CLASSIFICATION OF LOG DEL PEZZO SURFACES OF INDEX ≤ 2

VALERY ALEXEEV AND VIACHESLAV V. NIKULIN

Abstract. This is an expanded version of our work [AN88], 1988, in Russian.

We classify del Pezzo surfaces over ℂ with log terminal singularities (equivalently, quotient singularities) of index ≤ 2. By classification, we understand a description of the intersection graph of all exceptional curves on an appropriate (somewhat stronger than minimal) resolution of singularities together with the subgraph of the curves which are contracted to singular points.

The final results are similar to classical results about classification of non-singular del Pezzo surfaces and use the usual finite root systems. However, the intermediate considerations use the theory of K3 surfaces (especially of K3 surfaces with non-symplectic involutions) and the theory of reflection groups in hyperbolic spaces (especially of reflection groups of hyperbolic lattices).

As an “elementary” application, our results permit one to classify sextics in ℙ^{2} with simple singularities and a component of geometric genus ≥ 2.

Contents

Introduction 2 1. Log del Pezzo surfaces of index ≤ 2 and Smooth Divisor

Theorem. 9 1.1. Basic definitions and notations 9 1.2. Log terminal singularities of index 2 10 1.3. Basic facts about log del Pezzo surfaces 11 1.4. Smooth Divisor Theorem 12 1.5. Reduction to DPN surfaces of elliptic type 15 2. General Theory of DPN surfaces 16 2.1. General remarks 16 2.2. The lattice S, and the main invariants (r, a, δ) (equivalently

(k, g, δ)) of pairs (X, θ) and (Y, C) 17

2.3. Exceptional curves on (X, θ) and Y 19 2.4. Root invariants of pairs (X, θ) 26 2.5. Finding the locus X^{θ} 30 2.6. Conditions for the existence of root invariants 31 2.7. Three types of non-symplectic involutions of K3 surfaces

and DPN surfaces. 34

3. Classification of DPN surfaces of elliptic type 35 3.1. Fundamental chambers of W^{(2,4)}(S) for elliptic type. 35 3.2. Root invariants, and subsystems of roots in Δ^{(4)}(M^{(2)}) for

elliptic case 40

3.3. Classification of non-symplectic involutions (X, θ) of

elliptic type of K3 surfaces 43

3.4. Proof of Theorem 3.6 52 3.5. Classification of DPN surfaces of elliptic type 69 3.6. Application: On classification of plane sextics with simple

singularities 79

4. Classification of log del Pezzo surfaces of index ≤ 2 and applications 83

4.1. Classificaion of log del Pezzo surfaces of index ≤ 2 83 4.2. Application: Minimal projective compactifications of

affine surfaces in ℙ^{2} by relatively minimal log del Pezzo surfaces of index ≤ 2. 87

4.3. Some questions 87 References 89

Introduction

This is an expanded version of our work [AN88], 1988, in Russian. See also [AN89], 1989, for a short exposition of the same results.

A complete algebraic surface Z over ℂ with log terminal singularities is a del Pezzo surface if its anticanonical divisor −K_{Z} is ample. A log terminal singularity in this case is analytically equivalent to a quotient singularity ℂ^{2}/G, where G ⊂ GL(2, ℂ) is a finite subgroup. The index k of z ∈ Z is the minimal positive integer for which the divisor kK_{Z} is a Cartier divisor in a neighborhood of z.

The aim of this work is to classify del Pezzo surfaces with log terminal singularities of index ≤ 2. We shortly call them log del Pezzo surfaces of index ≤ 2.

DEL PEZZO SURFACES OF INDEX ≤ 2 3

Classification of nonsingular del Pezzo surfaces is well known, and they are classical examples of rational surfaces (see, f.e. [Nag60, Man86, MT86]. Classification of log del Pezzo surfaces of index 1, i.e. Gorenstein del Pezzo surfaces, is the subject of a classical work of du Val [DV34] and papers of Nagata [Nag60], Demazure [Dem80], Urabe [Ura83], Bindschadler-Brenton-Drucker [BBD84], Coray-Tsfasman [CT88]. See also survey [MT86].

By classification, we understand a description of the set of exceptional curves on an appropriate (see Section 4 for the precise definition) resolution of singularities σ : Y → Z. Indeed, knowing the dual graph of exceptional curves on Y, we can describe all the ways to obtain Y and Z from the minimal nonsingular surfaces ℙ^{2} and 𝔽_{n}, n = 0, 2, 3 . . .. In particular, we obtain a complete list of 46 minimal, i.e. with Pic Z = ℤ, log del Pezzo surfaces of index ≤ 2 (see Section 4). Of those, 28 are Gorenstein and were previously known (see [BBD84]), and the other 18 are not Gorenstein. Note that minimal log del Pezzo surfaces play a special role in log Minimal Model Program, see f.e. [Sho85, Kaw84, Mor82, Rei83].

The main method for obtaining our classification is to reduce it to a classification of non-symplectic involutions on K3 surfaces (and to K3 surface theory). The main points of the latter are contained in [Nik80.1, Nik79, Nik83, Nik84.1, Nik87].

In Section 1, we show that for log del Pezzo surfaces Z of index ≤ 2 the linear system |−2K_{Z}| contains a nonsingular curve, and that there exists an appropriate resolution of singularities σ : Y → Z for which the linear system | − 2K_{Y} | contains a nonsingular divisor X (i.e. Y is a “good” DPN surface) such that the component of C that belongs to σ^{∗}| − 2K_{Z}| has genus ≥ 2 (i.e. the DPN surface Y is of elliptic type).

In Section 2, following [Nik80.1, Nik79, Nik83, Nik84.1, Nik87], we build a general theory of DPN surfaces Y. Here, we use the fact that the double cover X of Y branched along C is a K3 surface with a non-symplectic involution θ. In this way, the classification of DPN surfaces (Y, C) is equivalent to the classification of K3 surfaces with non-symplectic involution (X, θ). The switch to K3 surfaces is important because it is easy to describe exceptional curves on them and there are powerful tools available: the global Torelli theorem [PS-Sh71] and surjectivity of the period map [Kul77].

In Section 3, we extend this theory to a classification of DPN surfaces Y of elliptic type, i.e. when one of components of C has genus ≥ 2, by describing dual diagrams of exceptional curves on Y. In Section 3.6, we give an application of this classification to a classification of curves D of degree 6 on ℙ^{2} with simple singularities in the case when one of components of D has geometric genus ≥ 2.

