover the Albanese variety as (H^{0,1})^{∗}/ Hom(H_{ℤ}^{1}, ℤ). Since a torus is isomorphic to its Albanese, the Hodge structure determines the isomorphism type.

Since a Kummer surface is built out of a torus, it is reasonable to expect a similar phenomenon for Kummer surfaces. However, H^{1} of a Kummer surface is trivial; we must use H^{2}, which contains H^{2}(T, ℤ) = Λ^{2}H^{1}(T, ℤ) as a subgroup. The difficulties in the Torelli problem for Kummer surfaces come from the necessity of passing from an isomorphism Λ^{2}H^{1}(T) → Λ^{2}H^{1}(T′) to an isomorphism H^{1}(T) → H^{1}(T′).

In order to solve the Torelli problem, we must consider some extra structure on the cohomology; we describe this extra structure for an arbitrary smooth K3 surface X. First, notice that H^{2,0}(X) is necessarily 1-dimensional: if we let ω be a nowhere-vanishing holomorphic 2-form and α be any other holomorphic 2-form, then \frac{α}{ω} defines a global holomorphic function and is therefore constant. It follows that dim_{ℂ} H^{2,0}(X) = dim_{ℂ} H^{0,2}(X) = 1, and thus that dim_{ℂ} H^{1,1}(X) = 20. (Recall that b_{2} = 22).

Next, there are certain compatibilities between the intersection form and the Hodge structure which are guaranteed by the Hodge index theorem: we have H^{p,q}(X) ⊥ H^{p′,q′}(X) unless p+p′ = q+q′ = 2; moreover, the intersection form is positive definite on (H^{2,0}(X)⊕H^{0,2}(X))∩H^{2}(X, ℝ), and has signature (1,19) on H^{1,1}(X)∩H^{2}(X, ℝ). (This last statement is a more general version of the Hodge index theorem.) It follows that it has signature (3,19) overall.

Given an isomorphism Φ : X′ → X between 2 smooth Kähler K3 surfaces, the induced isomorphism Φ^{∗} : H^{2}(X, ℤ) → H^{2}(X′, ℤ) has several properties:

(1) Φ^{∗} preserves the intersection form, i. e., Φ^{∗}(x) · Φ^{∗}(y) = x · y.

(2) Φ^{∗} preserves the Hodge structure, i. e., Φ^{∗}(H^{p,q}(X)) = H^{p,q}(X′).

(3) Φ^{∗} preserves the effective classes, i. e., Φ^{∗}(E(X)) = E(X′), where

E(X) = {cohomology classes in H^{2}(X, ℤ) of effective divisors},

(4) for some Kähler class κ on X and some Kähler class κ′ on X′, Φ^{∗}(κ) · κ′ > 0. (Here, a Kähler class is the cohomology class in H^{1,1}(X) ∩ H^{2}(X, ℝ) of a Kähler metric on X. Property (4) could have been stated: for every κ and every κ′, but it is in fact enough to check it for 1 as we will see in the next section.)

Definition. Let X, X′ be smooth Kähler surfaces. An effective Hodge isometry is an isomorphism φ : H^{2}(X, ℤ) → H^{2}(X′, ℤ) such that

(1) φ(x) · φ(y) = x · y if x, y ∈ H^{2}(X, ℤ)

(2) φ(H^{p,q}(X)) = H^{p,q}(X′)

(3) φ(E(X)) = E(X′)

(4) for some Kähler classes κ on X and κ′ on X′, φ(κ) · κ′ > 0.

Remark. If X is the minimal desingularization of a Kummer surface, then X is Kähler. In fact, if r : H^{2}(T, ℤ) → H^{2}(X, ℤ) is the natural map from the cohomology of the torus, and κ is any Kähler class on T then r(κ) + ε ∑_{i=1}^{16} e_{i} is a Kähler class on X for sufficiently small ε > 0.

Theorem (The Torelli theorem for Kummer surfaces). Let X be the minimal desingularization of a Kummer surface, and let X′ be a smooth Kähler K3 surface.

If φ : H^{2}(X, ℤ) → H^{2}(X′, ℤ) is an effective Hodge isometry, then there is an isomorphism Φ : X′ → X such that Φ^{∗} = φ.

Proof. There are several steps.

Step 1. We first check that X′ is also the minimal desingularization of a Kummer surface. Let e_{1}, …, e_{16} be the classes of the 16 disjoint smooth rational curves on X coming from the resolution of the Kummer surface’s singularities. Since φ preserves effective classes, each φ(e_{i}) is effective; we must check that it is irreducible and then it will follow that X′ is also the desingularization of a Kummer surface. If we write φ(e_{i}) = ∑ f_{ij} where each f_{ij} is the class of an irreducible component of φ(e_{i}) (with repetitions allowed), then φ^{−1}(f_{ij}) must be an effective curve on X and e_{i} = ∑ φ^{−1}(f_{ij}). But the curve E_{i} does not move; hence there can be only one f_{ij} so that φ(e_{i}) is irreducible.

