DOUBLE COVERS OF KUMMER SURFACES

AFSANEH MEHRAN

Abstract. Besides its construction as a quotient of an abelian surface, a Kummer surface can be obtained as the quotient of a K3 surface by a ℤ/2ℤ-action. In this paper, we classify all such K3 surfaces. Our classification is expressed in terms of period lattices and extends Morrison’s criterion of K3 surfaces with a Shioda-Inose structure. Moreover, we list all the K3 surfaces associated to a general Kummer surface and provide very geometrical examples of this phenomenon.

1. Introduction

The definition of a Kummer surface unravels the natural link between K3 surfaces and abelian surfaces. Indeed, let A denote an abelian surface and let A →^{i} A be its involution automorphism, i.e. i(a) = −a. The quotient, A/{1, i}, is a singular surface with sixteen ordinary double points. Its minimum resolution, Kum(A), is a K3 surface called the Kummer surface associated to A. Closely related to Kummer surfaces, are the K3 surfaces with a Shioda-Inose structure ([19], [8]). A K3 surface X is said to have a Shioda-Inose structure if it admits a rational map of degree two into a Kummer surface, X −−→^{τ} Kum(A), which induces a Hodge isometry of the period lattices,

T_{X}(2) →^{τ_{∗}} T_{Kum(A)}.

Morrison proved that X admits a Shioda-Inose structure if and only if there exists a Hodge isometry, T_{X} ≃ T_{A}, between the period lattices of X and an abelian surface A ([8]).

However, in general, a rational map of degree two between a K3 surface and a Kummer surface, does not need to induce a Hodge isometry of period lattices.

The aim of this paper is to treat this more general situation, i.e. it classifies all the K3 surfaces that admit a rational map of degree two into a given Kummer surface. As a result, we prove in theorem 3.1 that if X is a K3 surface such that its period lattice admits an embedding into the period lattice of an abelian surface A, T_{X} ↪ T_{A}, with the property that

T_{A}/T_{X} ≃ (ℤ/2)^{α} with 0 ≤ α ≤ 4,

then there exists a rational map of degree two,

X −−→^{τ} Kum(A).

Note that the case where α = 0 recovers Morrison’s criterion for K3 surfaces with a Shioda-Inose structure. Moreover, we give a complete list of all the K3 surfaces covering a general Kummer surface.

In the second part of this paper, we introduce the notion of an even eight on a K3 surface. We use this concept to construct explicit examples of the surfaces appearing in our list (theorem 6.4, theorem 7.3). Interestingly in these cases, the rational map X −−→^{τ} Kum(A) decomposes as

\xymatrix{X \ar@{-->}[d]_{τ} \ar[r]^{ζ} & Ω \ar[d]_{φ} \\ Kum(A) \ar[r]^{ρ} & ℙ^{2}}

where ρ is the double cover of ℙ^{2} branched along a reducible sextic Σ decomposable as a (possibly degenerate) quartic Γ and a conic Δ and where φ and ζ are double covers branched along Δ and φ^{∗}(Γ) respectively (theorem 6.5, theorem 7.4).

Acknowledgements 1. This paper is part of the author’s thesis work. The author would like to thank her adviser, Igor Dolgachev for all his guidance, suggestions and encouragement. The author is also very grateful to Bert van Geemen for his precious help in establishing the list of K3 surfaces covering a general Kummer surface.

2. Lattices and involutions on K3 surfaces

The theory of K3 surfaces is intimately connected to the study of integral quadratic forms or lattices. In this section, we recall the basic properties of lattices and discuss their fruitful applications to involutions on K3 surfaces. A (non-degenerate) lattice S is a free ℤ-module of finite rank together with a (non-degenerate) bilinear form, b : S × S → ℤ. The lattice S is even if the associated quadratic form takes only even values. An embedding of lattices M ↪ S is an homomorphism of ℤ-modules preserving the bilinear forms. We say that the embedding is primitive if the quotient S/M is torsion free. The signature of S, (t_{+}, t_{−}) is the signature of the bilinear form b, over the real vector space S ⊗ ℝ. The lattice S(m) for m ∈ ℤ denotes the same free ℤ-module as S, with the bilinear form b_{mS}(x, y) = mb_{S}(x, y). If S is an non-degenerate even lattice, then the ℚ-valued quadratic form on S^{∗} = Hom(S, ℤ) induces a quadratic form on the finite abelian group D_{S} = S^{∗}/S,

q_{S} : D_{S} → ℚ/2ℤ,

which is well defined mod 2ℤ and a bilinear form

b_{S} : D_{S} × D_{S} → ℚ/ℤ.

The pair (D_{S}, q_{S}) is called the discriminant form of S, and l(D_{S}) is the minimum number of generators of D_{S}. A non-degenerate lattice S is unimodular if D_{S} ≃ {0}. An embedding S ↪ S′ of even lattices with a finite cokernel is called an overlattice. Let S ↪ S′ be an overlattice of even non-degenerate lattices, then

|D_{S}| = [S′ : S]^{2}|D_{S′}|.

Consider the chain of embeddings S ↪ S′ ↪ S^{′∗} ↪ S^{∗} and let H_{S′} = S′/S be the subgroup of D_{S}. Then the correspondence, S′ ↔ H_{S′}, determines a bijection between even overlattices of S and isotropic subgroups of D_{S}.

A p-adic lattice and its discriminant form are defined in the same way as for an integral lattice, only replacing ℤ by ℤ_{p}. Let S be a lattice over ℤ, for every prime number p, q_{p} denotes the restriction of the discriminant form q_{S} to (D_{S})_{p}, the p-component of D_{S}. There exists a unique p-adic lattice K(q_{p}) whose discriminant form is isomorphic to ((D_{S})_{p}, q_{p}) and whose rank is equal to l((D_{S})_{p}). The p-adic number |K(q_{p})| is then the determinant of a symmetric matrix associated to the lattice K(q_{p}); it is therefore well defined mod (ℤ_{p}^{∗})^{2}.

Theorem 2.1. [13] There exists a primitive embedding of the even lattice S, with invariants (t_{+}, t_{−}, q_{S}) into the unimodular lattice L with signature (l_{+}, l_{−}) if and only if the following conditions are simultaneously satisfied:

(1) l_{+} − l_{−} ≡ 0 (mod 8),

(2) l_{−} − t_{−} ≥ 0, l_{+} − t_{+} ≥ 0, l_{+} + l_{−} − t_{+} − t_{−} ≥ l(D_{S}),

(3) (−1)^{l_{+}−t_{+}}|D_{S}| ≡ ±|K(q_{p})|(mod ℤ_{p}^{∗})^{2} for all odd primes p for which l_{+} + l_{−} − t_{+} − t_{−} = l((D_{S})_{p}),

(4) |D_{S}| ≡ ±|K(q_{2})|(mod ℤ_{2}^{∗})^{2} if l_{+} + l_{−} − t_{+} − t_{−} = l((D_{S})_{2}) and q_{2} ≇ \frac{x^{2}}{θ·2} ⊕ q′_{2} for some θ ∈ ℤ_{2}^{∗}, and some q′_{2}.

A K3 surface X is a smooth projective complex surface satisfying

K_{X} = 0 and H^{1}(X, O_{X}) = 0.

Let X be a K3 surface. The second cohomology group, H^{2}(X, ℤ), equipped with the cup product, is an even unimodular lattice of signature (3, 19) and is isomorphic to U^{3} ⊕E_{8}^{2}(−1), where U is the hyperbolic plane and E_{8}(−1) is the unique even negative definite lattice of rank eight. The period lattice of a surface X, denoted by T_{X}, is defined by

T_{X} = S_{X}^{⊥} ⊂ H^{2}(X, ℤ)

where S_{X} is the Néron-Severi group of X. Recall that the lattice H^{2}(X, ℤ) admits a Hodge decomposition of weight two

H^{2}(X, ℂ) ≃ H^{2,0} ⊕ H^{1,1} ⊕ H^{0,2}.

Similarly, the period lattice T_{X} inherits a Hodge decomposition of weight two denoted by

T_{X} ⊗ ℂ ≃ T^{2,0} ⊕ T^{1,1} ⊕ T^{0,2}.

An isomorphism between two lattices that preserves their bilinear forms and their Hodge decomposition is called a Hodge isometry. K3 surfaces have the unusual property that they satisfy a ”‘Torelli theorem”’, namely

Theorem 2.2. [16] Let X and X′ be two K3 surfaces and let ϕ : H^{2}(X′, ℤ) → H^{2}(X, ℤ) be a Hodge isometry. If ϕ preserves effective classes, then there exists a unique isomorphism u : X → X′ such that u^{∗} = ϕ.

Mukai improved this result for K3 surfaces of large picard number.

Theorem 2.3. [9] Let X and X′ be K3 surfaces and let ϕ : T′_{X} → T_{X} be a Hodge isometry. If ρ(X) ≥ 12, there exists an isomorphism u : X → X′ such that ϕ is induced by u^{∗}.

Definition 2.4. Let X be a K3 surface, an involution X →^{ι} X is called a Nikulin involution if it preserves any holomorphic two-form on X, i.e. ι^{∗}(ω) = ω for all ω ∈ H^{2,0}(X).

Theorem 2.5. [12] Let X →^{ι} X be a Nikulin involution on X. Then 1) ι has eight isolated fixed points and the surface X/{1, ι} has eight ordinary double points,

2) the minimal resolution Y of X/{1, ι} is a K3 surface.

Let Z →^{ϵ} X be the blow-up of the eight fixed points of ι. The involution ι lifts to an involution ι̂on Z whose fixed points consist of the eight exceptional curves. Thus the quotient surface, Z →^{p} Z/{1, ι̂} = Y, contains eight disjoint smooth rational curves, C_{1}, …, C_{8}, which make up the branch locus of the map p. Consequently they satisfy

C_{1} + ⋯ + C_{8} ∈ 2S_{Y} .

The commutative resolution diagram

\xymatrix{Z \ar[r]^{ϵ} \ar[d]_{p} & X \ar[d]_{/ι} \\ Y \ar[r] & X/\{1, ι\}}

induces a rational map of degree two between K3 surfaces, X −−→^{τ} Y.

Definition 2.6. The Nikulin lattice is an even lattice N of rank eight generated by {c_{i}}_{i=1}^{8} and d = \frac{1}{2} ∑_{i=1}^{8} c_{i}, with the bilinear form c_{i} · c_{j} = −2δ_{ij}.

Conversely, any rational map of degree two between K3 surfaces, X −−→^{τ} Y, is obtained from a similar construction and therefore it implies the existence of a Nikulin involution on X.

Let X →^{ι} X be a Nikulin involution. The two invariant lattices

T^{ι} = {x ∈ H^{2}(X, ℤ)|ι^{∗}(x) = x} and S_{ι} = {x ∈ H^{2}(X, ℤ)|ι^{∗}(x) = −x}

define the primitive embeddings

T_{X} ↪ T^{ι} ↪ H^{2}(X, ℤ), S_{ι} ↪ S_{X} ↪ H^{2}(X, ℤ)

and the isomorphisms

T^{ι} ≃ U^{3} ⊕ E_{8}(−2) and S_{ι} ≃ E_{8}(−2).

Let X −−→^{τ} Y be the rational map associated to ι. The maps induced on the period lattices

T_{Y} →^{τ^{∗}} T_{X} and T_{X} →^{τ_{∗}} T_{Y}

preserve the Hodge decompositions and satisfy

τ_{∗}τ^{∗}(x) = 2x, τ^{∗}τ_{∗}(x) = 2x, τ^{∗}(x) · τ^{∗}(y) = 2xy, τ_{∗}(x) · τ_{∗}(y) = 2xy.

The image of T_{Y} by τ^{∗} is then a sublattice of T_{X} isomorphic to T_{Y} (2) with two-elementary quotient, i.e.

T_{X}/T_{Y} (2) ≃ (ℤ/2ℤ)^{β} for some β ≥ 0.