In obtaining results of Sections 2 and 3, a big role is played by the arithmetics of quadratic forms and reflection groups in hyperbolic spaces. From this point of view, the success of our classification hinges mainly on the fact that we explicitly describe subgroups generated by all reflections in automorphism groups of certain integral hyperbolic quadratic forms (even hyperbolic 2-elementary lattices of small rank, see Theorem 3.1). These computations are also important by themselves for the arithmetics of quadratic forms.

In Section 4, the results of Sections 1–3 are applied to the classification of log del Pezzo surfaces of index ≤ 2. Also, following [BBD84], we give an application of our classification to a description of rational compactifications of certain affine surfaces. In Section 4.3, we discuss some open questions. In particular, we give a formula for the dimension of moduli of log del Pezzo surfaces of index ≤ 2 and corresponding “good” DPN surfaces of elliptic type.

Below, we try to give an explicit and as elementary as possible exposition of our final results on classification of log del Pezzo surfaces of index ≤ 2.

Let Z be a log del Pezzo surface of index ≤ 2. Its singularities of index 1 are Du Val singularities classified by their minimal resolution of singularities. They are described by Dynkin diagrams A_{k}, D_{m} or E_{n}, with each vertex having weight −2. Singularities of Z of index 2 are singularities K_{n} which have minimal resolutions with dual graphs shown below:

To get the “good” resolution of singularities σ : Y → Z, additionally one has to blow up all points of intersection of components in preimages of singular points K_{n}. Then the “good” resolution of a singular point K_{n} is described by the graph In these graphs every vertex corresponds to an irreducible non-singular rational curve F_{i} with F_{i}^{2} equal to the weight of the vertex. Two vertices are connected by an edge if F_{i} · F_{j} = 1, and are not connected if F_{i} · F_{j} = 0. Thus, the “good” resolution of singularities σ : Y → Z of a log del Pezzo surface of index ≤ 2 consists of minimal resolutions of singular points of index 1 and the “good” resolutions shown on the graphs above of singular points K_{n} of index 2.

Our classification of log del Pezzo surfaces Z of index ≤ 2 and the corresponding DPN surfaces of elliptic type which are “good” resolutions of singularities of Z is contained in Table 3 (see Sect. 3.5).

All cases of Table 3 are labeled by a number 1 ≤ N ≤ 50. For N = 7, 8, 9, 10, 20 we add some letters and get cases: 7a,b, 8a–c, 9a–f, 10a–m, 20a–d. Thus, altogether, Table 3 contains

50 + (2 − 1) + (3 − 1) + (6 − 1) + (12 − 1) + (4 − 1) = 72

cases.

The labels N = 1, …, 50 enumerate the so called main invariants of log del Pezzo surfaces Z. They are triplets (r, a, δ) (equivalently (k = (r − a)/2, g = (22 − r − a)/2, δ)) where r, a, δ are integers: r ≥ 1, a ≥ 0, δ ∈ {0, 1}, g ≥ 2, k ≥ 0. Thus, there exist exactly 50 possibilities for the main invariants (r, a, δ) (equivalently, (k, g, δ)) of log del Pezzo surfaces of index ≤ 2.

The main invariants have a very important geometric meaning. Any log del Pezzo surfaces Z of index ≤ 2 and its “good” resolution of singularities Y are rational. The number

r = rk Pic Y

is the Picard number of Y, i. e. Pic Y = ℤ^{r}. We prove that | − 2K_{Z}| contains a non-singular irreducible curve C_{g} of genus g ≥ 2 which shows the geometric meaning of g. This is equivalent to saying that there is a curve

C = C_{g} + E_{1} + ⋯ + E_{k} ∈ | − 2K_{Y} |,

where E_{i} are exceptional (i. e. irreducible with negative self-intersec-tion) curves on Y with (E_{i})^{2} = −4. (The inequality g ≥ 2 means that Y is of elliptic type). All these curves E_{i} come from the “good” resolution of singularities of Z described above. Thus, the invariant k equals the number of exceptional curves on Y with square −4. All of them are nonsingular and rational. E.g., if k = 0 then Z is Gorenstein and all of its singularities are Du Val. See Sect. 1.

Let us describe the invariant δ ∈ {0, 1}. The components C_{g}, E_{1}, …, E_{k} are disjoint. Since C is divisible by 2 in Pic Y, it defines a double cover π : X → Y ramified in C. Let θ be the involution of the double cover. Then the set of fixed points X^{θ} = C. Here, X is a K3 surface and

(1)  δ = 0 ⟺ X^{θ} ∼ 0 mod 2 in H_{2}(X, ℤ) ⟺

there exist signs (±)_{i} for which

(2)  \frac{1}{4} ∑_{i} [(±)_{i}cl(C^{(i)})] ∈ Pic Y,

where C^{(i)} are all irreducible components (i. e. C_{g}, E_{1}, …, E_{k}) of C.

The connection with K3 surfaces with non-symplectic involutions is the main tool of our classification.

As promised, our classification describes all intersection (or dual) graphs Γ(Y) of exceptional curves on Y and also shows exceptional curves which must be contracted by σ : Y → Z to get the log del Pezzo surface Z of index ≤ 2 from Y. All these graphs can be obtained from graphs Γ in the right column of Table 3 and with the same main invariants (r, a, δ). Let describe this in more detail.

All exceptional curves E on Y are irreducible non-singular and rational. They are of three types:

(1) E^{2} = −4, equivalently E is a component of genus 0 of C ∈ | − 2K_{Y} |. In graphs of Table 3 these correspond to double transparent vertices;

(2) E^{2} = −2. In graphs of Table 3 these correspond to black vertices;

(3) E^{2} = −1 (the 1st kind). In graphs of Table 3 these correspond to simple transparent vertices.

All exceptional curves E_{i} with (E_{i})^{2} = −4, i = 1, … k, together with all exceptional curves F of the 1st kind such that there exist two different curves E_{i}, E_{j}, i ≠ j, with (E_{i})^{2} = (E_{j})^{2} = −4 and F · E_{i} = F · E_{j} = 1 define the logarithmic part Log Γ(Y) ⊂ Γ(Y) of Y. Since F · (−2K_{Y}) = F · C = 2, the curves F are characterized by the property C_{g} · F = 0. The logarithmic part Log Γ(Y) can be easily seen on graphs Γ of Table 3: curves E_{i}, i = 1, …, k, are shown as double transparent, the curves F of the first kind of Log Γ(Y) are shown as simple transparent vertices connected by two edges with (always two) double transparent vertices. This part of Γ is denoted by Log Γ and is also called the logarithmic part of Γ. Thus, we have:

(3)  Log(Γ(Y)) = Log Γ

(with the same main invariants (r, a, δ)). The logarithmic part Log Γ(Y) gives precisely the preimage of singular points of Z of index two.