Step 2. Now we have tori T and T′ whose Kummer surfaces are X and X′. We use the notation of section 8. The map φ induces an isomorphism between the singular sets Sing(X) ≅ T_{2} and Sing(X′) ≅ T′_{2} and an isomorphism L^{#}/L →^{≅} L′^{#}/L′ which sends M/L to M′/L′ and M^{#}/L to M′^{#}/L′. Thus, we get an isomorphism of codes V ⊂ 𝔽_{2}^{T_{2}} ≅ V′ ⊂ 𝔽_{2}^{T′_{2}}. The natural structures of affine spaces on T_{2} and T′_{2} must be preserved by this isomorphism; hence, if we fix origins i_{0} ∈ T_{2} and i′_{0} ∈ T′_{2} for T and T′ such that φ(e_{i_{0}}) = e_{i′_{0}} we have an induced isomorphism φ_{1} = H^{1}(T, 𝔽_{2}) → H^{1}(T′, 𝔽_{2}) (as 𝔽_{2}-vector spaces).

Now we also have, by considering N = M^{⊥} mapping to φ(N) = N′ = M′^{⊥}, a natural isomorphism φ_{2} : H^{2}(T, ℤ) → H^{2}(T′, ℤ). The compatibility condition between N^{#}/N and M^{#}/M then guarantees (since M^{#}/M is mapped to M′^{#}/M′) that φ_{2} ≡ φ_{1} ∧ φ_{1} mod 2. (In fact, we only checked this for reducible elements in H^{2}(T, 𝔽_{2}) = Λ^{2} Hom(T_{2}, 𝔽_{2}), but the reducible elements generate Λ^{2} Hom(T_{2}, 𝔽_{2}) as an 𝔽_{2}-vector space.)

Step 3. Consists of the following.

Proposition. Let H and H′ be two free ℤ-modules of rank 4 on which an orientation has been chosen. (The orientations determine isomorphisms Λ^{4}H ≅ ℤ and Λ^{4}H′ ≅ ℤ and thereby determine symmetric bilinear forms on Λ^{2}H resp. Λ^{2}H′ by

Λ^{2}H × Λ^{2}H → Λ^{4}H ≅ ℤ

and similarly for H′.) Let ψ : Λ^{2}H → Λ^{2}H′ be an isomorphism preserving this bilinear form. The following are equivalent:

(i) There exists an isomorphism λ : H → H′ such that ψ = ±λ ∧ λ (ii) There exists an isomorphism λ : G ⊗ 𝔽_{2} → H′ ⊗ 𝔽_{2} such that ψ ≡ λ ∧ λ mod

A proof of this proposition can be found on pp. 103–105 of the seminar notes “Geometrie des surfaces K3 … ” edited by Beauville et al., or on p. 138 of the book by Barth-Peters-Van de Ven.

To finish the proof, by steps 2 and 3 there is an isomorphism λ : H^{1}(T, ℤ) → H^{1}(T′, ℤ) such that λ ∧ λ = ±φ_{2} : H^{2}(T, ℤ) → H^{2}(T′, ℤ). If λ ∧ λ = −φ_{2}, let κ and κ′ be Kähler classes on T and T′. Since λ is induced by an isomorphism T′ →^{≅} T we have (λ ∧ λ)(κ) · κ′ > 0. On the other hand,

(λ ∧ λ)(κ) · κ′ = −φ_{2}(κ) · κ′
= −φ(κ + ε ∑ e_{i}) · (κ′ + ε′ ∑ e_{i})
< 0.

Since κ + ε ∑ e_{i}, κ′ + ε ∑ e′_{i} are Kähler classes on X and X′; this is a contradiction. Thus, λ ∧ λ = φ_{2}.

There is an isomorphism Λ : T′ → T inducing λ; by composing with a translation we may assume Λ(i′_{0}) = i_{0}. Since the 𝔽_{2}-space structures on T_{2}, T′_{2} are preserved, we see that the induced isomorphism Φ : X′ → X between the Kummer surfaces satisfies Φ^{∗} = φ. Q.E.D.

10. Nef and ample bundles, Kähler classes, and the Weyl group

We consider in this section only smooth K3 surfaces X. Recall that every (nonzero) effective irreducible divisor D on X satisfies D^{2} ≥ −2, and if D^{2} ≥ 0 then D moves in a nontrivial linear system (i. e. h^{0}(O_{X}(D)) ≥ 2); in particular, D^{2} ≥ 0 implies that D is nef. Now every effective divisor can be written as a nonnegative linear combination of irreducible ones, so the effective divisors form a cone generated by the smooth rational curves (i. e. effective irreducibles with D^{2} = −2) and the nef divisors.

Conversely, if D^{2} ≥ −2 then either D or −D is effective. But even when D^{2} ≥ 0 it may fail to be nef, and even if D^{2} = −2 it may fail to be irreducible. So the characterization of nef divisors and of smooth rational curves requires further work.