Lemma 2.7. [15] Let X be a K3 surface and let T_{X} ↪ T ≃ U^{3} ⊕ E_{8}(−2) be a primitive embedding of lattices. Then there exists a Nikulin involution ι of X such that for the corresponding rational map of degree two X −−→^{τ} Y

τ^{∗}T_{Y} ≃ T_{Y} (2) ≃ 2(T^{∗} ∩ (T_{X} ⊗ ℚ)) ⊂ T_{X}.

Finally, the following argument is due to Nikulin ([15]) and will be crucial for the proof of the theorem classifying the K3 surfaces covering a given Kummer surface. Let X be a K3 surface with a Nikulin involution ι, denote by T the period lattice T_{X} and by S, the invariant lattice S_{ι}. Let M be the primitive sublattice of H^{2}(X, ℤ) generated by T ⊕ S. Then the subgroup Γ = M/(T ⊕ S) ⊂ D_{T} ⊕ D_{S} is isotropic and satisfies

Γ ∩ (D_{T} ⊕ {0}) = 0 and Γ ∩ ({0} ⊕ D_{S}) = 0.

Let π_{T} and π_{S} be the projections in D_{T} and D_{S} respectively. Let ℌ = π_{T} (Γ) ⊂ D_{T}. If α = l(ℌ) ≤ 4, then the discriminant group of M is of the form D_{M} ≃ D_{T} ⊕ u(2)^{4−α}, where u(2) denotes the discriminant form of the lattice U(2). Note also that ℌ is not necessarily isotropic. The map ξ = π_{S} ∘ π_{T}^{−1} defines an inclusion of quadratic forms (i.e. an injection of groups preserving the bilinear form)

ξ : ℌ → (D_{S}, −q_{S}).

Moreover, Nikulin proved the equality

((S^{⊥})^{∗} ∩ (T ⊗ ℚ))/T = ℌ ⊂ D_{T} .

3. Classification Theorem

In this section, we give necessary and sufficient conditions for a K3 surface to admit a rational map of degree two into a given Kummer surface. As a corollary, we show that there exist finitely many such K3 surfaces and give an exhaustive list of all the K3 surfaces covering a general Kummer surface.

Theorem 3.1. Let Kum(A) be a Kummer surface. A K3 surface X admits a rational map of degree two, X −−→^{τ} Kum(A), if and only if there exists an embedding of period lattices ϕ : T_{X} ↪ T_{A} such that

(1) ϕ_{⊗ℂ} preserves the Hodge decomposition,

(2) T_{A}/T_{X} ≃ (ℤ/2ℤ)^{α} with 0 ≤ α ≤ 4,

(3) |D_{T_{X}}| ≡ ±|K(q_{T_{X}})_{2}| mod (ℤ_{2}^{∗})^{2}, if α = \frac{rank(T_{X})+l((D_{T_{X}})_{2})}{2} −3 and (q_{T_{X}})_{2} ⊕u^{4−α}(2) ≇ \frac{x^{2}}{θ·2} ⊕q′_{2} for some θ ∈ ℤ_{2}^{∗} and some q′_{2}.

Proof. Let Y denote the Kummer surface Kum(A) and suppose that X admits a rational map of degree two into Y, X −−→^{τ} Y. The covering involution ι : X → X is then a Nikulin involution. Consider the induced Hodge isometry of lemma 2.7,

T_{Y} (2) →^{≃} 2(T_{X} ⊗ ℚ ∩ (T^{ι})^{∗}) ⊂ T_{X}.

Denote by ℌ̃ the rational overlattice of T_{X} defined by T_{X} ⊗ ℚ ∩ (T^{ι})^{∗}. Since 2ℌ̃ ⊂ T_{X}, the quotient ℌ̃/ T_{X} is two-elementary. Since the period lattice of Kum(A) is Hodge isometric to T_{A}(2) ([11]), T_{A}(4) is Hodge isometric to T_{Y} (2). Thus there is a Hodge isometry, T_{A}(4) →^{≃} 2ℌ̃ ⊂ T_{X}. The obvious identification of T_{A}(4) with 2T_{A} shows that T_{A} ≃ ℌ̃ and that the inclusion preserving the Hodge decomposition T_{X} ↪ ℌ̃ ≃ T_{A} satisfies

T_{A}/T_{X} ≃ ℌ̃/T_{X} ≃ (ℤ/2ℤ)^{α} for some α ≥ 0.

Moreover, the lattice ℌ̃ must be an integral overlattice of T_{X}, and conseqently the quotient ℌ̃/T_{X}, is an isotropic subgroup of (D_{T_{X}}, q_{T_{X}}). Note that the group ℌ̃/T_{X} corresponds to the group ℌ introduced in the last part of the previous section. Therefore ℌ admits an inclusion of quadratic forms into (D_{S_{ι}}, −q_{|S_{ι}}), where l(D_{S_{ι}}) = 8. Because ℌ is isotropic, we obtain the upper bound α ≤ 4. Finally, let M be the primitive closure of T_{X} ⊕ S_{ι} in H^{2}(X, ℤ), then D_{M} ≃ D_{T_{X}} ⊕ u(2)^{4−α}. The lattice M admits a primitive embedding into the unimodular lattice in H^{2}(X, ℤ), so it satisfies the hypothesis of the theorem 2.1. In particular, it satisfies

|D_{T_{X}}| ≡ ±|K(q_{T_{X}})_{2}| mod (ℤ_{2}^{∗})^{2}

if α = \frac{rank(T_{X})+l(D_{T_{X}})_{2}}{2} − 3 and (q_{T_{X}})_{2} ⊕ u^{4−α}(2) ≇ \frac{x^{2}}{θ·2} ⊕ q′_{2} for some θ ∈ ℤ_{2}^{∗}.

Conversely, assume that X satisfies conditions (1), (2) and (3). Claim: There exists a primitive embedding T_{X} ↪ U^{3} ⊕ E_{8}(−2) such that

(T_{X} ⊗ ℚ) ∩ (U^{3} ⊕ E_{8}(−2)^{∗}) ≃ T_{A}.

Assuming the claim, by lemma 2.7, there exists a Nikulin involution on X such that the corresponding rational map of degree two X −−→^{τ} Ỹ satisfies

TỸ (2) ≃ 2(U^{3} ⊕ E_{8}(−2)^{∗} ∩ T_{X} ⊗ ℚ) ≃ 2T_{A} ⊂ T_{X}.

The Hodge isometry TỸ (2) ≃ 2T_{A} implies that TỸ ≃ T_{A}(2) and hence that T_{Y} ≃ TỸ. According to theorem 2.3, Ỹ ≃ Y.

Proof of the claim: Let ℌ ≃ (ℤ/2ℤ)^{α} be the isotropic subgroup of D_{T_{X}} corresponding to the overlattice T_{X} ⊂ T_{A}. Since α ≤ 4, there exists an embedding of quadratic forms

ℌ ↪^{ξ} (D_{S}, −q_{S}),

where S ≃ E_{8}(−2). Consider the subgroup

Γ_{ξ} = {h + ξ(h)|h ∈ ℌ} ⊂ D_{T_{X}} ⊕ D_{S}.

Clearly Γ_{ξ} is an isotropic subgroup of D_{T_{X}} ⊕ D_{S}. Denote by M the overlattice it defines and observe that

rank(T_{M}) = rank(T_{X}) + 8 and D_{M} ≃ q_{T_{X}} ⊕ u(2)^{4−α}.

In order for M to admit a primitive embedding into the unimodular lattice H^{2}(X, ℤ), it must satisfy the hypothesis of the theorem 2.1. In our setting it involves checking that

l((D_{M})_{p}) + rank(T_{X}) ≤ 14 for all p prime,

since for any finite abelian group, l(D) = max_{p}l(D_{p}). If p is odd, then (D_{M})_{p} ≃ (D_{T_{X}})_{p} and the strict inequality l((D_{T_{X}})_{p}) + rank(T_{X}) < 14 always holds, as l((D_{T_{X}})_{p}) ≤ rank(T_{X}) and 2 ≤ rank(T_{X}) ≤ 5. If p is 2, then l((D_{M})_{2}) = l((D_{T_{X}})_{2}) + 8 − 2α. The facts that l((D_{T_{X}})_{2}) − 2α ≤ l((D_{A})_{2}) (see the technical lemma 3.2 below), and that the lattice T_{A} embeds primitively into U^{3} imply the inequalities

l((D_{T_{X}})_{2})+8−2α+rank(T_{X}) ≤ l((D_{T_{A}})_{2})+8+rank(T_{A}) ≤ l(D_{T_{A}})+8+rank(T_{A}) ≤ 6+8 = 14.

Thus the inequality l((D_{M})_{p}) + rank(T_{X}) ≤ 14 holds for all p. The assumption

|D_{T_{X}}| ≡ ±|K(q_{T_{X}})_{2}| mod (ℤ_{2}^{∗})^{2}

if α = \frac{rank(T_{X})+l(D_{T_{X}})_{2}}{2} − 3 and (q_{T_{X}})_{2} ⊕ u^{4−α}(2) ≇ \frac{x^{2}}{θ·2} ⊕ q′_{2} for some θ ∈ ℤ_{2}^{∗}, is therefore enough to guarantee the existence of the primitive embedding of M into H^{2}(X, ℤ). Consider the embedding

T_{X} ⊕ S ⊂ M ⊂ H^{2}(X, ℤ).

By construction, its restriction to T_{X} and to S are primitive. It follows that S is primitively embedded into S_{X} or equivalently T_{X} is primitively embedded into S^{⊥} ≃ U^{3} ⊕ E_{8}(−2). Finally, from the last equality of section 2, we find that

ℌ ≃ (T_{X} ⊗ ℚ) ∩ (U^{3} ⊕ E_{8}(−2)^{∗})/T_{X}

and therefore

(T_{X} ⊗ ℚ) ∩ (U^{3} ⊕ E_{8}(−2)^{∗}) ≃ T_{A}.
□

Lemma 3.2. Let D be a finite abelian 2-group with a bilinear form b(x, y) ∈ ℚ/ℤ and a quadratic form q(x) = b(x, x) ∈ ℚ/2ℤ. Let ℌ ⊂ D be an isotropic subroup, then

l(D) − 2l(ℌ) ≤ l(ℌ^{⊥}/ℌ).

Proof. It is enough to show that l(D/ℌ) ≤ l(ℌ^{⊥}). Indeed it would imply that

l(D) ≤ l(D/ℌ) + l(ℌ) ≤ l(ℌ^{⊥}) + l(ℌ) ≤ l(ℌ^{⊥}/ℌ) + l(ℌ) + l(ℌ)

where the first and last inequalities are obvious.

We assume first that ℌ ≃ ℤ/2 with generator h. There exist y_{1}, ⋯, y_{l(D)} generators of D, where y_{1} is of the form \frac{1}{k}h for some k ∈ ℤ_{>0}. Let 0 ≤ l ≤ l(D) be such that

y_{1}, …, y_{l} ∈ ℌ^{⊥} and y_{l+1}, …, y_{l(D)} ∉ ℌ^{⊥}.

Note that any element of order two y ∈ D satisfies b(y, x) = 0 or \frac{1}{2} for all x ∈ D. Indeed, if b(y, x) ≠ 0 ∈ ℚ/ℤ, then 2b(x, y) = 0 ∈ ℚ/ℤ, hence b(x, y) = \frac{1}{2} ∈ ℚ/ℤ. Thus

y_{1}, …, y_{l}, y_{l+1} + y_{l+2}, …, y_{l+1} + y_{l(D)} ∈ ℌ^{⊥}

and l(ℌ^{⊥}) ≥ l(D) − 1. If l(ℌ^{⊥}) = l(D), clearly l(ℌ^{⊥}) ≥ l(D/ℌ). If l(ℌ^{⊥}) = l(D) − 1, then there exists a generator y_{j} of D with b(h, y_{j}) = \frac{1}{2}. Moreover y_{j} must be of order two, since if 2y_{j} ≠ 0, then 2y_{j} ∈ ℌ^{⊥} and l(D) = l(ℌ^{⊥}). Hence

\frac{1}{2} = b(y_{1}, y_{j}) = b(\frac{1}{k}h, y_{j}) = \frac{1}{2k} ,

or equivalently k = 1 and y_{1} = h. It follows that l(D/ℌ) ≤ l(D) − 1 = l(ℌ^{⊥}).