All exceptional curves E on Y with E^{2} = −2 define the Du Val part Duv Γ(Y) ⊂ Γ(Y) of Γ(Y). Its connected components are Dynkin graphs A_{n}, D_{n} or E_{n} and they correspond to all Du Val singularities of Z. Thus the Du Val part Duv Γ(Y) ⊂ Γ(Y) gives precisely the preimage of all Du Val (i. e. of index one) singular points of Z. The Du Val part Duv Γ of a graphs Γ of Table 3 is defined by all its black vertices. We have:

(4)  D = Duv Γ(Y) ⊂ Duv Γ

(for the same main invariants (r, a, δ)). Any subgraph D of Duv Γ can be taken.

Let us describe the remaining part of Γ(Y). Each graph Γ of Table 3 defines a lattice S in the usual way. It is

S = ( ⨁_{v∈V (Γ)} ℤe_{v}) / Ker

defined by the intersection pairing: e_{v}^{2} = −1, if v is simple transparent, e_{v}^{2} = −2, if v is black, e_{v}^{2} = −4, if v is double transparent, e_{v} · e_{v′} = k if the vertices v ≠ v′ are connected by k edges. “Ker” denotes the kernel of this pairing. We denote E_{v} = e_{v} mod Ker. In all cases except N = 1, 2, 11, the lattice S gives the Picard lattice Pic Y of Y. Thus, Log Γ(Y) = Log Γ and D = Duv Γ(Y) ⊂ Duv Γ define divisor classes E_{v}, v ∈ V (Log Γ(Y) ∪ Duv Γ(Y)), of the corresponding exceptional curves on Y. Each exceptional curve E is evidently defined by its divisor class.

Black vertices v ∈ V (Duv Γ) define roots E_{v} ∈ S with E_{v}^{2} = −2 and define a finite Weyl group W ⊂ O(S) generated by reflections in all these roots. The remaining part

Var Γ(Y) = Γ(Y) − (Log Γ(Y) ∪ Duv Γ(Y))

(it is called the varying part of Γ(Y)) is defined by the varying part

Var Γ = Γ − (Duv Γ ∪ Log Γ)

of the graph Γ of Table 3. Further, we identify exceptional curves v ∈ V (Γ(Y)) with their divisor classes E_{v} ∈ S. We have

(5) V (Var Γ(Y)) = {E ∈ W ({E_{v} | v ∈ V (Var Γ)}) | E_{v} ·D ≥ 0} ⊂ S.

The intersection pairing on S then defines the full graph Γ(Y) of Y. This completes description of possible graphs Γ(Y) of exceptional curves of log del Pezzo surfaces Z of index ≤ 2.

Thus, to find all possible graphs Γ(Y) of exceptional curves of σ : Y → Z, one has to choose one of the graphs Γ of Table 3 (this also defines main invariants of Y), then one has to choose a subgraph D = Duv Γ(Y) ⊂ Duv Γ. Then Γ(Y) consists of D, Log Γ(Y) = Log Γ and the remaining part Var Γ(Y) defined by (5). See Theorems 3.16, 3.17 and 4.1.

We note two important opposite cases.

Extremal case. This is the case when D = Duv Γ(Y) = Duv Γ. Then Γ(Y) = Γ is completely calculated in Table 3. This case is called extremal and gives log del Pezzo surfaces Z with Du Val singularities of the highest rank, respectively rk Pic Z = r − #V (Log Γ(Y)) − #V (Duv Γ(Y)) is minimal for the fixed main invariants. In particular, this case delivers all cases of minimal log del Pezzo surfaces of index ≤ 2 when rk Pic Z = 1. See Theorems 3.16, 4.2.

No Du Val singularities. This is the case when D = Duv Γ(Y) = ∅. Equivalently, all singularities of Z have index 2, if they exist. Then Γ(Y) = Log Γ ∪ Var Γ(Y) where

V (Var Γ(Y)) = W ({E_{v} | v ∈ V (Var Γ)}) .

Here, all the multiple cases 7a,b, 8a–c, 9a–f, 10a–m, 20a–d give the same surfaces, and one can always take cases 7a, 8a, 9a, 10a, 20a for the main invariants. This case is very similar to and includes the classical case of non-singular del Pezzo surfaces corresponding to the cases 1— 10. See Theorem 4.4 about this (without Du Val singularities) case. Log del Pezzo surfaces of this case are defined by their main invariants (r, a, δ) up to deformation. Du Val parts Duv Γ of graphs Γ of Table 3 can be considered as analogs to root systems (or Dynkin diagrams) which one usually corresponds to non-singular del Pezzo surfaces. In cases 7 — 10, 20, one should take graphs Γ of cases 7a — 10a, 20a.

The Root invariant. It is possible that two different subgraphs D ⊂ Γ, D ⊂ Γ′ of graphs of Table 3 (with the same main invariants (r, a, δ)) give isomorphic graphs Γ(Y) and Γ(Y′) for the corresponding “good” resolutions, and then they give similar log del Pezzo surfaces Z and Z′ of index ≤ 2, according to our classification. The root invariant

(6)  ([D], ξ)

gives the necessary and sufficient condition when thisc happens.

To define the root invariant (6), we first remark that the main invariants (r, a, δ) define a unique hyperbolic (i. e. with one positive square) even 2-elementary lattice S with these invariants. Here r = rk S, S^{∗}/S ≅ (ℤ/2)^{a}, and δ = 0, if and only if (x^{∗})^{2} ∈ ℤ for any x ∈ S^{∗}. In (6), the [D] is the root lattice generated by D, and ξ : [D]/2[D] → S^{∗}/S a homomorphism preserving finite forms (x^{2})/2 mod 2, x ∈ [D], and y^{2} mod 2, y ∈ S^{∗}. The construction of the root invariant (6) uses the double cover π : X → Y by K3 surface X (see above) with the non-symplectic involution θ of the double cover. Then S = H^{2}(X, ℤ)^{θ} is the sublattice where θ^{∗} is identical. The root invariant (6) is considered up to automorphisms of S and the lattice D. See Sect. 2.4 about this construction and a very easy criterion (the kernel H of ξ is almost equivalent to ξ) about isomorphism of root invariants. The root invariant was first introduced and used in [Nik84.1] and [Nik87].

In practice, to calculate the root invariant for log del Pezzo surfaces of index ≤ 2, one should just go from the graphs Γ of Table 3 to the equivalent graphs Γ(P (M^{(2,4)})) or Γ(P (X)_{+}) of Tables 1 or 2 of exceptional curves for the K3 pairs (X, θ) (see Sects 3.2, 3.5).