If X has an ample line bundle L, this bundle can be used to distinguish the effective divisors. For L · D > 0 whenever D is effective, and thus the sign of L · D determines whether D or −D is effective (when D^{2} ≥ −2). More generally, if X has a Kähler metric with cohomology class κ [something which certainly holds in the case of an ample bundle, whose class becomes κ] then again κ · D > 0 for all effective divisors. Note that κ^{2} > 0 (and L^{2} > 0 in the ample line bundle case) so that this is also a necessary condition for a class to be Kähler. A first attempt at identifying the set of Kähler class is then to consider the set

(*) {x ∈ H^{1,1}(X) ∩ H^{2}(X, ℝ) | x^{2} > 0, x · d > 0 for all cohomology classes d of effective divisors}.

The Kähler classes certainly lie in this set; but as we will see below, for non-algebraic K3 surfaces one additional piece of information must be added.

Dually, we can hope to use the set (*) (or some refinement of it) to identify the nef divisors and the smooth rational curves: a nonzero divisor D with D^{2} ≥ −2 will be effective (we hope!) exactly when D · x > 0 for some (and hence for every) x in the set (*).

For algebraic K3 surfaces, all of this works without further modification: this is guaranteed by the Nakai-Moishezon criterion for ampleness. In fact, that criterion (combined with the Lefschetz (1,1)-theorem) says exactly that

{classes of ample divisors on X} = {x ∈ H^{1,1}(X) ∩ H^{2}(X, ℤ) |

x^{2} > 0, x · d > 0 for all cohomology classes d of effective divisors}.

(The difference between the right hand side of this equation and (*) is that this time we have required the class to be integral, not just real.) The set of classes of nef divisors will simply be the closure of the set of classes of ample divisors.

To introduce the refinement I mentioned in the non-algebraic case, we need some notation: let H_{ℝ}^{1,1}(X) = H^{1,1}(X) ∩ H^{2}(X, ℤ). The intersection form has signature (1,19) when restricted to this space, and this implies that the set

C(X) = {x ∈ H_{ℝ}^{1,1}(X) | x^{2} > 0}

has 2 connected components. The picture is this:

x · x > 0 (first component)

x · x = 0

x · x < 0 (second component)

(x · x = 0 is sometimes called the light cone in analogy with the theory of special relativity, where forms with this signature [or rather, the opposite signature to this] appear.) The two components are interchanged by the map x ↦ −x.

There is a convenient characterization of the components of C(X): x and y (both in C(X)) belong to the same component if and only if x · y > 0. To see this, consider the line segment tx + (1 − t)y for 0 ≤ t ≤ 1; since

(tx + (1 − t)y)^{2} = t^{2}x^{2} + 2t(1 − t)x · y + (1 − t)y^{2} > 0

this never leaves the component. Moreover, if x · y < 0 then x and −y are in the same component so x and y cannot be.

Similarly if x ∈ C(X), y ∈ \overline{C(X)} (the closure), then y belongs to the closure of the component containing x if and only if x · y ≥ 0.

Now fix a Kähler class κ on X; any other Kähler class must satisfy κ · x > 0, and so belongs to the same component of C(X) as does κ. (This is because any convex combination tα_{κ} + (1 − t)α_{x} of Kähler forms α_{κ} and α_{x} with 0 ≤ t ≤ 1 is again a Kähler form.) Furthermore, any class d of an effective divisor satisfies κ · d > 0; for such classes d with d^{2} ≥ 0 we see that d belongs to the closure of the component of C(X) containing κ: It automatically follows that x · d > 0. So consider the set

(**) {x ∈ H_{ℝ}^{1,1}(x) | x^{2} > 0, x · κ > 0 and for all classes of

irreducible effective divisors d with d^{2} = −2 we have x · d > 0}

This is a subset of the previous one; the only “new” condition that has been added is x · κ > 0, and all previous conditions continue to hold.

To describe this set more efficiently, we define

Δ(X) = {δ ∈ H^{1,1}(X) ∩ H^{2}(X, ℤ) | δ^{2} = −2}

Δ^{+}(X) = {δ ∈ Δ(X) | δ is the class of an effective divisor}.

(So for each δ ∈ Δ(X), either δ ∈ Δ^{+}(X) or −δ ∈ Δ^{+}(X).) We can now re-describe the set (**) as:

V^{+}(X) = {x ∈ H_{ℝ}^{1,1}(X) | x^{2} > 0, x · κ > 0 and x · δ > 0 for all δ ∈ Δ^{+}(X)}.

This set is independent of the choice of Kähler metric κ; it only depends on the component of C(X) in which κ lies. The remainder of the section is devoted to studying properties of this set.

For δ ∈ Δ(X), define the reflection in δ to be the mapping

s_{δ} : x ↦ x + (x · δ)δ.