If |ℌ| > 2, then there exists a subgroup ℌ′ ⊂ ℌ, such that ℌ/ℌ′ ≃ ℤ/2. Clearly ℌ/ℌ′ is an isotropic subgroup of D_{S}/ℌ′. Since a minimal set of generators of (ℌ/ℌ′)^{⊥} define a set of independent elements of ℌ^{⊥}, we get the inequality l((ℌ/ℌ′)^{⊥}) ≤ l(ℌ^{⊥}). By induction, we obtain

l(D/ℌ) = l((D/ℌ′)/ℌ/ℌ′) ≤ l((ℌ/ℌ′)^{⊥}) ≤ l(ℌ^{⊥}).
□

Let A be an abelian surface. If X is a K3 surface such that its period lattice T_{X} admits an embedding into T_{A} with T_{A}/T_{X} ≃ (ℤ/2ℤ)^{α}, then there is an embedding of 2T_{A} into T_{X}. The finite group T_{X}/2T_{A} is a subspace of the 𝔽_{2}-symplectic space (T_{A}/2T_{A}, q_{T_{A}/2T_{A}}). Conversely, any subspace U of T_{A}/2T_{A} defines a sublattice of T_{A}, T_{U}, containing 2T_{A} and hence satisfying T_{A}/T_{U} ≃ (ℤ/2ℤ)^{α}. Moreover, the lattice T_{U} inherits a Hodge decomposition from the one on T_{A} and if T_{U} satisfies the condition (3) of the theorem 3.1, it determines a unique K3 surface. Note that any automorphism of T_{A} preserving its Hodge decomposition must be ± identity, [16], and therefore if U and U′ are two subspaces of T_{A}/2T_{A} such that there exists an orthogonal transformation ϕ ∈ O(T_{A}/2T_{A}) mapping U to U′, then (if it exists) the lifting of ϕ to T_{A} cannot define a Hodge isometry between T_{U} and T_{U′}. Clearly, we have the following corollary.

Corollary 3.3. Let Kum(A) be a Kummer surface. There exist finitely many isomorphism classes of K3 surfaces admitting a rational map of degree two into Kum(A).

We say that a Kummer surface Y is general if it is associated to a general abelian surface A, that is to say A is the jacobian of a curve of genus two and its period lattice T_{A} is isomorphic to U ⊕ U ⊕ ⟨−2⟩. In order to determine all the sublattices of T_{A} satisfying the conditions of 3.1, we classify all the subspaces of (T_{A}/2T_{A} ≃ 𝔽_{2}^{5}, q(x) = x_{1}x_{2} +x_{3}x_{4} +x_{5}^{2}). We obtain in this way a short and exhaustive list of all K3 surfaces satisfying the conditions of theorem 3.1. In total, we find 373 K3 surfaces that are listed below. The number in the first column indicates the number of non-isomorphic K3 surfaces within a given type of period lattice. In other words, it represents the size of the orbit of a given subspace of T_{A}/2T_{A} for the action of the orthogonal group O(𝔽_{2}^{5}, q).

\begin{matrix}♯ K3’s & T_{X} & D_{T_{X}} & q_{T_{X}} \\ 1 & U ⊕ U ⊕ ⟨−2⟩ & ℤ/2ℤ & \frac{1}{2}x^{2} \\ 15 & U(2) ⊕ U ⊕ ⟨−2⟩ & ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ & x_{1}x_{2} − \frac{1}{2}x_{3}^{2} \\ 10 & U ⊕ U ⊕ ⟨−8⟩ & ℤ/8ℤ & −\frac{1}{8}x^{2} \\ 6 & U ⊕ N & ℤ/8ℤ & \frac{3}{8}x^{2} \\ 20 & U(4) ⊕ U ⊕ ⟨−2⟩ & ℤ/4ℤ ⊕ ℤ/4ℤ ⊕ ℤ/2ℤ & \frac{1}{2}x_{1}x_{2} − \frac{1}{2}x_{3}^{2} \\ 15 & U(2) ⊕ U(2) ⊕ ⟨−2⟩ & ℤ/2ℤ^{⊕5} & x_{1}x_{2} + x_{3}x_{4} − \frac{1}{2}x_{5}^{2} \\ 45 & U(2) ⊕ U ⊕ ⟨−8⟩ & ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/8ℤ & x_{1}x_{2} − \frac{1}{8}x_{3}^{2} \\ 60 & U ⊕ ⟨−2⟩ ⊕ ⟨2⟩ ⊕ ⟨−8⟩ & ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/8ℤ & −\frac{1}{2}x_{1}^{2} + \frac{1}{2}x_{2}^{2} − \frac{1}{8}x_{3}^{2} \\ 15 & U(2) ⊕ N & ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/8ℤ & x_{1}x_{2} + \frac{3}{8}x_{3}^{2} \\ 15 & U(4) ⊕ U(2) ⊕ ⟨−2⟩ & ℤ/4ℤ^{⊕2} ⊕ ℤ/2ℤ^{⊕3} & \frac{1}{2}x_{1}x_{2} + x_{3}x_{4} − \frac{1}{2}x_{5}^{2} \\ 60 & U(4) ⊕ U ⊕ ⟨−8⟩ & ℤ/4ℤ ⊕ ℤ/4ℤ ⊕ ℤ/8ℤ & \frac{1}{2}x_{1}x_{2} − \frac{1}{8}x_{3}^{2} \\ 20 & U(4) ⊕ N & ℤ/4ℤ ⊕ ℤ/4ℤ ⊕ ℤ/8ℤ & \frac{1}{2}x_{1}x_{2} − \frac{1}{2}x_{1}^{2} + x_{1}x_{3} + \frac{3}{8}x_{3}^{2} \\ 15 & U(2) ⊕ U(2) ⊕ ⟨−8⟩ & ℤ/2ℤ^{⊕4} ⊕ ℤ/8ℤ & x_{1}x_{2} + x_{3}x_{4} − \frac{1}{8}x_{5}^{2} \\ 45 & ⟨2⟩^{2} ⊕ ⟨−2⟩^{2} ⊕ ⟨−8⟩ & ℤ/2ℤ^{⊕4} ⊕ ℤ/8ℤ & \frac{1}{2}(x_{1}^{2} + x_{2}^{2} − x_{3}^{2} − x_{4}^{2}) − \frac{1}{8}x_{5}^{2} \\ 1 & U(4) ⊕ U(4) ⊕ ⟨−2⟩ & ℤ/4ℤ^{⊕4} ⊕ ℤ/2ℤ & \frac{1}{2}x_{1}x_{2} + \frac{1}{2}x_{3}x_{4} − \frac{1}{2}x_{5}^{2} \\ 15 & U(4) ⊕ U(2) ⊕ ⟨−8⟩ & ℤ/2ℤ^{⊕2} ⊕ ℤ/4ℤ^{⊕2} ⊕ ℤ/8ℤ & x_{1}x_{2} + \frac{1}{2}x_{3}x_{4} − \frac{1}{8}x_{5}^{2} \\ 15 & U(4) ⊕ ⟨2⟩ ⊕ ⟨−2⟩ ⊕ ⟨−8⟩ & ℤ/2ℤ^{⊕2} ⊕ ℤ/4ℤ^{⊕2} ⊕ ℤ/8ℤ & \frac{1}{2}x_{1}^{2} − \frac{1}{2}x_{2}^{2} + \frac{1}{2}x_{3}x_{4} − \frac{1}{8}x_{5}^{2} \end{matrix}

The bilinear form of the lattices N is given by the symmetric integral matrix

N = ( \begin{matrix}2 & 1 & 2 \\ 1 & −2 & 0 \\ 2 & 0 & 0 \end{matrix} ) .

4. Even Eight

The remaining of this paper will be devoted to the geometrical description of the K3 surfaces appearing in the above list that have two-elementary period lattices. The study of such K3 surfaces is made easier by the existence on them of another involution.

Proposition 4.1. [13] Let X be a K3 surface. The period lattice T_{X} is two-elementary if and only if X admits an involution θ : X → X such that θ_{S_{X}}^{∗} = id_{S_{X}} and θ_{T_{X}}^{∗} = −id_{T_{X}}.

As a first step towards our geometrical description, we introduce the notion of an even eight. Secondly, in order to provide examples of different even eights on a Kummer surface, we give explicit generators for the Néron-Severi lattice of a general Kummer surface. We follow Naruki’s notation in [10].

Definition 4.2. Let Y be a K3 surface, an even eight on Y is a set of eight disjoint smooth rational curves, C_{1}, ⋯, C_{8}, such that C_{1} + ⋯ + C_{8} ∈ 2S_{Y} .

Recall, from section 2, that a rational map of degree two between K3 surfaces, X −−→^{τ} Y factors as

\xymatrix{Z \ar[rr]^{ϵ} \ar[dr]_{p} &  & X \ar@{-->}[dl]^{τ} \\  & Y}

where ϵ is the blow-up of X at eight points and p is a regular map of degree two, branched along eight disjoint smooth rational curves, hence an even eight. Conversely, given an even eight, C_{1}, ⋯, C_{8}, on a K3 surface Y, there is exists a double cover Z →^{p} Y branched along ∑_{i=1}^{8} C_{i}. Let p^{∗}(C_{i}) = 2E_{i}, then each E_{i} is an exceptional divisor of the first kind. We may therefore blow down ∑_{i=1}^{8} E_{i} to a smooth surface X, which is known to be a K3 surface. We recover, in this way, a rational map of degree two X −−→^{τ} Y between the K3 surfaces X and Y (see [11]).

Let Y ≃ Kum(A) be a general Kummer surface and let C be the genus two curve of which

A is the jacobian. The degree two map given by the linear system |2C|, A →^{|2C|} ℙ^{3}, factors through the involution i, where i(a) = −a, and hence defines an embedding A/{1, i} ↪ ℙ^{3}. The image of this map, Y_{0}, is a quartic in ℙ^{3} with sixteen nodes. Denote by L_{0} the class of a hyperplane section of Y_{0}. Projecting from a node, Y_{0} admits a rational map of degree two into the projective plane which induces the map Y →^{ϕ} ℙ^{2}, given by the linear system |L − E_{0}|, where L is a the pullback of L_{0} on Y and E_{0} is the exceptional curve resolving the center of projection. The branch locus of the map ϕ is a reducible plane sextic Ω, which is the union of six lines, l_{1}, ⋯, l_{6}, all tangent to a conic W.