Thus, two Du Val subgraphs D ⊂ Γ, D ⊂ Γ′ of graphs of Table 3 give isomorphic full graphs Γ(Y) and Γ(Y′) of their log del Pezzo surfaces, if and only if their root invariants (6) are isomorphic. Moreover, we constantly use the root invariant to prove existence of the corresponding K3 pairs (X, θ) and log del Pezzo surfaces Z. The main invariants (r, a, δ) and the root invariants (6) are the main our tools in classification. They are equivalent to the full graphs Γ(Y) of exceptional curves on Y, but they are much more convenient to work with. For non-singular del Pezzo surfaces and log del Pezzo surfaces of index ≤ 2 without Du Val singularities the root invariant is zero. It is why, in this cases, we have so simple classification above.

Acknowledgment. We are grateful to V.A. Iskovskikh for useful discussions.

1. Log del Pezzo surfaces of index ≤ 2 and Smooth Divisor Theorem.

1.1. Basic definitions and notations. Let X be a normal algebraic surface over the field ℂ, and K_{X} be a canonical Weil divisor on it. The surface X is called ℚ-Gorenstein if a certain positive multiple of K_{X} is Cartier, and ℚ-factorial if this is true for any Weil divisor D. These properties are local: one has to require all singularities to be ℚ-Gorenstein, resp. ℚ-factorial.

Let us denote by Div(X), Z^{1}(X) respectively the groups of Weil, resp. Cartier, divisors on X. Assume that X is ℚ-factorial. Then the groups Div(X) ⊗ ℚ, resp. Z^{1}(X) ⊗ ℚ of ℚ-Cartier divisors and ℚ-Weil divisors coincide. The intersection form defines natural pairings

Div(X) ⊗ ℚ × Div(X) ⊗ ℚ → ℚ,
Div(X) ⊗ ℝ × Div(X) ⊗ ℝ → ℝ.

Quotient groups modulo kernels of these pairings are denoted N_{ℚ}(X) and N_{ℝ}(X) respectively; if the surface X is projective, they are finitedimensional linear spaces. The Mori cone is a convex cone \overline{NE}(X) in N_{ℝ}(X), the closure of the cone generated by classes of effective curves.

Let D be a ℚ-Cartier divisor on X. We will say that D is ample if some positive multiple is an ample Cartier divisor in the usual sense. By Kleiman’s criterion, for this to hold it is necessary and sufficient that D, as a liner function on N_{ℝ}(X) were strictly positive on \overline{NE}(X)−\{0\}.

One say that the surface X has only log terminal singularities if it is ℚ-Gorenstein and for a resolution of singularities π : Y → X, in a natural formula K_{Y} = π^{∗}K_{X} +∑ α_{i}F_{i}, where F_{i} are simple divisors and α_{i} ∈ ℚ, one has α_{i} > −1. The least common multiple of denominators of α_{i} is called the index of X.

It is known that two-dimensional log terminal singularities in characteristic zero are exactly the quotient singularities. They are listed in [Bri68]. They are all rational and ℚ-factorial. We can now formulate the following

Definition 1.1. A normal complete surface X is called a log del Pezzo surface of index ≤ k if it has only log terminal singularities of index ≤ k and the anticanonical divisor −K_{X} is ample. Equivalently, X is a del Pezzo surface (i. e. −K_{X} is ample) with log-terminal singularities of index ≤ k.

We will use the following notations. If D is a ℚ-Weil divisor, D = ∑ c_{i}C_{i}, c_{i} ∈ ℚ, then ⌜D⌝ will denote the ceiling ∑⌜c_{i}⌝D_{i}, and {D} = ∑{c_{i}}C_{i} its fractional part. A divisor D is nef if for any curve C, D · C ≥ 0; D is big if D^{2} > 0.

Below we will frequently use the following

Theorem 1.2 (Kawamata, Viehweg, [Kaw82, Vie82]). Let Y be a smooth surface and let D be a ℚ-divisor on Y such that

(1) supp{D} is a divisor with normal crossings;

(2) D is big and nef. Then H^{i}(K_{Y} + ⌜D⌝) = 0 for i > 0.

1.2. Log terminal singularities of index 2. Let (X, p) be a twodimensional log terminal singularity of index ≤ 2, and π : X̃ → X be its minimal resolution. We have KX̃ = π^{∗}K_{X} + ∑ α_{i}F_{i}, where −1 < α_{i} ≤ 0 and F_{i}^{2} ≤ −2. Therefore, for each i one has α_{i} = −1/2 or 0. One can rewrite the set of equations KX̃ · F_{i} = −F_{i}^{2} − 2 in a matrix form:

M · (α_{1}, …, α_{n})^{t} = (−F_{1}^{2} − 2, …, −F_{n}^{2} − 2)^{t},

where M = (F_{i} · F_{j}) is the intersection matrix. M is nondegenerate and negative definite.

Let us give some easy consequences of this formula.

(1) If for some i_{0}, α_{i_{0}} = 0 then all α_{i} = 0, and the singularity (X, p) is Du Val, of type A_{n}, D_{n}, E_{6}, E_{7} or E_{8}.

(2) If all α_{i} = −1/2 then we get the following list of singularities:

In these graphs every curve F_{i} corresponds to a vertex with weight F_{i}^{2}, two vertices are connected by an edge if F_{i} · F_{j} = 1 and are not connected if F_{i} · F_{j} = 0.

1.3. Basic facts about log del Pezzo surfaces.

Lemma 1.3. All log del Pezzo surfaces X are rational.

Proof. Let π : X̃ → X be the minimal resolution of singularities, KX̃ = π^{∗}K_{X} + ∑ α_{i}F_{i}, −1 < α_{i} ≤ 0. Then

\begin{matrix}h^{1}(OX̃) = & h^{1}(KX̃ + (−KX̃)) = h^{1}(KX̃ + ⌜−π^{∗}K_{X} + ∑(−α_{i})F_{i}⌝) = \\ h^{1}(KX̃ + ⌜−π^{∗}K_{X}⌝) = 0 \end{matrix} (7)

by Kawamata-Viehweg theorem.

Also, h^{0}(nKX̃) = 0 for any positive integer n since −π_{∗}KX̃ = −K_{X} is an effective nonzero ℚ-Weil divisor. Therefore, by Castelnuovo criterion the surface X̃, and hence also X, are rational. □

Lemma 1.4. In the above notations, if X ≠ ℙ^{2} or 𝔽_{n} then the Mori cone of the surface X̃ is generated by the curves F_{i} and exceptional curves of the 1st kind. The number of these curves is finite. There are no other curves with negative self-intersection number on X̃.