This acts on H^{2}(X, ℤ), preserving the Hodge decomposition and the intersection form, since (x+(x·δ)δ)^{2} = x^{2}. (The Hodge decomposition is preserved since δ ∈ H^{1,1} and H^{2,0} ⊕H^{0,2} ⊂ δ^{⊥}.) It therefore acts on H_{ℝ}^{1,1}(x) as well. We define the Weyl group of X to be the group W (X) generated by {s_{δ} | δ ∈ Δ}; this can be regarded as a subgroup of Aut(H^{2}(X, ℤ)) or Aut(H_{ℝ}^{1,1}(X)) ≅ O(1, 19), where the automorphisms in question are those preserving the intersection form. Note that the action of W (X) (even of all of O(1, 19)) on H_{ℝ}^{1,1}(X) preserves the subset C(X).

Lemma 1. W (X) is a discrete group which acts properly discontinuously on C(X).

Proof. There is a W (X)-equivariant isomorphism

C(X) ≅ C^{1}(X) × ℝ^{+}

where C^{1}(X) = {x ∈ H_{ℝ}^{1,1}(X) | x^{2} = 1} and ℝ^{+} denotes the positive reals. It suffices to show that the action on C^{1}(X) is properly discontinuous. Now Aut(H_{ℝ}^{1,1}(X)) ≅ O(1, 19) acts transitively on C^{1}(X), and the stabilizer of a point is a compact group isomorphic to O(19).

W (X), being a subgroup of Aut(H^{2}(X, ℤ)), is discrete in Aut(H^{2}(X, ℝ)) and hence also in the subgroup Aut(H_{ℝ}^{1,1}(X)) in which it lies. Thus, the action of W (X) on O(1, 19) is properly discontinuous, which implies that the induced action on C^{1}(X) ≅ O(1, 19)/O(19) is also properly discontinuous. Q.E.D.

Lemma 2. If a discrete group W acts properly discontinuously on a space C and if S is a subset of W then

F = ⋃_{s∈S}{x | s(x) = x}

is closed in C.

Proof. For y ∈ X − F let W_{y} = {w ∈ W | w(y) = y} be the stabilizer; we have W_{y} ∩ S = ∅. Since the action is properly discontinuous, there is a neighborhood U of y such that wU ∩ U = ∅ for all w ∈ W − W_{y}; in particular, for all w ∈ S. But then no point of U is fixed by any element of S, so U ⊂ X − F. Q.E.D.

Corollary. ⋃_{δ∈Δ(x)} δ^{⊥} is closed in C(X).

(Because δ^{⊥} is the fixed locus of the reflection S_{δ}.)

The hyperplanes δ^{⊥} are called the walls in C(X), and the connected components of C(X) − ⋂_{δ∈Δ(x)} δ^{⊥} are called the chambers of C(X). Chambers are open subsets of C(X) (and of H_{ℝ}^{1,1}(X).).

Theorem (it’s in Bourbaki …). The group W (X) × {±1} acts transitively on the set of chambers of C(X).

The use of this theorem in the study of K3 surfaces is this: a chamber V is determined by the set Δ_{V} = {δ ∈ Δ(x) | δ · x > 0 for x ∈ V } which says which side of each wall V lies on. (V^{+}(X) is one of the chambers). If we happen to have an isomorphism of Hodge structures which does not preserve effective classes and Kähler classes (i. e. does not preserve V^{+}), then this group W (X) × {±1} can be used to alter the isomorphism so that these things are preserved.

The theorem can also be used to derive conclusions about the irreducible classes in Δ^{+}(X): These are the ones whose walls actually meet the closure of V^{+}(X). Unfortunately, we do not have sufficient time to explore this topic.

Proof of the Theorem. Since ±1 interchanges the two connected components of C(X) while W (X) preserves them (because: x · s_{δ}(x) = x^{2} + (x · δ)(x · δ) > 0), it suffices to check the transitivity of the action of W (X) on the chambers in one of the components of C(X).

Let x, y ∈ C(x) such that x·y > 0 and x·δ ≠ 0, y·δ ≠ 0 for all δ ∈ Δ(X). We must show that for some w ∈ W (X), w(x) and y lie in the same connected component of C(x) − ⋃_{δ∈Δ(x)} δ^{⊥}. Let ℓ = x^{2}.

For each a ∈ ℝ, the set

{z ∈ C(x) | 0 ≤ y · z ≤ a, z^{2} = ℓ}

is compact. (Pictorially, z^{2} = ℓ, y · z > 0 y · z = a y · z = 0

Since the action of W (X) on C(X) is properly discontinuous, it follows that

{w ∈ W (x) | y · w(x) ≤ a}

is a finite set. Note that 0 ≤ y · w(x) and w(x)^{2} = ℓ. Thus, the function z ↦ y · z on the orbit W x of x attains its minimum at a point z_{0} = w_{0}x. But then for all δ ∈ Δ we have

y · w_{δ}(w_{0}x) ≥ y · w_{0}x

i. e.

y · (z_{0} + (δ · z_{0})δ) ≥ y · z_{0}

so that

(δ · z_{0})(y · δ) ≥ 0.