[figure: l_{1}; l_{2}; l_{3}; l_{4}; l_{5}; l_{6}; W]

Figure 1. The sextic Ω

Let p_{ij} = l_{i} ∩ l_{j} ∈ ℙ^{2}, where 1 ≤ i < j ≤ 6. Index the ten (3, 3)-partitions of the set {1, 2, …, 6}, by the pair (i, j) with 2 ≤ i < j ≤ 6. Each pair (i, j) defines a plane conic l_{ij} passing through the sixtuplet p_{1i}, p_{1j}, p_{ij}, p_{lm}, p_{ln}, p_{mn}, where {l, m, n} is the complement

of {1, i, j} in {1, 2, …, 6} and where l ≤ m ≤ n. The map ϕ factors as Y ⟶^{ϕ̃} ℙ̃^{2} ⟶^{η} ℙ^{2} where η is the blowup of ℙ^{2} at the p_{ij}’s and where ϕ̃ is the double cover of ℙ̃^{2} branched along the strict transform in ℙ̃^{2} of the plane sextic Ω. Denote by E_{ij} ⊂ Y the preimage of the exceptional curves of ℙ̃^{2}. The ramification of the map ϕ̃ consists of the union of six disjoint smooth rational curves, C_{0} + C_{12} + C_{13} + C_{14} + C_{15} + C_{16}. The preimage of the ten plane conics l_{ij} defines ten more smooth disjoint rational curves on Y, C_{ij}, 2 ≤ i < j ≤ 6. Finally, note that ϕ(E_{0}) = W. The sixteen curves E_{0}, E_{ij} 2 ≤ i < j ≤ 6 are called the nodes of Y and the sixteen curves C_{0}, C_{ij} 2 ≤ i < j ≤ 6 are called the tropes of Y. These two sets of smooth rational curves satisfy a beautiful configuration called the 16_{6}-configuration, i.e. each node intersects exactly six tropes and vice versa. It is now possible to fully describe the Néron-Severi lattice S_{Y} of a general Kummer surface. Indeed, it is generated by the classes of E_{0}, E_{ij}, C_{0}, C_{ij} and L, with the relations:

(1) C_{0} = \frac{1}{2}(L − E_{0} − ∑_{i=2}^{6} E_{1i}),

(2) C_{1j} = \frac{1}{2}(L − E_{0} − ∑_{i≠j} E_{ij}), where 2 ≤ j ≤ 6,

(3) C_{jk} = \frac{1}{2}(L − E_{1j} − E_{1k} − E_{jk} − E_{lm} − E_{ln} − E_{mn}) where 2 ≤ i < j ≤ 6, and \{l, m, n\}

are as described above. The intersection pairing is given by:

(1) the E_{0}, E_{ij} are mutually orthogonal,

(2) ⟨L, L⟩ = 4, ⟨L, E_{0}⟩ = ⟨L, E_{ij}⟩ = 0,

(3) ⟨E_{0}, E_{0}⟩ = ⟨E_{ij}, E_{ij}⟩ = −2,

(4) the C_{0}, C_{ij} are mutually orthogonal,

(5) ⟨L, C_{0}⟩ = ⟨L, C_{ij}⟩ = 2. The action on S_{Y} of the covering involution α of the map ϕ is given by:

\begin{matrix}α(C_{0}) = C_{0} & & α(C_{1j}) = C_{1j} & 2 ≤ j ≤ 6 \\ α(E_{ij}) = E_{ij} & 1 ≤ i < j ≤ 6 & α(L) = 3L − 4E_{0} \\ α(E_{0}) = 2L − 3E_{0} & & α(C_{ij}) = C_{ij} + L − 2E_{0} & 2 ≤ i < j ≤ 6. \end{matrix}

Remark 4.3. Conversely, the minimal resolution of the double cover of ℙ^{2} branched along the sextic Ω in figure 1 is a Kummer surface (see [4] for a proof).

Consider the divisors e_{1} := (L − E_{0}) − (E_{12} + E_{46}), e_{2} := 2(L − E_{0}) − (E_{12} + E_{13} + E_{24} + E_{46} + E_{56}), e_{3} := 3(L − E_{0}) − 2E_{12} − (E_{13} + E_{24} + E_{36} + E_{45} + E_{46} + E_{56}), e_{4} := 4(L − E_{0}) − 2(E_{12} + E_{13} + E_{46}) − (E_{24} + E_{25} + E_{36} + E_{45} + E_{56}), e_{5} := 5(L − E_{0}) − 3E_{12} − 2(E_{13} + E_{46} + E_{56}) + (E_{24} + E_{25} + E_{34} + E_{36} + E_{45}),

e_{6} := C_{23}, e_{7} := α(C_{23}), e_{8} := E_{35}.

Theorem 4.4. [10] The eight (−2)-classes, e_{1}, …, e_{8} represent an even eight on Y. The geometric meaning of the (−2)-classes can be explained in terms of the double plane model of the general Kummer surface, Y →^{ϕ} ℙ^{2}. The class e_{1} is represented by the proper inverse image of a line passing through p_{12} and p_{46}. The class of e_{2} is represented by the proper inverse image of a conic passing through p_{12}, p_{13}, p_{24}, p_{46}, p_{56}. The class of e_{3} is represented by the proper inverse image of a cubic passing through p_{13}, p_{24}, p_{36}, p_{45}, p_{46}, p_{56} and having a double point at p_{12}. The class of e_{4} is represented by the proper inverse image of a quartic passing through p_{24}, p_{25}, p_{36}, p_{45}, p_{56} and having double points at p_{12}, p_{13}, p_{46}. The class of e_{5} is represented by the proper inverse image of a quintic passing through p_{24}, p_{25}, p_{34}, p_{36}, p_{45} and having double points at p_{13}, p_{46}, p_{56} and a triple point at p_{12}.

5. K3 surface with a Shioda-Inose structure

A particular case in the list of page 8 is given by the unique K3 surface X_{0} that has its period lattice isomorphic to U ⊕ U ⊕ ⟨−2⟩. In fact, Naruki showed that the map X_{0} −−→^{τ} Y is the rational map of degree two associated to the even eight of the theorem 4.4. Moreover, according to Morrison’s criterion, X_{0} has a Shioda-Inose structure ([8]).

Prior to describing the geometry of X_{0}, we will state some basics about elliptic fibrations on K3 surfaces (see [7] for details).

Definitions 5.1. (1) A smooth surface X is elliptic if it admits a regular map into a

curve C, X →^{f} C where the general fiber is a smooth connected curve of genus one.

(2) When nonempty, the set of sections of f, MW_{f}(X), form a finitely generated group called the Mordell-Weil group, which can be identified with the set of k(C)-rational points of the generic fiber X_{η}.

(3) Let F denote the class of a fiber. The set {B · F |B ∈ S_{X}} is an ideal of ℤ. The multisection index l of f is the positive generator of this ideal. An l−section is a divisor B on X for which B · F = l.

We refer the reader to [7] for a table of Kodaira’s classification of singular fibers on an elliptic fibration.

Theorem 5.2. [18] Let X →^{f} ℙ^{1} be an elliptic K3 surface with a section S_{0}. Let A be the subgroup of S_{X} generated by the section S_{0}, the class of a fiber and the components of the reducible fibers that do not meet S_{0}. There is an short exact sequence of groups

0 → A →^{α} S_{X} →^{β} MW_{f}(X) → 0

where β is the restriction to the generic fiber. Let Θ_{i} (i = 1, …, k) be all the singular fibers of f. Then

(1) 24 = χ_{top}(X) = ∑_{i} χ_{top}(Θ_{i}),

(2) rank(MW_{f}(X)) = rank(S_{X}) − 2 − ∑_{i}(m(Θ_{i}) − 1), where m(Θ_{i}) denotes the number of irreducible components of Θ_{i}.

(3) When the rank of MW_{f}(X) is zero, then

|D_{S_{X}}| = \frac{∏_{i} s_{i}}{n^{2}}

where s_{i} is the number of simple components of the singular fiber Θ_{i} and n is the order of the finite group MW_{f}(X).

Pjateckiĭ-Šapiro and Šafarevič showed that on a K3 surface, if D is a non-zero effective nef divisor with D^{2} = 0, then the system |D| is of the form |mD_{0}|, m ≥ 1 and the linear system

|D_{0}| defines a morphism X →^{|D_{0}|} ℙ^{1}, with general fiber a smooth connected curve of genus one ([16]).

Proposition 5.3. Let Y be a general Kummer surface. Then there exists an elliptic fibration on Y whose singular fibers are of the type 6I_{2} + I_{5}^{∗} + I_{1} and whose Mordell-Weil group is finite of order two.

Proof. Consider the divisor class

D = 5(L − E_{0}) − 3E_{12} − 2(E_{13} + E_{46} + E_{56}) − (E_{24} + E_{25} + E_{36} + E_{45}).

Geometrically, D can be represented by the proper inverse image, under the map Y →^{ϕ} ℙ^{2}, of a quintic passing trough p_{24}, p_{25}, p_{36}, p_{45}, having double points at p_{13}, p_{46}, p_{56} and a triple point at p_{12}. Since D ∼ e_{5} + E_{34} and e_{5} · E_{34} = 2, D must be a reduced nef effective divisor with D^{2} = 0. Let Y →^{f} ℙ^{1} be the elliptic fibration defined by the linear system |D|. The divisors

(1) F_{1} = e_{5} + E_{34},

(2) F_{2} = e_{4} + \underbrace{(L − E_{0}) − (E_{12} + E_{56})}_{a_{1}},

(3) F_{3} = e_{3} + \underbrace{2(L − E_{0}) − (E_{12} + E_{13} + E_{46} + E_{56} + E_{25})}_{a_{2}},

(4) F_{4} = e_{2} + \underbrace{3(L − E_{0}) − 2E_{12} − (E_{13} + E_{46} + E_{56} + E_{25} + E_{36} + E_{45})}_{a_{3}},

(5) F_{5} = e_{1} + \underbrace{4(L − E_{0}) − 2(E_{12} + E_{13} + E_{56}) − (E_{46} + E_{24} + E_{25} + E_{36} + E_{45})}_{a_{4}},

(6) F_{6} = e_{8} + \underbrace{5(L − E_{0}) − 3E_{12} − 2(E_{13} + E_{46} + E_{56}) − (E_{24} + E_{25} + E_{36} + E_{45} + E_{35})}_{a_{5}}

represent six fibers of type I_{2}, as under the genericity assumption, the (−2)-cycles, a_{1}, … a_{5} are represented by smooth rational curves. Their geometric meaning can be explained exactly in the same way as in the theorem [10]. The divisor

F_{7} = e_{6} + e_{7} + 2(E_{23} + C_{12} + E_{26} + C_{16} + E_{16} + C_{0}) + E_{14} + E_{15}

defines a fiber of type I_{5}^{∗}. Note also that C_{14} ·D = C_{15} ·D = 1, i.e. the curves C_{14} and C_{15} are sections of f. The fifteen components of the singular fibers, e_{1}, …, e_{8}, C_{12}, E_{23}, E_{26}, C_{16}, E_{16}, C_{0}, E_{14} are independent over ℚ. Consequently, by the theorem 5.2, there are no other reducible fibers and the Mordell-Weil group is finite of order two. □

We note that the curves e_{1}, ⋯, e_{8} defining the even eight of theorem [10] are components of the singular fibers of this elliptic fibration. In fact, we see that the surface X_{0} admits also an elliptic fibration, X_{0} →^{|τ^{∗}D|} ℙ^{1}, with singular fibers of type I_{2} + I_{10}^{∗} + 6I_{1} and with a Mordell-Weil group of order two. The reducible fibers and the sections of this latter fibration form the diagram of curves on X_{0}, shown in figure 2. Nikulin proved that those curves are the only smooth rational curves lying on X_{0} and that the group Aut(X_{0}) is generated by two commuting involutions θ and ι. The involution θ satisfies θ_{S_{X_{0}}}^{∗} = id_{S_{X_{0}}}, θ_{T_{X_{0}}}^{∗} = −id_{T_{X_{0}}}, while the involution ι is the Nikulin involution that gives the Shioda-Inose structure on X_{0} ([14]).

Let S be a reducible plane sextic, decomposable as a cubic C with a cusp, a line L intersecting the cubic only at the cusp and a conic Q tangent to the line, (see figure 3).

Proposition 5.4. [3] The map X_{0} −−→^{τ} Y factors as

\xymatrix{X_{0} \ar@{-->}[d]^{/ι} \ar[r]^{/θ} & ℙ^{×}ℙ^{1} \ar[d]^{π_{1}} \\ Y \ar[r]^{π} & ℙ^{2}}

where the map π_{1} is the double cover of ℙ^{2} branched along the conic Q and the map X_{0} →^{/θ} ℙ^{1} × ℙ^{1} is the double cover of ℙ^{1} × ℙ^{1} branched along π_{1}^{∗}(C + L).

The map π is given by the divisor

Δ = 9(L − E_{0}) − 5E_{12} − 4(E_{13} + E_{46} − 3E_{56} − 2(E_{24} + E_{36} + E_{45}) − (E_{25} + E_{34})

[figure: •; S_{1}; •; R_{2}; •; S_{2}; •; R_{3}; S_{3}; •; R_{4}; •; •; R; S_{4}; •; •; R_{5}; •; R′_{5}; •; S; •; S′; •; S_{5}; •; •; T; R_{6}; •; S_{6}; •; R_{7}; •; S_{7}; •; R_{1}]

Figure 2. Rational curves on X_{0}

L C Q

Figure 3. The sextic S

and it contracts the e_{i}’s. Conversely,

Proposition 5.5. [10] The canonical resolution of the double cover of ℙ^{2} branched along the sextic S is a jacobian Kummer surface.