Proof. Let us show that on the surface X̃ there exists a ℚ-divisor Δ with Δ ≥ 0, [Δ] = 0 and such that the divisor −(KX̃ + Δ) is ample.

Choose Z = ∑ β_{i}F_{i} so that Z · F_{i} > 0. Since the matrix (F_{i} · F_{j}) is negative definite and F_{i} · F_{j} ≥ 0 for i ≠ j, one has β_{i} < 0. Let us show that for a small 0 < ϵ ≪ 1 the divisor T = −π^{∗}K_{X} +ϵZ is ample. First of all T^{2} = K_{X}^{2} + ϵ^{2}Z^{2} > 0 if 0 < ϵ < \sqrt{(K_{X}^{2})/(−Z^{2})}. We shall assume that. Let C be an irreducible curve on X̃. Let us show that for small ϵ we have C · T > 0.

If C^{2} ≥ 0, this follows from the fact that the intersection form on N_{ℝ}(X) is hyperbolic.

If C = F_{i} then F_{i} · T = ϵF_{i} · Z = 0.

If C^{2} < 0 and C ≠ F_{i} then

C · KX̃ = C · (π^{∗}KX̃ + ∑ α_{i}F_{i}) < 0.

On the other hand, p_{a}(C) = \frac{C^{2} + C · KX̃}{2} + 1 ≥ 0. So, C^{2} < 0 and

C · KX̃ < 0 implies that C^{2} = −1, p_{a}(C) = 0, i.e. C is an exceptional curve of the 1st kind. The conditions nC · π^{∗}K_{X} ∈ ℤ (where n is the index of X), C^{2} = −1 and 0 < −C · π^{∗}K_{X} = 1 + ∑ α_{i}F_{i} · C ≤ 1 imply that the number of exceptional curves of the 1st kind is finite. Hence, C(−π^{∗}K_{X}) > 0 implies that for sufficiently small 0 < ϵ ≪ 1 we have T · C = (−π^{∗}K_{X} + ϵZ) · C > 0. Therefore the divisor T = −π^{∗}K_{X} + ϵZ is ample. But −π^{∗}K_{X} + ϵZ = − (KX̃ + ∑(−α_{i} − ϵβ_{i})F_{i}). Therefore, Δ ≥ 0, and for sufficiently small ϵ, we have [Δ] = 0 because α_{i} > −1.

Now, by [Kaw84, Thm.4.5], NE(X) = ∑ R_{j}, where R_{j} are “good extremal rays”. The curves F_{i} and exceptional curves of the 1st kind are obviously extremal rays. On the other hand, let R_{j} be a “good extremal ray”, generated by an irreducible curve C. If C ∉ {F_{1}, …, F_{k}} then C · K_{X} = C · (π^{∗}K_{X} + ∑ α_{i}F_{i}) < 0. Hence, by [Mor82] the curve C is an exceptional curve of the 1st kind, if X is different from ℙ^{2} or 𝔽_{n}. □

1.4. Smooth Divisor Theorem.

Theorem 1.5. Let X be a log del Pezzo surface of index ≤ 2. Then the linear system | − 2K_{X}| is nonempty, has no fixed components and contains a nonsingular element D ∈ | − 2K_{X}|.

Proof. Let π : X̃ → X be the minimal resolution of singularities. It is sufficient to prove the statement for the linear system | − π^{∗}2K_{X}| on X̃. We have 2KX̃ = π^{∗}2K_{X} − ∑ a_{i}F_{i}, all a_{i} = 0 or 1 (a_{i} = −2α_{i}).

1. Nonemptiness.

−π^{∗}2K_{X} = KX̃ + (−3KX̃ − ∑ a_{i}F_{i}) = KX̃ + ⌜D⌝, where D = \frac{3}{2}(−2KX̃ − ∑ a_{i}F_{i}) is big, nef, since D = \frac{3}{2}(−π^{∗}2K_{X}).

Hence, by Kawamata-Viehweg theorem1.2, H^{i}(−π^{∗}2K_{X}) = 0 for i > 0 and h^{0}(−π^{∗}2K_{X}) = χ(−π^{∗}2K_{X}) = 3K_{X}^{2} + 1 > 0.

2. No fixed components. Let E be the fixed part, so that |−π^{∗}2K_{X} − E| is a movable linear system. Then

h^{0}(−π^{∗}2K_{X}) = h^{0}(−π^{∗}2K_{X} − E),

−π^{∗}2K_{X} − E = KX̃ + (−3KX̃ − ∑ a_{i}F_{i} − E) = KX̃ + ⌜D⌝,

D = \frac{3}{2}(−2KX̃ − ∑ a_{i}F_{i}) − E = (−π^{∗}2K_{X} − E) + (−π^{∗}K_{X}) =

D_{1} + D_{2} .

Here, D_{1} is movable and D_{2} is big, nef, hence D_{1} + D_{2} is big, nef. By Kawamata-Viehweg Theorem 1.2,

h^{i}(−π^{∗}2K_{X} − E) = 0, i > 0,
χ(−π^{∗}2K_{X}) = χ(−π^{∗}2K_{X} − E),
(8) 2χ(−π^{∗}2K_{X}) − 2χ(−π^{∗}2K_{X} − E) = E · (−2π^{∗}2K_{X} − KX̃ − E).

Let us show that this expression (8) is not equal to zero. Suppose

−π^{∗}K_{X} · (KX̃ + E) = π^{∗}K_{X} · E − K_{X}^{2} < 0.

Then the divisor KX̃ + E cannot be effective. Therefore,

χ(−E) = h^{0}(−E) − h^{1}(−E) + h^{0}(KX̃ + E) ≤ 0.

Hence, E · (KX̃ + E) = 2χ(−E) − 2 < 0, and the expression (8) is strictly positive. So, we can assume that −π^{∗}K_{X} · E ≥ K_{X}^{2}. Let us write

E = β(−π^{∗}K_{X}) + F, F ∈ (π^{∗}K_{X})^{⊥}

in N_{ℚ}(X̃). One has β ≤ 2 since −π^{∗}K_{X} · (−π^{∗}2K_{X} − E) ≥ 0. Then

E(−2π^{∗}2K_{X} − KX̃ − E) = (5 − β)βK_{X}^{2} − F · (∑ α_{i}F_{i} + F).