Thus, z_{0} and y are on the same side of every wall ((δ·z_{0}) > 0 if and only if (y·δ) > 0) and so belong to the same connected component of C(X) − ⋃_{δ∈Δ(x)} δ^{⊥}. Q.E.D.

10.1. Addendum to section 10. Two things should be noticed about the application of this result to K3 surfaces: (1) the set

V^{+}(X) = {x ∈ H_{ℝ}^{1,1}(X) | x^{2} > 0, x · κ > 0, x · δ > 0 for all δ ∈ Δ^{+}(X)}

is a chamber (independent of the choice of Kähler class κ), and (2) a map φ : H^{2}(X, ℤ) → H^{2}(X′, ℤ) satisfies φ(V^{+}(X)) = V^{+}(X′) if and only if it satisfies both (a) φ(E(X)) = E(X′) and (b) φ(κ) · κ′ > 0 for some Kähler classes κ and κ′. In particular, the definition of “effective Hodge isometry” can be reformulated as: preserves Hodge structures and intersection forms, and maps V^{+}(X) isomorphically to V^{+}(X′).

11. The period mapping for K3 surfaces

Recall two key properties of the intersection form on the second cohomology group of a K3 surface.

Property 1. If X is a K3 surface, the intersection form on H^{2}(X, ℤ) is unimodular, by Poincaré duality. (We first encountered this property in section 8.)

Property 2. If X is a K3 surface, the intersection form on H^{2}(X, ℤ) has signature (3,19), by the extended version of the Hodge index theorem. (We first encountered this property in section 9.)

There are two additional key facts about K3 surfaces which we have not yet mentioned.

Topological Fact. Let X be a (smooth) compact complex surface. Suppose that H^{1}(X, ℤ/2ℤ) = 0. Then for all γ ∈ H^{2}(X, ℤ), γ · γ − c_{1}(X) · γ ≡ 0 mod 2. (This is clear for algebraic classes, i. e. those coming from divisors D: D · D + K_{X} · D = 2g(D) − 2 must be even. The proof in general uses the Wu formula and Stiefel-Whitney classes; a good reference is the book of Milnor and Stasheff.) In particular, for a smooth K3 surface X we have γ · γ ≡ 0 mod 2 for all γ ∈ H^{2}(X, ℤ).

Number-Theoretic Fact. There is a unique (up to isomorphism) free ℤ-module Λ with a unimodular symmetric bilinear form Λ × Λ → ℤ of signature (3,19) such that γ · γ ≡ 0 mod 2 for all γ ∈ Λ. This form Λ is isomorphic to (−E_{8})^{⊕2} ⊕ U^{⊕3} where E_{8} is the unimodular even positive definite form of rank 8, and U is the hyperbolic plane. We call Λ the K3 lattice. (A good reference for this is Serre’s Cours d’Arithmétique.) In particular, for every smooth K3 surface there exists an isomorphism α : H^{2}(X, ℤ) → Λ preserving the bilinear forms; a choice of such an isomorphism is called a marking of X.

By using a marking α : H^{2}(X, ℤ) → Λ, we get a Hodge structure on Λ; we want to describe the set of all Hodge structures of the appropriate type.

More generally, suppose we have a free ℤ-module L with bilinear form of signature (2h^{2,0} + 1, h^{1,1} − 1), and we want to consider Hodge decompositions L ⊗ ℂ = H^{2,0} ⊕ H^{1,1} ⊕ H^{0,2} such that H^{p,q} ⊥ H^{p′,q′} unless p + p′ = q + q′ = 2, and such that the form is positive definite on (H^{2,0} ⊕ H^{0,2}) ∩ (L ⊗ ℝ), and has signature (1, h^{1,1} − 1) on H^{1,1} ∩ (L ⊗ ℝ). Such a Hodge structure is completely specified by H^{2,0}, since H^{0,2} = \overline{H^{2,0}} and H^{1,1} = (H^{2,0} ⊕ H^{0,2})^{⊥}. In fact, the natural parameter space for all such Hodge structures is the space

Ω_{L} = {μ ∈ Gr(h^{2,0}, L ⊗ ℂ) | μ is totally isotropic for the bilinear form, and for each x ∈ μ, x ≠ 0 we have x · x̄ > 0}

(H^{2,0} = μ, H^{0,2} = μ̄, H^{1,1} = (μ ⊕ μ̄)^{⊥} is the Hodge structure).

Now it is not difficult to see that the tangent space to Gr(h^{2,0}, L ⊗ ℂ) at μ is given by

T_{μ} Gr(h^{2,0}, L ⊗ ℂ) ≅ Hom(μ, (L ⊗ ℂ)/μ)

in a natural way. Slightly more difficult is the isomorphism

T_{μ}Ω_{L} ≅ Hom(μ, μ^{⊥}/μ).