6. K3 Surfaces with T_{X} ≃ U(2) ⊕ U ⊕ ⟨−2⟩

Building on the results of the previous section and using the same notation, we describe the geometry of the K3 surfaces, appearing in our list, that have their period lattices isomorphic to U(2) ⊕ U ⊕ ⟨−2⟩.

In the proposition 5.3, we defined the curves a_{1}, ⋯ a_{5}, which were components of the singular fibers of the elliptic fibration Y →^{f} ℙ^{1}. Set a_{6} := e_{6}, a_{7} := e_{7}, a_{8} := E_{34}.

Lemma 6.1. The a_{i}’s form an even eight on Y. If X −−→^{μ} Y is the associated rational map of degree two, then T_{X} ≃ U(2) ⊕ U ⊕ ⟨−2⟩.

Proof. A direct computation shows that

∑_{i=1}^{8} a_{i} = 2C_{13} +2E_{34} +2·{8(L−E_{0})−(5E_{12} +4E_{46} +3E_{13} +3E_{56} +E_{36} +E_{25} +E_{45})} ∈ 2S_{Y} .

Recall that the rational double cover X −−→^{μ} Y associated to the even eight a_{1}, ⋯, a_{8} factors as

\xymatrix{Z \ar[r]^{ϵ} \ar[d]_{p} & X \ar@{-->}[dl]^{μ} \\ Y}

where Z →^{p} Y is a degree two map branched along the a_{i}’s and Z →^{ϵ} X contracts the eight exceptional curves p^{−1}(a_{i}). Let X →^{i} X be the covering Nikulin involution. We will

describe the proper inverse image under μ of the elliptic fibration f : Y →^{|D|} ℙ^{1} introduced in the proposition 5.3. Each of the curves e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{8} meet the a_{i}’s at exactly two points, thus their pullback under p are smooth rational curves, each meeting an exceptional curve at two points. After blowing down the exceptional curves, they become six disjoint nodal curves f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{8}. It follows from the properties of μ^{∗} that the divisor μ^{∗}D is a nonzero effective nef divisor with self-intersection 0. Thus the linear system |μ^{∗}D| is of the form |mD_{0}|, m ≥ 1, where D_{0} is a smooth curve of genus one. Clearly the f_{i}’s are singular fibers of the elliptic fibration defined by |D_{0}|, X →^{g} ℙ^{1}. Moreover as i(f_{i}) = f_{i}, the involution i must act trivially on ℙ^{1} under the map g.

Claim: The general member of the linear system |mD_{0}| is a smooth curve of genus one. Proof of the claim: Note that, since mD_{0} · μ^{∗}C_{14} = μ^{∗}D · μ^{∗}C_{14} = 2D · C_{14} = 2, we must have m ≤ 2. If m = 2, then the preimage of the general member of |D| would have to split into two disjoint smooth curves of genus one, μ^{−1}(D) = D_{0} + D_{1}, where D_{0} ∼^{lin} D_{1}, and i(D_{0}) = D_{1}. The last assertion contradicts the assumption that i acts trivially on ℙ^{1}. We deduce that the preimage of the I_{1} fiber does not split into two I_{1} fibers but instead it becomes a singular fiber of type I_{2}. We denote its two components by S and i^{∗}S. Finally, the I_{5}^{∗} fiber pulls back to a I_{10}^{∗}-fiber whose components will be denoted by

\begin{aligned}E′_{14} + E′_{15}+ & 2(C′_{0} + E′_{16} + C′_{16} + E′_{26} + C′_{12} + E′_{23} \\ & +i^{∗}C′_{12} + i^{∗}E′_{26} + i^{∗}C′_{16} + i^{∗}E′_{16} + i^{∗}C′_{0}) + i^{∗}E′_{14} + i^{∗}E′_{15}. \end{aligned}

The section C_{14} meets the a_{i}’s at two points so it pulls back to a two-section C′_{14}. We conclude that X contains the diagram of (−2)-curves of figure 4.

Let M be the lattice of rank seventeen, generated by all the (−2) curves of figure 4. A direct computation shows that

D_{M} ≃ ℤ/2ℤ ⊕ ℤ/4ℤ ⊕ ℤ/4ℤ.

Claim: The lattice S_{X} is two-elementary, i.e. D_{S_{X}} ≃ (ℤ/2ℤ)^{a} for some a ≥ 0. Assuming the claim and letting e be the index [S_{X} : M], we get that

|D_{M}| = 2^{5} = e^{2} · 2^{a} ⇒ e^{2} = 2^{5−a} ⇒ a = 1, 3 or 5.

If a = 5, then e = 1 and M = S_{X}, which contradicts the assumption that S_{X} is twoelementary. If a = 1, then by the classification of two-elementary lattices ([13]), T_{X} would be isomorphic to U ⊕U ⊕⟨−2⟩. We have mentioned in the previous section that a K3 surface with such a period lattice contains finitely many smooth rational curves forming the figure 2, which does not contain a subdiagram similar to the one in figure 4. We conclude that a = 3, or equivalently that T_{X} ≃ U ⊕ U(2) ⊕ ⟨−2⟩.

Proof of the claim: In order to prove the claim, it is enough to show the existence of an involution X →^{θ} X satisfying θ_{S_{X}}^{∗} = id_{S_{X}} and θ_{T_{X}}^{∗} = −id_{T_{X}} (see proposition 4.1). Observe

[figure: •; E′_{23}; •; C′_{12}; •; E′_{26}; •; C′_{26}; E′_{16}; •; C′_{0}; •; •; E′_{15}; E′_{14}; •; •; C′_{14}; •; S; •; i^{∗}S; •; i^{∗}E′_{14}; •; •; i^{∗}E′_{15}; i^{∗}C′_{0}; •; i^{∗}E′_{16}; •; i^{∗}C′_{16}; •; i^{∗}E′_{26}; •; i^{∗}C′_{12}]

Figure 4. Generators of the lattice M

that the curves C_{13} and C_{15} pull back to a curve of genus two W, and a curve of genus one E, respectively, both invariant under i. The cycle

C′_{14} + E′_{14} + C′_{0} + E′_{16} + C′_{16} + E′_{26} + C′_{12} + E′_{23} + i^{∗}C′_{12} + i^{∗}E′_{26} + i^{∗}C′_{16} + i^{∗}E′_{16} + i^{∗}C′_{0} + i^{∗}E′_{14}

is disjoint from the genus one curve E. Consequently it defines the I_{14} singular fiber of the elliptic fibration given by the linear system |E|, which we will denote by v : X →^{|E|} ℙ^{1}. The curves S, i^{∗}S, E′_{15} and i^{∗}E′_{15} are sections of v. Fix i^{∗}E′_{15} as the zero section. Let X →^{σ} X be the map induced by the canonical involution on X_{η}, the generic fiber of v, i.e. σ_{X_{η}}(x) = −x

and let X →^{t_{E{′ \atop 15}}} X be the translation by the section E′_{15}. Set γ := σ ∘ t_{E′_{15}}. The action of γ on the period lattice is given by

γ_{T_{X}}^{∗} = σ_{T_{X}}^{∗} ∘ (t_{E′_{15}})_{T_{X}}^{∗} = −id_{T_{X}} ∘ id_{T_{X}} = −id_{T_{X}}.

Moreover its action on the Néron-Severi lattice satisfies γ_{S_{X}}^{∗} = i_{S_{X}}^{∗}. Indeed, to show this, we note that the divisor S + i^{∗}S − E′_{15} is linearly equivalent to the divisor

E′_{14} +2(C′_{0} +E′_{16} +C′_{16} +E′_{26} +C′_{12} +E′_{23} +i^{∗}C′_{12} +i^{∗}E′_{26} +i^{∗}C′_{16} +i^{∗}E′_{16} +i^{∗}C′_{0})+i^{∗}E′_{14} +i^{∗}E′_{15}

which belongs to the kernel of the map β of the theorem 5.2. In other words −S + E′_{15} = i^{∗}S in MW_{f_{E}}(X), or equivalently γ^{∗}S = i^{∗}S and γ^{∗}E′_{15} = i^{∗}E′_{15}. It is now clear that γ_{M}^{∗} = i_{M}^{∗} and consequently that γ_{S_{X}}^{∗} = i_{S_{X}}^{∗}. Define the map θ = i ∘ γ. By construction θ satisfies θ_{T_{X}⊕S_{X}}^{∗} = −id_{T_{X}} ⊕id_{S_{X}}, which implies, by the proposition 4.1, that S_{X} is two-elementary. □

Definition 6.2. Define the lattice E_{8}^{twist}(−1) to be the lattice E_{8}(−1) with the following modification

b(e_{i}, e_{7}) = \{ \begin{matrix}−4 & if e_{i} = e_{7} \\ 2 & if e_{i} = e_{6} \\ 0 & otherwise \end{matrix} \}

where the label of the vertices of the dynkin diagram associated to E_{8}(−1) is as follows

• e_{1} • e_{2} • e_{3} • e_{4} • e_{5} • e_{6} • e_{7} . • e_{8}

Note that |D_{E_{8}^{twist}(−1)}| = 4.

Proposition 6.3. Let Y be a general Kummer surface and N be the Nikulin lattice (see definition 2.6 in section 2) generated by the a′_{i}s, then there is a primitive embedding E_{8}^{twist}(−1) ↪ N^{⊥} ⊂ S_{Y} .

Proof. The elements E_{14}, C_{12}, E_{26}, C_{16}, E_{16}, C_{0}, E_{14}, 2C_{14} + a_{5} + a_{8} generate a sublattice of N^{⊥} isomorphic to E_{8}^{twist}(−1). To prove that the embedding is primitive, it is enough to show that this set can be completed to a basis of N^{⊥}. Let T be the lattice generated by the elements E_{14}, C_{12}, E_{26}, C_{16}, E_{16}, C_{0}, E_{14}, 2C_{14} + a_{5} + a_{8} and 2E_{23} + a_{6} + a_{7}. T is a sublattice of N^{⊥} of finite index e. By explicitly taking the determinant of a matrix representing the bilinear form of T, we find that |D_{T} |=4 and therefore, 4 = e^{2}|D_{N^{⊥}}|. Either e = 1 or N^{⊥} is unimodular, but according to Milnor’s classification [6], there is no even unimodular lattice of rank nine. We conclude that T = N^{⊥} and therefore the embedding is primitive. □

Theorem 6.4. Let X be a K3 surface with T_{X} ≃ U(2) ⊕ U ⊕ ⟨−2⟩. Then X has a Nikulin involution such that if X −−→^{μ} Y is the rational quotient map,

1) Y is a Kummer surface,

2) there is a primitive embedding E_{8}^{twist}(−1) ↪ N^{⊥}.

According to the theorem 3.1, we already know that X admits a rational map of degree two into a Kummer surface, so we need to show that the even eight associated to such a map is given by the a_{i}’s. Before proving this, we will show, using very geometrical arguments, that any K3 surface with T_{X} ≃ U(2)⊕U ⊕⟨−2⟩ must contain a diagram of smooth rational curves meeting as in figure 4. Let θ be the involution that satisfies θ_{T_{X}}^{∗} = −id_{T_{X}} and θ_{S_{X}}^{∗} = id_{S_{X}} which exists by proposition 4.1. Nikulin showed that

X^{θ} := {x ∈ X|θ(x) = x} = E ⨿ ∐_{i=1}^{7} R_{i},

where E is a curve of genus one and the R_{i}’s are disjoint rational curves. The system |E| defines an elliptic fibration X →^{v} ℙ^{1}, with a section C and exactly one reducible singular fiber 𝔉 of type I_{14}. Moreover θ acts non trivially on ℙ^{1} under v ([14] p.1424-1430). The R_{i} are components of 𝔉. Since θ preserves all the rational curves, any section of v must meet the singular fiber at a R_{i} component. Thus the surface X contains the following configuration of (−2)-curves

[figure: S_{4}; R_{4}; S_{3}; R_{3}; S_{2}; R_{2}; S_{1}; R_{5}; S_{5}; R_{6}; S_{6}; R_{7}; S_{7}; R_{1}; C; .]