The first term in this sum is ≥ 3 since βK_{X}^{2} = −π^{∗}K_{X} · E ≥ K_{X}^{2} and K_{X}^{2} = χ(π^{∗}2K_{X})−1 is a positive integer. The second term achieves the

minimum for F = −\frac{1}{2} ∑ α_{i}F_{i} and equals −\frac{m}{4}, where m is the number of non-Du Val singularities. Therefore, all that remains to be shown is that surface X has less than 12 non-Du Val singularities. This will be obvious from the following Proposition.

Proposition 1.6. Let X be a log del Pezzo surface of index ≤ 2 and π : X̃ → X be its minimal resolution of singularities. Then K2 ; X̃ ≥ 0.

Proof. One has KX̃ = π^{∗}K_{X} +∑ α_{i}F_{i}. Let us set \overline{K} = π^{∗}K_{X} −∑ α_{i}F_{i}. Let us show that −\overline{K} is nef. By Lemma 1.4, one has to show that −\overline{K} · F_{i} ≥ 0 and −\overline{K} · C ≥ 0 if C is an exceptional curve of the 1st kind. Indeed, −\overline{K} · F_{i} = KX̃ · F_{i} = −F_{i}^{2} − 2 ≥ 0 since resolution π is minimal. Next,

−π^{∗}K_{X} · C = −KX̃ · C + ∑ α_{i}F_{i} · C = 1 + ∑ α_{i}F_{i} · C > 0.

Since this number is half-integer,

−\overline{K} · C = 1 + 2 ∑ α_{i}F_{i} · C ≥ 0.

So, −\overline{K} is nef and K2 ; X̃ = \overline{K}^{2} ≥ 0. Finally, if X̃ = ℙ^{2} or 𝔽_{n} then K2 ; X̃ = 9 or 8 respectively. □

By Lemma 1.3, surface X̃ is rational. By Noether formula, rk Pic X̃ ≤ 10; hence in any case X has no more than 10 singular points.

3. Existence of a smooth element. Assume that all divisors in the linear system | − π^{∗}2K_{X}| are singular. Then there exists a fixed pint P, and for a general element D ∈ | − π^{∗}2K_{X}| the multiplicity of D at point P is k ≥ 2. This point does not lie on F_{i} since −π^{∗}2K_{X} · F_{i} = 0. Let ϵ : Y → X̃ be the blowup at P, and f = πϵ : Y → X, L be the exceptional divisor of ϵ. We have: h^{0}(−f^{∗}2K_{X}) = h^{0}(−f^{∗}2K_{X} − L), the linear system | − f^{∗}2K_{X} − kL| is movable, and

2K_{Y} = f^{∗}2K_{X} − ∑ a_{i}F_{i} + 2L,

−f^{∗}2K_{X} = K_{Y} + (−3K_{Y} − ∑ a_{i}F_{i} + 2L) = K_{Y} + ⌜D⌝,

D = \frac{3}{2}(−2K_{Y} − ∑ a_{i}F_{i} + L) = \frac{3}{2}(−f^{∗}2K_{X} − L) =

\frac{3}{2}[(−f^{∗}2K_{X} − kL) + (k − 1)L].

The divisor D is nef since for any irreducible curve C ≠ L, C · D ≥ 0, and also D · L = 1. It is big since (−f^{∗}2K_{X} − L)^{2} = 4K_{X}^{2} − 1 > 0. Now,

−f^{∗}2K_{X} − L = K_{Y} + (−3K_{Y} − ∑ a_{i}F_{i} + L) = K_{Y} + ⌜D⌝,

D = \frac{3}{2}(−2K_{Y} − ∑ a_{i}F_{i} + \frac{2}{3}L) = \frac{3}{2}[(−f^{∗}2K_{X} − kL) + (k − \frac{4}{3})L].

This divisor D is nef since for any irreducible curve C ≠ L, C · D ≥ 0, and also D · L = 2; D is big since

(−f^{∗}2K_{X} − \frac{4}{3}L)^{2} = 4K_{X}^{2} − \frac{16}{9} > 0.

Now, again by Kawamata-Viehweg theorem, h^{i}(−f^{∗}2K_{X}) = h^{i}(−f^{∗}2K_{X} − L) = 0 for i > 0, and one must have χ(−f^{∗}2K_{X}) = χ(−f^{∗}2K_{X} − L). But

χ(−f^{∗}2K_{X}) − χ(−f^{∗}2K_{X} − L) = \frac{1}{2}L · (−2f^{∗}2K_{X} − K_{X} − L) =
1 + L · (−f^{∗}2K_{X}) > 0.

The contradiction thus obtained completes the proof of the theorem. □

Remark 1.7. In the same way, parts 1 and 3 are proved for a log del Pezzo surface of arbitrary index n and the linear system −π^{∗}nK_{X}. Part 2 is easy to prove under the assumption that π(E) passes only through Du Val singularities.

1.5. Reduction to DPN surfaces of elliptic type. Let X be a log del Pezzo surface of index ≤ 2. Consider a resolution of singularities f : Y → X for which every Du Val singularity is resolved by inserting a usual tree of (−2)-curves, and the singularity K_{n} – by the following chain

(9)

The later resolution is obtained by blowing up all intersection points of exceptional curves on the minimal resolution of K_{n} points, see their diagrams in Section 1.2. In contrast to the minimal resolution, we will call this the “good” resolution of singularities. Consider a smooth element C_{g} ∈ | − 2K_{X}|. It does not pass through singularities of the surface X. If we identify the curve C_{g} with its image under the morphism f, then it is easy to see from the formulas of Section 1.2 that −f^{∗}2K_{X} coincides with C_{g}, and −2K_{Y} – with the disjoint union of C_{g} and curves in the above diagrams which have self-intersection −4. Moreover, it is easy to compute the genus of the curve C_{g}, and it equals g = K_{X}^{2} + 1 ≥ 2. This shows that the surface Y is a “good” DPN surface of elliptic type in the sense of the next Section (see Sects 2.1 and 2.7).

Vice versa, the results of Sects 2 and 3 will imply (see Sect. 4) that a “good” DPN surface Y of elliptic type admits a unique contraction of exceptional curves f : Y → X to a log del Pezzo surface of index ≤ 2.

In this way, classification of log del Pezzo surfaces of index ≤ 2 is reduced to classification of “good” DPN surfaces of elliptic type.

2. General Theory of DPN surfaces

2.1. General remarks. As it was shown in Section 1, a description of log del Pezzo surfaces of index ≤ 2 is reduced to a description of rational surfaces Y containing a nonsingular curve C ∈ | − 2K_{Y} | and a certain configuration of exceptional curves. Such surfaces Y and exceptional curves on them were studied in the papers [Nik79, Nik83, Nik84.1, Nik87] of the second author. They are one of possible generalizations of del Pezzo surfaces.