In fact, since μ^{⊥} = μ ⊕ H^{1,1} we have

T_{μ}Ω_{L} ≅ Hom(H^{2,0}, H^{1,1}).

Returning to the K3 lattice Λ, we let Ω = Ω_{Λ}; in this case, the Grassmannian is a projective space and we can write

Ω = {[ω] ∈ ℙ(Λ ⊗ ℂ) | ω · ω = 0, ω · ω̄ > 0}

with associated Hodge structure

H^{2,0} = ℂ(ω), H^{0,2} = ℂ(ω̄), H^{1,1} = <ω, ω̄>^{⊥}.

Thus we see that Ω is an open subset in a quadric in ℙ^{21}.

The next topic to be discussed is deformation theory; for lack of time, I refer you to the chapter by Gauduchon “Théorème de Torelli locale pour les surfaces K3” in the seminar notes “Géometrie des surfaces K3 … ” edited by Beauville et al.

For a smooth K3 surface X, the sheaf of holomorphic vector fields Θ_{X} is naturally isomorphic to the sheaf of holomorphic 1-forms Ω_{X}^{1} via contraction with the nowhere vanishing holomorphic 2-form ω. Thus, H^{2}(Θ_{X}) = H^{2}(Ω_{X}^{1}) = 0; it follows that there is a smooth local universal deformation of X: this is a map π : X → S with smooth fibers such that π^{−1}(0) = X and the Kodaira-Spencer map is an isomorphism. (The Koraira-Spencer map sends T_{0}S to H^{1}(Θ).)

We choose a marking α : H^{2}(X, ℤ) → Λ, and extend this to a continuous family of markings α_{s} : H^{2}(π^{−1}(s), ℤ) → Λ by using the differentiable triviality of the family X → S. (That is, there is a C^{∞} isomorphism X ≅_{C^{∞}} X ×S.) We then get the period mapping of the family S → Ω which sends s to the Hodge structure on Λ given by α(H^{2}(π^{−1}(S), ℤ).

Now we have

T_{0}S

⟶^{differential}

of period map

T_{μ}Ω
Kodaira-Spencer↓≅  ↓
H^{1}(Θ_{X})  Hom(H^{2,0}(X), H^{1,1}(X)).

Theorem (The local Torelli theorem for K3 surfaces). The differential of the period map S → Ω is an isomorphism.

The proof is based on the more general fact that in the diagram above, the natural map H^{1}(Θ_{X}) → Hom(H^{2,0}(X), H^{1,1}(X)), or H^{1}(Θ_{X}) → Hom(H^{0}(Ω_{X}^{2}), H^{1}(Ω_{X}^{1})) is in fact given by the mapping

H^{1}(Θ_{X}) ⊗ H^{0}(Ω_{X}^{2}) → H^{1}(Ω_{X}^{1})

induced by contraction of a vector field with a 2-form to produce a 1-form. In the case of a K3 surface, as has already been pointed out, this map is an isomorphism. Q.E.D.

As a consequence of the local Torelli theorem, the set of Hodge structures corresponding to K3 surfaces is an open set in the 20-dimensional complex manifold Ω. We can discover a lot about the moduli of K3 surfaces by examining the period space Ω.

Where are the algebraic K3 surfaces in our picture? If we choose a class λ ∈ Λ with λ^{2} > 0 (corresponding to an ample divisor under a marking of X, say), then we can look at the subset

Ω_{λ} = {[ω] ∈ Ω | ω · λ = 0}.

This has codimension 1 in Ω, and parametrizes all K3 surfaces for which a marking α exists such that α^{−1}(λ) is the class of a divisor. As a consequence, we see that the set of algebraic K3 surfaces is a union of codimension 1 subvarieties in the set of all K3 surfaces. In particular, every algebraic K3 surface has arbitrarily close deformations which are non-algebraic.

More generally, given a Hodge structure on Λ, we may consider Λ ∩ H^{1,1}; if α : H^{2}(X, ℤ) → Λ is a marking of X preserving the Hodge structures, then α^{−1}(Λ∩H^{1,1}) is the Neron-Severi group of X (by the Lefschetz (1,1)-theorem). We can try to preassign the structure of the Neron-Severi group by finding submodules M ⊂ Λ and looking at Hodge structures for which M lies in H^{1,1}, as follows:

Given M ⊂ Λ such that the signature of M is (1, r − 1) or (0, r), define

Ω_{M} = {[ω] ∈ Ω | ω · μ = 0 for all μ ∈ M}.

It turns out that Ω_{M} is non-empty and has dimension equal to 22−r = 22−rank(M).