Consider the divisor

F := 2R_{2} + 4S_{1} + 6R_{1} + 3C + 5S_{7} + 4R_{7} + 3S_{6} + 2R_{6} + S_{5}.

The system |F | defines another elliptic fibration, X →^{q} ℙ^{1}. By construction, q has a singular fiber of type II^{∗}, a section R_{5} and a 2-section S_{2}. Moreover, the curves S_{4}, R_{4}, S_{3} and R_{3} must be part of another reducible fiber of q, 𝔍. We will show by elimination that 𝔍 must be of type I_{2}^{∗}. First, suppose that 𝔍 was of type I_{b} for b ≥ 4 :, then there would exist a rational smooth curve R meeting S_{4} at exactly one point, p:

[figure: S_{4}; R; R_{4}; S_{3}; R_{3}; .; R_{5}]

Since θ^{∗} acts as the identity on S_{X}, θ(p) must also belong to S_{4} ∩ R. But R meets S_{4} at exactly one point, so p must be fixed by θ. Either p ∈ R_{5} ∩ S_{4} ∩ R, which contradicts the fact that R_{5} is a section of q, or p ∈ R_{4} ∩ S_{4} ∩ R, which contradicts the assumption that S_{4}, R_{4}, S_{3} and R_{3} are part of a fiber of type I_{b}. The fiber 𝔍 can neither be of type I_{1}^{∗} : Again, suppose it is, then S_{4} must be a component of multiplicity one (S_{4} meets the section R_{5}). Denote by R the other component of multiplicity one of 𝔍 that meets R_{4}. Two cases are possible:

(1) R does not meet the curve S_{2}:

[figure: S_{4}; R_{4}; R; S_{3}; R_{3}; S; S_{2}; R_{2}; R_{5}; .]

Then the sixth component of 𝔍, S would meet S_{2} at exactly one point, p. Because p ∈ S ∩ S_{2}, its image θ(p) belongs to S ∩ S_{2}. So θ(p) = p and consequently p ∈ R_{3} ∩ S_{2} or p ∈ R_{2} ∩ S_{2}. The first case contradicts the fact that 𝔍 is of type I_{1}^{∗}, while in the second case, S would meet a component of another singular fiber, which is absurd.

(2) R does meet the curve S_{2} :

[figure: S_{4}; R_{4}; R; S_{3}; R_{3}; S_{2}; R_{2}; R_{5}; .]

Since S_{2} is a 2-section of q and S_{2} ·R_{3} = 1, necessarily R·S_{2} = 1. Let p ∈ R∩S_{2}, then θ(p) = p. It follows that either p ∈ R_{3} ∩ S_{2}, which contradicts again the assumption that 𝔍 is of type I_{1}^{∗}. Or p ∈ R_{2}∩S_{2}∩R, which implies again that R meets a component of another singular fiber.

We exclude also the case where the fiber 𝔍 is of type I_{3}^{∗} as follows. Suppose it is, then S_{4} has to be a component of multiplicity one and R_{4}, S_{3} and R_{3} components of multiplicity two. Let R be the fourth component of multiplicity two of 𝔍 and L_{1}, L_{2} the two components of multiplicity one intersecting it:

[figure: S_{4}; R_{4}; S_{3}; R_{3}; R; L_{2}; L_{1}; S_{2}; R_{5}; .]

Since L_{1} and L_{2} do not meet any R_{i}’s nor any S_{i}’s, they must be part of a reducible fiber of the fibration defined by |E|. But this fibration is known to have a unique reducible singular fiber. The fiber 𝔍 cannot be of type I_{b}^{∗}, with b ≥ 4, as 𝔍 would have b + 5 components. But by theorem 5.2, the number of components of 𝔍 cannot exceed 8, so b ≤ 3. The same counting argument shows that 𝔍 cannot be of type II^{∗}. The 𝔍 cannot be of type III^{∗}. If it was, the group of sections of q would be finite of order n, where n would satisfy by the theorem 5.2

2^{3} = |D_{S_{X}}| = \frac{1 · 2}{n^{2}}

but no integer satisfies this equation. Finally 𝔍 cannot be of type IV^{∗}. Indeed in this case one of the components with multiplicity one will have to be disjoint from every R_{i} and S_{i}. Such a component must be part of a reducible fiber of the fibration defined by |E|. But again, this fibration is known to have a unique reducible singular fiber. By elimination, we have shown that 𝔍 must be of type I_{2}^{∗}. Moreover we have shown that X contains the following configuration of smooth rational curves

[figure: S_{4}; R_{4}; R; S_{3}; R_{3}; S_{2}; R_{2}; S_{1}; R_{5}; S_{5}; R_{6}; S_{6}; R_{7}; S_{7}; R_{1}; C; .]

By symmetry, the divisor class

F′ = 2R_{7} + 2S_{7} + 6R_{1} + 3C + 5S_{1} + 4R_{2} + 3S_{2} + 2R_{3} + S_{3}

defines an elliptic fibration, X →^{h} ℙ^{1}, for which the curves R_{6} + S_{5} + R_{5} + S_{4} are part of a reducible fiber of type I_{2}^{∗}, ℌ := S + S_{4} + 2R_{5} + 2S_{5} + 2R_{6} + T + T′. Since R · F′ = 0, the curve R is also part of a reducible fiber clearly disjoint from ℌ. It follows from theorem 5.2 that R can only be part of a fiber with two components so generically it is of type I_{2}. We conclude that the R_{i}’s, the S_{i}’s and the smooth rational curves R, S, T define the diagram on X

[figure: •; S_{1}; •; R_{2}; •; S_{2}; •; R_{3}; S_{3}; •; R_{4}; •; •; R; S_{4}; •; •; R_{5}; •; S; •; S_{5}; •; •; T; R_{6}; •; S_{6}; •; R_{7}; •; S_{7}; •; R_{1}; .]

The I_{10}^{∗} configuration apparent in this diagram defines an elliptic fibration for which S is part of a I_{2} fiber. By adding the other component of this fiber, we obtain a diagram of smooth rational curves meeting exactly as in the figure 4.

We can now prove theorem 6.4.

Proof. In order to explicitly construct the Nikulin involution i we apply the inverse process used in lemma 6.1 to construct the involution θ. The curves R, S and T, constructed above, are sections of the elliptic fibration X →^{v} ℙ^{1}. Fix T as the zero section. Let σ be the map induced by inversion on the generic fiber and t_{R} be the translation by R. The map γ = σ ∘t_{R} is an involution on X such that γ_{T_{X}}^{∗} = −id_{T_{X}}. Set i := θ∘γ. As the maps θ^{∗} and γ^{∗} commute on H^{2}(X, ℤ), i is an involution. Moreover it follows from the fact that i_{T_{X}}^{∗} = id_{T_{X}} that i is a Nikulin involution. Its eight fixed points can be easily described. Indeed a point p is fixed by i if and only if γ(p) = θ(p). Since γ acts trivially on the fibers of v and θ acts non trivially on the same fibers, the fixed points of i must lie on the only two fibers fixed by θ, i.e E and 𝔉. The points lying on E satisfy γ(p) = θ(p) = p and therefore correspond to the four fixed points of γ restricted to E. The four remaining fixed points must lie on 𝔉. More precisely, two of them lie on S_{1} and the other two lie on R_{5}. Indeed, since S_{j} · i^{∗}(S_{j}) = 0 for j ≠ 1, a point p, lying on some S′_{j}s is fixed by i if and only if p ∈ S_{1} and θ(p) = p. Thus the two fixed points of θ restricted to S_{1} are fixed by i. Similarly, R_{j} · i^{∗}(R_{j}) = 0 for j ≠ 5 implies that the two fixed points of γ restricted to R_{5} are fixed by i. Consider X −−→^{μ} Y the rational quotient map associated to i and denote by A_{1}, …, A_{8} the induced even eight. The A_{i}’s and the image of the curves S_{i}, R_{i}, S, R and T under μ form the following diagram of rational curves on Y

[figure: •; •; •; •; •; •; •; •; •; A_{5}; •; A_{6}; •; A_{1}; •; A_{2}; •; A_{3}; •; A_{4}; •; •; •; A_{7}; •; A_{8}; .]

The surface Y admits therefore an elliptic fibration with a section and whose singular fibers are of the type 6I_{2} + I_{5}^{∗} + I_{1}. In the table of [17], we find that the Mordell Weil group of such a fibration must be of order two. It follows now from the proposition 5.3 that S_{Y} admits a primitive embedding of the Kummer lattice which implies in turn by [11] that Y is a Kummer surface. The primitive embedding E_{8}^{twist}(−1) ↪ N^{⊥} follows from the proposition 6.3. □

Let Σ be a reducible plane sextic, decomposable as an irreducible quartic Γ with an ordinary node and a cusp of type A_{4}, a line Λ_{1} passing trough the two singular points of Γ and another line Λ_{2} (see figure 5).

[figure: Λ_{2}; Γ; Λ_{1}]

Figure 5. The sextic Σ

Theorem 6.5. The rational quotient map X −−→^{μ} Y decomposes as

\xymatrix{X \ar@{-->}[d]^{μ} \ar[r]^{q_{2}} & Z \ar[d]^{q_{1}} \\ Y \ar[r]^{q_{0}} & ℙ^{2}}

where q_{0} is the canonical resolution of the double cover of ℙ^{2} branched along Σ. The maps q_{1} and q_{2} are the canonical resolutions of the double covers branched along Γ and q_{1}^{∗}(Λ_{1} + Λ_{2}) respectively.

Proof. We have shown in the course of proving theorem 6.4 that the map μ can be thought as the rational double cover associated to the even eight defined by the a_{i}’s. Therefore there exists on X a curve of genus two defined by W = μ^{∗}C_{13} that satisfies

W ∩ E = {p_{1}, p_{2}, p_{3}, p_{4}}, W ∩ R_{5} = {p_{5}, p_{6}} i(p_{i}) = p_{i} i = 1, … 6,

where the p_{i}’s are fixed points of the Nikulin involution. The quotient X/θ is a smooth rational surface. Applying Hurwitz formula we find that K_{X/θ}^{2} = −7. Note that the image in X/θ of the nine disjoint curves S_{i}, for i = 1 ⋯ 7, R and T have self intersection −1. We may therefore blow them down to a surface Z, whose canonical divisor has self-intersection 2. The map X →^{q_{2}} Z is ramified along the R_{i}’s and E and it sends W to a smooth rational curve C, with C^{2} = 2. Since the involution i commutes with θ on X, it induces an involution ī on Z. Note that ī has four fixed points on C, denoted by q_{2}(p_{1}), q_{2}(p_{2}), q_{2}(p_{3}), q_{2}(p_{4}), thus ī restricted to C is the identity and the quotient map, Z → Z/ī, is ramified along C. Using Hurwitz formula a second time, we find that K_{Z/ī}^{2} = 6. Moreover, under the composition,

X →^{q_{2}} Z →^{/ī} Z/ī, the curves R_{2}, R_{3} and R_{5} map to rational curves R′_{2}, R′_{3}, R′_{5}, with the intersection properties

R_{5}^{′2} = R_{2}^{′2} = −1, R_{3}^{′2} = −2, R′_{5} · R′_{2} = R′_{5} · R′_{3} = 0, R′_{2} · R′_{3} = 1.

We may successively blow down R′_{5}, R′_{2} and R′_{3} to obtain a rational surface whose canonical divisor has self intersection equal to 9, i.e. the projective plane. It is an easy exercise to verify that under the composition, X →^{q_{2}} Z →^{q_{1}} ℙ^{2}, the genus two curve W maps to a quartic Γ with an ordinary node and a cusp of type A_{4}, the R_{i}’s get contracted to a line Λ_{1} passing through the singularities of Γ, and the genus one curve E maps to another line Λ_{2}. The following proposition completes the proof. □

Proposition 6.6. A surface Y is a general Kummer surface if and only if Y is the canonical resolution of a double cover of ℙ^{2} branched along the reducible sextic Σ = Γ + Λ_{1} + Λ_{2}.