Many other generalizations of del Pezzo surfaces were proposed, see f.e. [Dem80, Har85.1, Har85.2, Loo81], and most authors call their surfaces “generalized del Pezzo surfaces”. Therefore, we decided following [Nik87] to call our generalization DPN surfaces.

Definition 2.1. A nonsingular projective algebraic surface Y over ℂ is called a DPN surface if its irregularity q(Y) = 0, K_{Y} ≠ 0 and there exists an effective divisor C ∈ | − 2K_{Y} | with only simple rational, i.e. A, D, E-singularities. Such a pair (Y, C) is called a DPN pair. A DPN surface Y is called “good” if there exists a nonsingular divisor C ∈ |−2K_{Y} |; in this case the pair (Y, C) is called “good” or nonsingular DPN pair.

The classification of surfaces implies that if C = ∅ then Y is an Enriques surface. If C ≠ ∅ then Y is rational.

Using the well-known properties of blowups, the following results are easy to prove. Let (Y, C) be a DPN pair, E ⊂ Y be an exceptional curve of the 1st kind on Y and σ : Y → Y′ the contraction of E. Then (Y′, σ(C)) is also a DPN pair. In this way, by contracting exceptional curves of the 1st kind, one can always arrive at a DPN pair (Y′, C′) where Y′ is a relatively minimal rational surface. In this case, the only possibilities for Y′ are ℙ^{2}, 𝔽_{0}, 𝔽_{2}, 𝔽_{3} and 𝔽_{4}, since only for them | − 2K_{Y′}| contains a reduced divisor. If Y′ = ℙ^{2} then C′ is a curve of degree 6; if Y′ = 𝔽_{0} = ℙ^{1} × ℙ^{1} then C′ is a curve of bidegree (4, 4); if Y′ = 𝔽_{2} then C′ ∈ |8f + 4s_{2}|; if Y′ = 𝔽_{3} then D′ = C_{1} + s_{3}, where C_{1} ∈ |10f + 3s_{3}|; if Y′ = 𝔽_{4} then C′ = D_{1} + s_{4}, where D_{1} ∈ |12f + 3s_{4}|.

Here, the linear system |f| is a pencil of rational curves on surface 𝔽_{n} with a section s_{n}, s_{n}^{2} = −n. Vice versa, if (Y′, C′) is a DPN pair, P is a singular point of C′ and σ : Y → Y′ is a blowup of P with an exceptional (−1)-curve E then (Y, C) is a DPN pair, where

C = \{ \begin{matrix}σ^{−1}(C′) & if P has multiplicity 2 on C′ \\ σ^{−1}(C′) + E & if P has multiplicity 3 on C′ \end{matrix}

In this way, by blowups, from an arbitrary DPN pair (Y′, C′) one can always pass to a “good” DPN pair (Y, C), i.e. with a nonsingular C. A description of arbitrary DPN pairs and surfaces is thus reduced to a description of nonsingular DPN pairs and “good” DPN surfaces and exceptional curves on Y, where a curve E ⊂ Y is called exceptional if E is irreducible and E^{2} < 0.

Let (Y, C) be a nonsingular DPN pair. Since C ∈ | − 2K_{Y} |, there exists a double cover π : X → Y defined by −K_{Y}, branched along C. From the classification of surfaces, it follows that X is a K3 surface, and the involution θ of the double cover π has the following property: θ^{∗}(ω_{X}) = −ω_{X}, where ω_{X} is a regular two-dimensional nowhere vanishing differential form on X. (Recall that K_{X} = 0 and q(X) = 0.) Such an involution θ on X is called non-symplectic. Vice versa, if (X, θ) is a K3 surface with a non-symplectic involution then the set X^{θ} of fixed points of the involution is nonsingular, and the pair (Y = X/{1, θ}, C = π(X^{θ})) is a “good” DPN pair where π : X → Y is the quotient morphism. Thus, a description of “good” DPN pairs (Y, C) is reduced to a description of K3 surfaces with a non-symplectic involution (X, θ).

2.2. The lattice S, and the main invariants (r, a, δ) (equivalently (k, g, δ)) of pairs (X, θ) and (Y, C). Let S_{X} ⊂ H^{2}(X, ℤ) be the Picard lattice of a K3 surface X with a non-symplectic involution θ. Set

(S_{X})_{±} = {x ∈ S_{X} | θ^{∗}(x) = ±x} and
H^{2}(X, ℤ)_{±} = {x ∈ H^{2}(X, ℤ) | θ^{∗}(x) = ±x}.

One has (S_{X})_{+} = H^{2}(X, ℤ)_{+}. A fundamental invariant of the pair (X, θ) and the corresponding DPN pair (Y, C) is the lattice

S = (S_{X})_{+} = H^{2}(X, ℤ)_{+}.

This is an even 2-elementary (see below) hyperbolic (it follows that X is algebraic) lattice. Up to an isomorphism, it is determined by a triple of invariants (r, a, δ). Here r = rk S is the rank of S. Here a ∈ ℕ is defined by the condition S^{∗}/S ≃ (ℤ/2ℤ)^{a} where S^{∗} = Hom(S, ℤ) is the dual lattice (such lattices S are called 2-elementary). The invariant δ takes values in {0, 1}. The δ = 0 ⟺ (x^{∗})^{2} ∈ ℤ for every x^{∗} ∈ S^{∗} ⟺ the discriminant quadratic form of S

q_{S} : 𝔄 = S^{∗}/S → ℚ/2ℤ, q_{S}(x^{∗} + S) = (x^{∗})^{2} + 2ℤ

is even: it takes values in (ℤ/2ℤ). See [Nik80.2] about discriminant forms of lattices. The triple (r, a, δ) admits an interpretation in terms of X^{θ} = C. If (r, a, δ) ≠ (10, 8, 0) or (10, 10, 0) then

X^{θ} = C = C_{g} + E_{1} + ⋯ + E_{k},

where C_{g} is a nonsingular irreducible curve of genus g, and E_{1}, … E_{k} are nonsingular irreducible rational curves, the curves are disjoint to each other,

g = (22 − r − a)/2, k = (r − a)/2

(we shall formally use the same formulae for g and k even in cases (r, a, δ) = (10, 8, 0) or (10, 10, 0)). If (r, a, δ) = (10, 8, 0) then X^{θ} ≃ C = C_{1}^{(1)} + C_{1}^{(2)}, where C_{1}^{(i)} are elliptic (genus 1) curves. If (r, a, δ) = (10, 10, 0) then X^{θ} ≃ C = ∅, i.e. in this case Y is an Enriques surface. One has (10) δ = 0 ⟺ X^{θ} ∼ 0 mod 2 in S_{X} (equivalently in H_{2}(X, ℤ)) ⟺

there exist signs (±)_{i} for which

(11)  \frac{1}{4} ∑_{i} [(±)_{i}cl(C^{(i)})] ∈ S_{Y} ,

where C^{(i)} go over all irreducible components of C. Signs (±)_{i} for δ = 0 are defined uniquely up to a simultaneous change and define the other (with respect to the complex) orientation of components of C. All possibilities for (r, a, δ) (equivalently, (k = (r − a)/2, g = (22 − r − a)/2, δ)) are known and are shown on Figure 1.