As an example of this, let e_{1}, …, e_{16} ∈ Λ be classes coming from some Kummer surface (so e_{i} · e_{j} = −2δ_{ij}), and let M be the saturation of the lattice L which they generate. (M/L ⊂ L^{#}/L is necessarily isomorphic to the Kummer code.) The resulting space Ω_{M} has dimension 4, and parametrizes Kummer surfaces. However, for any γ ∈ O(Λ) (the integral automorphism group of the unimodular form Λ), if we change the marking on the Kummer surfaces by using γ we discover that Ω_{γ(M)} also parametrizes Kummer surfaces.

Key Fact. ⋃_{γ∈O(Λ)} Ω_{γ(M)} is dense in Ω.

This “key fact” is an essential step in the proof of the global Torelli theorem for K3 surfaces. Using it, one proceeds like this: given an effective Hodge isometry φ : H^{2}(X, ℤ) → H^{2}(X′, ℤ), pick sequences of Kummer surfaces X_{i}, X′_{i} whose periods tend to those of X, X′ and show that isomorphisms φ_{i} can be chosen “converging” to φ. There exist isomorphisms Φ_{i} : X′_{i} → X_{i} with Φ_{i}^{∗} = φ_{i}, one needs to know that these converge in some appropriate sense.

The technical work needed to implement this method is somewhat difficult, and we will not go into it here.

12. The structure of the period mapping

For a smooth K3 surface X, we have identified a particular chamber of C(X)

V^{+}(X) = {x ∈ H_{ℝ}^{1,1}(X) | x^{2} > 0, x · λ > 0 and for all δ ∈ Δ^{+}(X), x · δ > 0}

which we call the Kähler chambers. (Here, κ is any Kähler class on X.) Now if X is projective, V^{+}(X) ∩ H^{2}(X, ℤ) is exactly the set of ample divisors on X. the set of nef and big divisors can be identified with \overline{V^{+}(X)} ∩ C(X) ∩ H^{2}(X, ℤ).

Now if λ is the class of a nef and big divisor, λ corresponds to a line bundle O_{X}(L). The linear system |3L| factors through the contraction π : X → X̄ of all curves C_{i} such that C_{i}^{2} = −2, C_{i} · L = 0, and embeds the resulting surface X̄. Moreover, O_{X}(L) = π^{∗}(OX̄(L̄)) for an ample linear system |L̄| on X̄. The ample line bundles on X̄ exactly correspond to classes λ ∈ \overline{V^{+}(X)}∩C(X)∩H^{2}(X, ℤ) such that λ^{⊥}∩Δ(X) exactly corresponds to the (-2)-classes supported on the exceptional set exc(π).

To generalize this to the Kähler clase, we should introduce the notion of a generalized Kähler metric on a singular surface. In fact, we have no time to do this—please believe that such a notion exists, that it determines a set of Kähler classes on every K3 surface, and that the integral Kähler classes always correspond exactly to the ample line bundles.

Suppose now that X is an arbitrary K3 surface (with rational double points allowed), fix a Kähler class κ on X, and let π : X̃ → X be the minimal desingularization. We make several definitions in parallel to the smooth case.

For R = ℤ, ℝ, or ℂ, let H_{R}^{2}(X) denote the orthogonal complement of the components of the exceptional set exc(π) in H^{2}(X̃, R). H_{ℤ}^{2}(X) inherits a Hodge structure from H^{2}(X̃, ℤ), since the components of exc(π) give classes in H_{ℝ}^{1,1}. H_{ℝ}^{1,1}(X) denotes H_{ℝ}^{2}(X) ∩ H^{1,1}(X). We define

Δ(X) = Δ(X̃)
Δ^{+}(X) = Δ^{+}(X̃)
R(x) = the root system of X = {δ ∈ Δ(X̃) | ±δ is supported on exc(π)}
V^{+}(x) = {x ∈ H_{ℝ}^{1,1}(X) | X^{2} > 0, x · κ > 0 and x · δ > 0
for all δ ∈ Δ^{+}(X) − (Δ^{+}(X) ∩ R(X)}

(That is, x · δ ≥ 0 for δ ∈ Δ^{+}(X) with equality if and only if δ ∈ R(X).) Note that when X is projective, V^{+}(X) ∩ H_{ℤ}^{2}(X) is the set of ample classes.

Definition.

(1) A map φ : H_{ℤ}^{2}(X) → H_{ℤ}^{2}(X′) is an effective Hodge isometry if and only if φ preserves the intersection forms and Hodge structures, and φ(V^{+}(X)) = V^{+}(X′).

(2) A map φ : H_{ℤ}^{2}(X) → H_{ℤ}^{2}(X′) is liftable if and only if there exists an isomorphism ψ : H^{2}(X̃, ℤ) → H^{2}(X̃′, ℤ) preserving intersection forms such that ψ|_{H_{ℤ}^{2}(X)} = φ.

We can now state the 3 main theorems about the structure of the period mapping for K3 surfaces. For the proofs, we refer to the book of Barth-Peters-Van de Ven, the seminar notes “Géométrie des surfaces K3 … ” edited by Beauville et al., and the references contained there.