Proof. Assume first that Y is a general Kummer surface. Recall that Y admits a double

plane model Y →^{ϕ} ℙ^{2} that factors as Y →^{ϕ̃} ℙ̃^{2} →^{ϵ} ℙ^{2} where ℙ̃^{2} is the blowup of ℙ^{2} at the fifteen singular points of the union of six lines (see figure 1). The image in ℙ̃^{2} of the curves a_{1}, ⋯ a_{8}, C_{15}, E_{15}, C_{0}, E_{16}, E_{26}, C_{12}, E_{26}, E_{14} C_{14} and C_{13} intersect as in the figure 6. The numbers in the parentheses denote the self-intersection.

[figure: •; C̃_{14}(-4); •; Ẽ_{14}(-1); •; C̃_{0}(-4); •; Ẽ_{16}(-1); C′_{16}(-4); •; Ẽ_{26}(-1); •; C̃_{12}(-4); •; •; Ẽ_{23}(-1); •; C̃_{23}(-2); •; •; ã_{5}(-1); ã_{8}(-1); •; Ẽ_{15}(-1); •; C̃_{15}(-4); •; ã_{1}(-1); •; ã_{2}(-1); •; ã_{3}(-1); •; ã_{4}(-1); •; C̃_{13}(-1)]

Figure 6. Diagram of curves in ℙ̃^{2}

The curve C̃_{23} corresponds to the images ϕ̃(a_{6}) = ϕ̃(a_{7}). We may blow down the curves ã_{1}, ã_{2}, ã_{3}, ã_{4}, ã_{5}, ã_{8}, Ẽ_{14}, Ẽ_{16}, Ẽ_{15}, Ẽ_{26}, Ẽ_{23}, C̃_{14}, C̃_{23}, C̃_{12} and C̃_{16}. This sequence of blow-downs defines a Cremona transformation

[figure: ℙ̃^{2}; ϵ; η; ℙ^{2}; ξ; ℙ^{2}]

which maps the six lines of figure 1 as follows

ξ(l_{0}) = Λ_{1} ξ(l_{3}) = Γ ξ(l_{5}) = Λ_{2}

while the lines l_{2}, l_{4} and l_{6} get contracted to the singular points of Γ.

Conversely, let Y_{0} →^{η_{1}} ℙ^{2} be the blow-up of the D_{9} singularity of Σ (line passing through a cusp of type A_{4}) and denote by Υ_{1} the exceptional divisor. By its recursive construction ([1] p.87), the canonical resolution of the double cover defined by Σ is the canonical resolution of the double cover defined by

Σ_{1} = η_{1}^{∗}Σ − 2Υ_{1} = Γ + 2Υ_{1} + Λ_{1} + Υ_{1} + Λ_{2} − 2Υ_{1} = Γ + Λ_{1} + Λ_{2} + Υ_{1}

where we use the same symbol to denote a curve and its proper transform. Recursively, if Y_{1} →^{η_{2}} Y_{0} is the blow-up of the point Γ ∩ Υ_{1}, then the canonical resolution of the double cover defined by Σ_{1} is the canonical resolution of the double cover defined by

Σ_{2} = η_{2}^{∗}Σ_{1} − 2Υ_{2} = Γ + Λ_{1} + Λ_{2} + Υ_{1} + Υ_{2}.

Blow up the simple node of Γ and denote the exceptional divisor by Υ_{3}. We must now take the double cover branched along

Σ_{3} = Γ + Λ_{1} + Λ_{2} + Υ_{1} + Υ_{2} + Υ_{3}.

Since the singularities of Σ_{3} are now all of type A_{1}, the exceptional divisors coming from their resolution will not be part of the branch locus along which we will take the double cover. The resolution of Σ_{3} gives a graph identical to the one of figure 6 in ℙ̃^{2} where Γ + Λ_{1} + Λ_{2} + Υ_{1} + Υ_{2} + Υ_{3} corresponds exactly to C̃_{13} + C̃_{0} + C̃_{15} + C̃_{16} + C̃_{12} + C̃_{14}. □

7. K3 Surfaces with T_{X} ≃ U(2) ⊕ U(2) ⊕ ⟨−2⟩

The last section of this paper is very similar to the two previous ones. It aims to describe the geometry of the K3 surfaces with period lattice isomorphic to U(2) ⊕ U(2) ⊕ ⟨−2⟩.

Proposition 7.1. Let Y be a general Kummer surface. There exists an elliptic fibration on Y whose singular fibers are of the type 4I_{2} + I_{2}^{∗} + I_{1}^{∗} + I_{1} and whose Mordell-Weil group is cyclic of order two.

Proof. Consider the divisor

B = 3(L − E_{0}) − 2E_{12} − (E_{13} + E_{24} + E_{45} + E_{46} + E_{56}).

Geometrically, B can be represented by the proper inverse image, under the map Y →^{ϕ} ℙ^{2}, of a cubic passing trough p_{13}, p_{24}, p_{45}, p_{46}, p_{56} having a double point at p_{12}. By counting parameters, we immediately see that such a cubic exists. Generically, its proper transform is smooth and irreducible and satisfies B^{2} = 0. Let Y →^{b} ℙ^{1} be the elliptic fibration defined by the linear system |B|.

The divisors

(1) F_{1} = \underbrace{(L − E_{0}) − (E_{12} + E_{45})}_{b_{1}} +2(L − E_{0}) − (E_{12} + E_{13} + E_{24} + E_{46} + E_{56}),

(2) F_{2} = \underbrace{2(L − E_{0}) − (E_{12} + E_{13} + E_{24} + E_{45} + E_{56})}_{b_{2}} +e_{1},

(3) F_{3} = \underbrace{3(L − E_{0}) − 2E_{12} − (E_{13} + E_{24} + E_{36} + E_{45} + E_{46} + E_{56})}_{b_{3}} +E_{36},

(4) F_{4} = \underbrace{E_{35}}_{b_{4}} +3(L − E_{0}) − 2E_{12} − (E_{13} + E_{24} + E_{45} + E_{56} + E_{46} + E_{35})

define four fibers of type I_{2}. The divisor

F_{5} = \underbrace{(L − E_{0}) − (E_{12} + E_{56})}_{b_{5}} +E_{14} + 2(C_{0} + E_{14} + C_{14}) + E_{15} + E_{16}

defines a fiber of type I_{2}^{∗}. The divisor

F_{6} = e_{6} + e_{7} + 2(C_{12} + E_{23}) + E_{26} + E_{25}

defines a fiber of type I_{1}^{∗}. Since C_{15} · B = C_{16} · B = 1, we see that the rational curves C_{15} and C_{16} are sections of b. Moreover, as ∑_{i=1}^{6} χ_{top}(F_{i}) = 23, there must be one more singular fiber F_{7} of type I_{1}. Finally, by theorem 5.2, we deduce that MW_{b}(X) ≃ ℤ/2ℤ. □

Set b_{6} := e_{6} b_{7} := e_{7} b_{8} := E_{34}.

Lemma 7.2. The b_{i}’s form an even eight on Y. If X −−→^{ν} Y is the associated rational map of degree two, then T_{X} ≃ U(2) ⊕ U(2) ⊕ ⟨−2⟩.

Proof. A direct computation shows that

∑_{i=1}^{8} b_{i} = 2C_{13} + 2E_{35} + 2E_{34} + 2{4(L − E_{0}) − (3E_{12} + 2E_{45} + 2E_{56} + E_{13} + E_{24} + E_{46})} ∈ 2S_{Y} .

We will apply the same strategy as in lemma 6.1 and analyze the fibration, b_{ν} : X →^{ν^{∗}B} ℙ^{1}. Let X →^{ι} X be the associated Nikulin involution. Likewise in lemma 6.1, ν^{∗}F_{i}, for i = 1, 2, 3, 4, are nodal curves preserved by ι. Hence ι preserves the fibers of b_{ν} (ν^{∗}B cannot split). As for the remaining singular fibers, ν^{∗}F_{5} becomes of type I_{4}^{∗}, ν^{∗}F_{6} becomes of type I_{2}^{∗} and ν^{∗}F_{7} becomes of type I_{2}. Finally ν^{∗}C_{15} and ν^{∗}C_{16} become 2-sections.. We conclude that there exists an embedding of lattices of finite cokernel U(2) ⊕ D_{8} ⊕ D_{6} ⊕ A_{1} ↪ S_{X}. Since D_{U(2)⊕D_{8}⊕D_{6}⊕A_{1}} ≅ (ℤ/2ℤ)^{⊕7}, the overlattice S_{X} must be a two-elementary lattice. Note also that b_{ν} does not have a section. Indeed, if C · ν^{∗}D = 1, then ι^{∗}C also would satisfy

ι^{∗}C · ι^{∗}(ν^{∗}D) = C · ν^{∗}D = 1.

Let C̃ ∈ S_{Y} be such that ι^{∗}C + C = ν^{∗}C̃. Then 2C̃ · D = ν^{∗}C̃ · ν^{∗}D = 2, thus C̃ · D = 1. So either C + ι^{∗}C = ν^{∗}C_{15} or C + ι^{∗}C = ν^{∗}C_{16}. But we have seen that C_{15} and C_{16} do not split as they each meet the b_{i}’s at two points. This proves that b_{ν} does not have a section. Let J(X) be the jacobian fibration of b_{ν} (see [2] for the definition), then there is an embedding ([5])

T_{X} ↪ T_{J(X)}, with T_{J(X)}/T_{X} ≃ ℤ/2ℤ.

Let a be the length of D_{S_{X}}, i.e. D_{S_{X}} ≃ (ℤ/2ℤ)^{a}. If a = 1, then by the classification of two-elementary lattices ([13]), T_{X} ≃ U ⊕ U ⊕ ⟨−2⟩ and X contains finitely many rational curves meeting as in figure 2 which do not define any I_{4}^{∗} + I_{2}^{∗} + I_{2} configuration. If a = 3, then T_{J(X)} has to be isomorphic to U ⊕U ⊕⟨−2⟩ and we get the same contradiction as in the previous case applied to the surface J(X). We conclude that T_{X} ≃ U(2) ⊕ U(2) ⊕ ⟨−2⟩. □

Theorem 7.3. Let X be a K3 surface with T_{X} ≃ U(2) ⊕ U(2) ⊕ ⟨−2⟩. Then the even eight associated to the rational map X −−→^{ν} Y is given by the b_{i}’s of lemma 7.2.

Proof. From a purely lattice theoretical point of view, we deduce from lemma 7.2 that there exists an embedding U(2)⊕D_{8}⊕D_{6}⊕A_{1} ↪ S_{X}. Thus X admits an elliptic fibration X →^{b_{ν}} ℙ^{1} with singular fibers of type I_{4}^{∗} + I_{2}^{∗} + I_{2}, i.e. it contains the following diagram of smooth rational curves

[figure: •; R_{1}; •; S_{3}; •; R_{2}; •; S_{4}; R_{3}; •; •; S_{7}; •; S_{8}; •; S_{10}; •; S_{11}; •; •; S_{9}; R_{4}; •; R_{5}; •; S; •; S′; •; S_{1}; •; S_{2}; •; S_{5}; •; S_{6}; .]

We shall see that the multisection index of b_{ν} is two. Note that by construction b_{ν} admits a 2-section that we call R_{6}. Without loss of generality, R_{6} ·S_{2} = R_{6} ·S_{6} = R_{6} ·S_{8} = R_{6} ·S_{11} = 1. Let X →^{θ} X be the involution acting trivially on S_{X}. Nikulin showed that

X^{θ} = {x ∈ X|θ(x) = x}

consists of the union of seven disjoint smooth rational curves [14]. In fact, the curves R_{1}, R_{2}, R_{3}, R_{4}, R_{5}, R_{6} belong to X^{θ}. Indeed, the equalities R_{1} · S_{1} = R_{1} · S_{2} = R_{1} · S_{3} = 1 imply that there exist three points on R_{1} fixed by θ, consequently R_{1} ⊂ X^{θ}. The same argument applies for R_{3}, R_{4} R_{5} and R_{6}. Assume that R_{2} ∉ X^{θ} and denote by p the point of R_{2} ∩ S_{3}. Because R_{2} · S_{3} = 1, θ(p) = p, so there exists C ⊂ X^{θ} such that p ∈ C ∩ R_{2} ∩ S_{3}. If F denotes the class of a fiber of b_{ν}, then C · F ≥ 4. The curves S and S′ cannot be fixed by θ as they don’t meet S_{1}, S_{5}, S_{7} and S_{10} which must intersect X^{θ}. It implies that C · (S + S′) ≤ 4, so C · (S + S′) = 4. Similarly, if q = R_{2} ∩ S_{4}, then there exists C′ ⊂ X^{θ}, such that C′ · F ≥ 4. If C ≠ C′, then C′ · (S + S′) = 0, which is absurd. If C = C′, then C · F ≥ 8, which contradicts C · (S + S′) = 4. So R_{2} ⊂ X^{θ}. Denote by R_{7} the remaining curve of X^{θ}. Necessarily

(R_{6} + R_{7}) · S = (R_{6} + R_{7}) · S′ = 2

and R_{7} is also a 2-section satisfying

R_{7} · S_{1} = R_{7} · S_{5} = R_{7} · S_{7} = R_{7} · S_{10} = 1.

Observe now that

D_{1} = 2R_{6} + S_{2} + S_{6} + S_{8} + S_{11} ∼^{lin} 2R_{7} + S_{1} + S_{5} + S_{7} + S_{10} = D_{2}.

The divisors D_{1} and D_{2} are two I_{0}^{∗} fibers of an elliptic fibration X →^{h} ℙ^{1}. The curves S_{3}, R_{2}, S_{4} and S_{9} are components of the other singular fibers of h and R_{1}, R_{3}, R_{4}, R_{5} are sections of h. Fix R_{4} as the zero section. Let σ be the map induced by inversion of the generic fiber and t_{R_{5}} be the translation by R_{5}. The map γ = σ ∘ t_{5} is an involution on X such that γ_{T_{X}}^{∗} = −id_{T_{X}}. Set ι := θ ∘ γ. As the maps θ^{∗} and γ^{∗} commute on H^{2}(X, ℤ) and γ^{∗} ∘ θ_{T_{X}}^{∗} = id_{T_{X}} we conclude that ι is a Nikulin involution. Note that

R_{1} + S_{3} + R_{2} + S_{4} + R_{3} + S_{3} + R_{7} + S_{1} ∼^{lin} R_{4} + S_{9} + R_{5} + S_{11} + R_{6} + S_{8}

as they both define two disjoint I_{8} and I_{6} fibers of some other fibration. Therefore −R_{1}+R_{5} = R_{3} in MW_{h}(X) and consequently ι^{∗}R_{1} = γ^{∗}R_{1} = R_{3}. Clearly ι^{∗}R_{4} = γ^{∗}R_{4} = R_{5}. The involution γ^{∗} acts trivially on the curves R_{6}, R_{7}, R_{2} and S_{9}. Consequently so does ι and each of those curves contains two of the eight fixed points of ι. Finally, observe that

ι^{∗}S_{1} = S_{5}, ι^{∗}S_{2} = S_{6}, ι^{∗}S_{3} = S_{4}, and ι^{∗}S_{7} = S_{10} ι^{∗}S_{8} = ι^{∗}S_{11}.

Consider X −−→^{ν} Y the rational quotient map by ι and denote by B_{1}, … B_{8} the induced even eight. Then Y contains the following diagram of −2-curves

[figure: •; •; •; •; •; •; B_{7}; •; B_{8}; •; •; •; B_{1}; B_{2}; •; •; •; B_{3}; B_{4}; •; •; •; B_{5}; •; B_{6}; •; •; .]

By inspection of the diagram, we see that Y admits an elliptic fibration with a section and whose singular fibers are of the type 4I_{2} + I_{2}^{∗} + I_{1}^{∗}. In the table of [17], we find that the Mordell Weil group of such a fibration must be of order two. It follows now from theorem 6.5 that S_{Y} admits a primitive embedding of the Kummer lattice which implies by Nikulin’s criterion [11] that Y is a Kummer surface. Finally, by proposition 7.1, the B_{i}’s correspond to the b_{i}’s. □

Let X be a reducible plane sextic, decomposable as three lines L_{1}, L_{2}, L_{3} meeting at a point, a line L_{4} and a conic W tangent to L_{4} (see figure 7).

[figure: L_{4}; L_{3}; L_{2}; L_{1}; W]

Figure 7. The sextic X

Theorem 7.4. The map X −−→^{ν} Y decomposes as

\xymatrix{X \ar@{-->}[d]^{ν} \ar[r]^{π_{2}} & ℙ^{1} × ℙ^{1} \ar[d]^{π_{1}} \\ Y \ar[r]^{π} & ℙ^{2}}

where π is the canonical resolution of the double cover of ℙ^{2} branched along X. The maps π_{1} and π_{2} are the canonical resolutions of the double covers branched along W and π_{1}^{∗}(L_{1} + L_{2} + L_{3} + L_{4}) respectively.

Proof. Let ℙ^{1} × ℙ^{1} →^{π_{1}} ℙ^{2} be the double cover branched along W. Note that the line L_{4} splits under the cover and that π_{1}^{∗}(L_{1}+L_{2}+L_{3}+L_{4}) is a divisor of bidegree (4, 4) and consequently it is an even divisor. The canonical resolution of the double cover defined by this even divisor is a K3 surface X which admits a rational map X −−→^{ν̃} Y, where Y is a general Kummer surface according to the proposition 7.5 below. Applying the same techniques as in theorem 6.5, we check that the corresponding even eight corresponds to the b_{i}’s of lemma 7.2 and therefore ν̃ = ν. □

Proposition 7.5. A surface Y is a general Kummer surface if and only if Y is the canonical resolution of the double cover of ℙ^{2} branched along the reducible sextic X. Moreover the double cover Y →^{π} ℙ^{2} contracts the b′_{i}s of lemma 7.2.

Proof. We proceed exactly as in proposition 6.6. Consider the images of the curves b_{1}, … b_{8},

C_{15}, E_{15}, C_{0}, E_{14}, C_{14}, C_{13}, E_{25}, E_{16}, E_{23}, C_{12}, E_{26}, C_{16} under the map Y →^{ϕ̃} ℙ̃^{2}. Blow down the

fourteen curves b̃_{1}, …, b̃_{8}, Ẽ_{14}, Ẽ_{15}, Ẽ_{23}, Ẽ_{26}, Ẽ_{25}, Ẽ_{16}, C̃_{12}, C̃_{0}. Since ℙ̃^{2} is the blow-up of ℙ^{2} at fifteen points, the resulting surface is, the blow-up of ℙ^{2} at a point, i.e. the Hirzenbruch surface 𝔽_{1}. Recall that Pic(𝔽_{1}) = ℤH ⊕ ℤF, where H^{2} = 1, H · F = 1, and F^{2} = 0. Under the sequence of blow-downs, C̃_{15} and C̃_{16} become linearly equivalent to H while C̃_{14} and C̃_{12} become linearly equivalent to F. Therefore the divisor C̃_{14} − C̃_{15} ∼^{lin} E_{p}, where is E_{p} is the exceptional divisor of 𝔽_{1} (we use the same symbol to denote the curves C̃_{13}, C̃_{14}, C̃_{15}, C̃_{16} and their images in 𝔽_{1}). By keeping track of the intersections all along the blow-downs, we see that, on 𝔽_{1}, C̃_{13} · C̃_{14} = C̃_{13} · C̃_{15} = 2. If follows that C̃_{13} · E_{p} = 0. Consider the contracting morphism 𝔽_{1} → ℙ^{2}, under this map C̃_{13} becomes a conic W, the curves C̃_{14}, C̃_{15}, C̃_{16} become three lines, L_{1}, L_{2}, L_{3}, meeting at a point and the curve C̃_{12} becomes a line L_{4} tangent to W. Again we have constructed a Cremona transformation

[figure: ℙ̃^{2}; ϵ; η; ℙ^{2}; ξ; ℙ^{2}]

which maps the six lines of the figure 1 as follows

ξ(l_{3}) = W, ξ(l_{2}) = L_{4}, ξ(l_{4}) = L_{1}, ξ(l_{5}) = L_{2}, ξ(l_{6}) = L_{3}.

and the line l_{0} gets contracted to the point of intersection L_{1} ∩ L_{2} ∩ L_{3}.

Conversely, let Y_{0} →^{η_{1}} ℙ^{2} be the blow-up of the point p ∈ L_{1} ∩L_{2} ∩L_{3} and denote by Υ_{1} the exceptional divisor. By construction, the canonical resolution of the double cover defined by X is the canonical resolution of the double cover defined by

X_{1} = η^{∗}X − 2Υ_{1} = W + L_{1} + L_{2} + L_{3} + L_{4} + Υ_{1}

(again we use the same symbol to denote a curve and its proper transform). Since the singularities of X_{1} are now all of type A_{1}, the exceptional divisors coming from their resolution will not be part of the branch locus along which we will take the double cover. Moreover, the canonical resolution of X_{1} gives rise to a graph of rational curves in ℙ̃^{2} meeting exactly as the curves b̃_{1}, ⋯, b̃_{8}, Ẽ_{14}, Ẽ_{15}, Ẽ_{23}, Ẽ_{26}, Ẽ_{25}, Ẽ_{16}, C̃_{12}, C̃_{0}, C̃_{13}, C̃_{14}, C̃_{15}, C̃_{16}. The divisor W + L_{1} + L_{2} + L_{3} + L_{4} + Υ_{1} corresponds to C̃_{13} + C̃_{14} + C̃_{15} + C̃_{16} + C̃_{12} + C̃_{0}. It is clear now that the double cover along such a divisor is a general Kummer surface. □

References

[1] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven. Compact complex surfaces, volume 4. Springer-Verlag, Berlin, second edition, 2004.

[2] F. R. Cossec and I. V. Dolgachev. Enriques surfaces. I, volume 76 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1989.

[3] F. Galluzzi and G. Lombardo. Correspondences between K3 surfaces. Michigan Math. J., 52(2):267–277, 2004. With an appendix by Igor Dolgachev.

[4] R. W. H. T. Hudson. Kummer’s quartic surface. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1990. With a foreword by W. Barth, Revised reprint of the 1905 original.

[5] J. Keum. A note on elliptic K3 surfaces. Trans. Amer. Math. Soc., 352(5):2077–2086, 2000.

[6] J. Milnor. On simply connected 4-manifolds. In Symposium internacional de topología algebraica International symposi um on algebraic topology, pages 122–128. Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958.

[7] R. Miranda. The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research]. ETS Editrice, Pisa, 1989.

[8] D. R. Morrison. On K3 surfaces with large Picard number. Invent. Math., 75(1):105–121, 1984.

[9] S. Mukai. Vector bundles on a K3 surface. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 495–502, Beijing, 2002. Higher Ed. Press.

[10] I. Naruki. On metamorphosis of Kummer surfaces. Hokkaido Math. J., 20(2):407–415, 1991.

[11] V. V. Nikulin. Kummer surfaces. Izv. Akad. Nauk SSSR Ser. Mat., 39(2):278–293, 471, 1975.

[12] V. V. Nikulin. Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch., 38:75–137, 1979.

[13] V. V. Nikulin. Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat., 43(1):111–177, 238, 1979.

[14] V. V. Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Algebro-geometric applications. In Current problems in mathematics, Vol. 18, pages 3–114. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981.

[15] V. V. Nikulin. On rational maps between K3 surfaces. In Constantin Carathéodory: an international tribute, Vol. I, II, pages 964–995. World Sci. Publishing, Teaneck, NJ, 1991.

[16] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič. Torelli’s theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat., 35:530–572, 1971.

[17] I. Shimada. On elliptic K3 surfaces. Michigan Math. J., 47(3):423–446, 2000.

[18] T. Shioda. On the Mordell-Weil lattices. Comment. Math. Univ. St. Paul., 39(2):211–240, 1990.

[19] T. Shioda and H. Inose. On singular K3 surfaces. In Complex analysis and algebraic geometry, pages 119–136. Iwanami Shoten, Tokyo, 1977.

Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case Postale 64, 1211 Genéve 4, Suisse.,

E-mail address: Afsaneh.Mehran@math.unige.ch