Global Torelli Theorem. If φ : H_{ℤ}^{2}(X) → H_{ℤ}^{2}(X′) is a liftable effective Hodge isometry between K3 surfaces X and X′, there exists an isomorphism Φ : X′ → X such that Φ^{∗} = φ.

Surjectivity Theorem. Given a submodule R ⊂ Λ generated by classes e_{i} with e_{i}^{2} = −2 on which the form is negative definite, a Hodge structure {H^{p,q}} on Λ of K3 type, and a connected component V^{+} of (R^{⊥} ∩ C(H^{1,1})) − ⋃_{δ∈Δ−R} δ^{⊥} (where C(H^{1,1}) = {x ∈ H^{1,1} | x^{2} > 0} and Δ = {δ ∈ H^{1,1} ∩ Λ | δ^{2} = −2}) there exists a K3 surface X and a marking α : H^{2}(X̃, ℤ) → Λ preserving Hodge structures and intersection forms, such that α(R(X)) = R and α(V^{+}(X)) = V^{+}.

Existence of Kähler Metrics Theorem. Every K3 surface is Kähler. Moreover, the set of Kähler classes is exactly V^{+}(X).

I want to describe how these theorems can be used to construct K3 surfaces whose Neron-Severi groups have specified properties. Typically, we are given a free ℤ-module M with an intersection matrix, and perhaps some elements μ_{1}, …, μ_{k} ∈ M and we would like to find a K3 surface X such that NS(X) = M, and the μ_{i} correspond to effective, or irreducible, or ample, or very ample divisors. If such is the case, we have M ⊂ Λ embedded in a saturated way; and the discriminant-groups give a condition which M must satisfy. Namely, let N = M^{⊥} so that M^{#}/M ≅ N^{#}/N; we must have

(minimum number of generators of M^{#}/M) ≤ rank(N).

The converse is a theorem of Nikulin.

Theorem (Nikulin). Let M be a free ℤ-module with a nondegenerate symmetric bilinear form of signature (1, r − 1) or (0, r). If

(minimum number of generators of M^{#}/M) ≤ 22 − r

then there is an embedding M ⊂ Λ with saturated image.

In any specific application, there are then several further steps to carry out. We illustrate all of this with an example: a free ℤ-module M generated by λ_{1}, λ_{2} with intersection matrix \binom{4 8}{8 4}.

Step 1. M can be generated with at most 2 generators, well less than 20. (In general, rank(M) ≤ 11 implies that M can be embedded, since r ≤ 22 − r.)

Step 2. We construct a Hodge structure on Λ such that Λ ∩ H^{1,1} = M, as follows. Choose ω ∈ Λ ⊗ ℂ such that ω · ω = 0, ω · ω̄ > 0 and the smallest ℚ-vector subspace of Λ ⊗ ℚ containing ω is M^{⊥}. (Essentially, this means choosing the coefficients in a basis of M^{⊥} to be algebraically independent transcendentals, except for the relation imposed by ω · ω = 0.) Then H^{1,1} ∩ (Λ ⊗ ℚ) = M^{⊥⊥} ⊗ ℚ = M ⊗ ℚ, so (since M ⊂ Λ is saturated) H^{1,1} ∩ Λ = M.

Step 3. Note that for a_{1}λ_{1} + a_{2}λ_{2} ∈ M we have (a_{1}λ_{1} + a_{2}λ_{2})^{2} ≡ 0 mod 4, so that there are no (-2)-classes in M (i. e. Δ = ∅). Let V^{+} be the chamber containing λ_{1}—this is simply the connected component of C(H^{1,1}) containing λ_{1}.

Step 4. By the surjectivity theorem, there is a smooth K3 surface X and an isomorphism α : H^{2}(X, ℤ) → Λ preserving Hodge structures and intersection forms, such that α(V^{+}(X)) = V^{+}. We have NS(X) = α^{−1}(M) ≅ M, and α^{−1}(λ_{1}) corresponds to an ample line bundle O_{X}(L_{1}). Since λ_{1} · λ_{2} > 0, α^{−1}(λ_{2}) also corresponds to an ample line bundle L_{2}.

(In more general examples, where M contains (−2)-classes, the analysis of Δ, V^{+}, the ampleness question, and whether or not X is smooth is much more complicated.)

Step 5. Are L_{1} and L_{2} very ample? Since L_{i}^{2} ≠ 2 or 8, to answer this we must know whether there can be a class d ∈ M such that d^{2} = 0, d · λ_{i} = 1 or 2. We showed in section 1 that this is not the case for this particular M.

It follows that |L_{1}| and |L_{2}| are very ample, so there is a smooth C ∈ |L_{2}|. If we embed X by φ_{|L_{1}|}, we get a smooth quartic surface X with a smooth curve C of degree 8 and genus 6. Thus, this set of lectures ends exactly where it began.

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