The invariants (r, a, δ) (equivalently (k, g, δ)) of K3 surfaces with non-symplectic involutions, and the corresponding DPN pairs and DPN surfaces play a crucial role in our classification, and henceforth we call them The Main Invariants. Pairs (X, θ) and DPN pairs (Y, C) with given main invariants (r, a, δ) form connected smooth families M_{(r,a,δ)} of surfaces without polarization. Here, dim M_{(r,a,δ)} = 20 − r. For a general point m ∈ M_{(r,a,δ)} the corresponding surface X = X_{m} has S = (S_{X})_{+} = S_{X}, involution θ is canonically defined by the surface X itself as an automorphism θ ∈ Aut X which acts identically on S_{X} and as multiplication by (−1) on ω_{X} (or T_{X} = (S_{X})_{H^{2}(X,ℤ)}^{⊥}).

The results described in this section were obtained in papers [Nik80.1, Nik80.2, Nik83] (see also [Nik87]).

Figure 1. All possible main invariants (r, a, δ)

2.3. Exceptional curves on (X, θ) and Y. A description of exceptional curves on Y can also be reduced to a description of K3 surfaces X with a non-symplectic involution θ considered above.

Let X be any algebraic K3 surface and F ⊂ X be an exceptional curve on X, i.e. F^{2} < 0 and F is irreducible. Then F^{2} = −2 and F is a nonsingular rational curve on X. The subset P (X) ⊂ S_{X} of divisor classes of all exceptional curves F on X has the following description. Let

V (X) = {x ∈ S_{X} ⊗ ℝ | x^{2} > 0}.

Since the lattice S_{X} is hyperbolic (i.e. has signature (1, t)), the cone V (X) has two convex halves. One of these halves V^{+}(X) is distinguished by the fact that it contains the ray ℝ^{+}h of a polarization h on X. It defines the hyperbolic space

L(X) = V^{+}(X)/ℝ^{+}.

The elements of this space are rays ℝ^{+}x, where x ∈ S_{X} ⊗ ℝ, x^{2} > 0 and x · h > 0. Each element β ∈ S_{X} ⊗ ℝ with square β^{2} < 0 defines a half-space

H_{β}^{+} = {ℝ^{+}x ∈ L(X) | β · x ≥ 0},

so that β is perpendicular to the bounding hyperplane

H_{β} = {ℝ^{+}x ∈ L(X) | β · x = 0},

and faces outward. The set

M(X) = ⋂_{f∈S_{X} ,f^{2}=−2 f is effective} H_{δ}^{+} = ⋂_{f∈P (X)} H_{δ}^{+}

is a locally finite convex polytope in L(X). The set P (M(X)) of vectors with square −2, perpendicular to the facets of M(X) and directed outward, is exactly the set P (X) of divisor classes of exceptional curves on X. Moreover, M(X) admits a description in terms of groups. Let O′(S_{X}) be the subgroup of index two of the full automorphism group O(S_{X}) of the lattice S_{X} which consists of automorphisms which preserve the half-cone V^{+}(S_{X}). Let W^{(2)}(S_{X}) ⊂ O′(S_{X}) be the subgroup of O′(S_{X}) generated by reflections s_{f} with respect to all elements f ∈ S_{X} with square (−2):

S_{f}(x) = x + (x · f)f, x ∈ S_{X}.

The action of the group W^{(2)}(S_{X}), as well as O′(S_{X}), on L(X) is discrete. W^{(2)}(S_{X}) is the group generated by reflections in hyperplanes H_{f}, f ∈ S_{X} and f^{2} = −2. The set M(X) is a chamber for this group, i.e. W^{2}(S_{X})(M(X)) defines a decomposition of L(X) into polytopes which are congruent to M(X), and W^{2}(S_{X}) acts transitively on this decomposition (cf. [PS-Sh71, Vin85]). The fundamental polytope M(X) is distinguished from other polytopes by the fact that it contains the ray ℝ^{+}h of polarization, since

M(X) = {x ∈ V^{+}(X) | x · f ≥ 0, ∀f ∈ P (X)}.

Now let (X, θ) be a K3 surface with a nonsymplectic involution and (Y = X/{1, θ}, C = π(X^{θ})) be the corresponding DPN pair. If E ⊂ Y is an exceptional curve then the curve F = π^{−1}(E) is either an irreducible curve on X with negative square, or F = F_{1} + θ(F_{1}), where

F_{1}^{2} + (θ(F_{1}))^{2} + 2F_{1} · θ(F_{1}) = 2E^{2} < 0.

The curves F_{1} and θ(F_{1}) are irreducible and have negative square (i.e. equal to (−2)). Using this, in an obvious way we get the following four possibilities for E and F (see Fig. 2 )

I E^{2} = −4, E is a component of C; resp. F is a component of X^{θ}, and F^{2} = −2.

IIa E^{2} = −1, E · C = 2 and E intersects C transversally at two

points; resp. F is irreducible and F^{2} = −2. IIb E^{2} = −1, E · C = 2 and E is tangent to C at one point; resp.

F = F_{1} + θ(F_{1}), where (F_{1})^{2} = −2 and F_{1} · θ(F_{1}) = 1. III E^{2} = −2, E ∩ C = ∅; resp. F = F_{1} + θ(F_{1}), where (F_{1})^{2} = −2

and F_{1} · θ(F_{1}) = 0.

Figure 2. Pictures of exceptional curves.

Let P (Y) ⊂ S_{Y} be the subset of divisor classes of all exceptional curves E on Y (here S_{Y} is the Picard lattice of Y modulo torsion, if Y is Enriques surface), and P (X)_{+} ⊂ S = (S_{X})_{+} be the set of divisor classes of all F = π^{−1}(E). We call them exceptional classes of the pair (X, θ). By what we said above, P (Y) and P (X)_{+} are divided into subsets:

P (Y) = P (Y)_{I} ∐ P (Y)_{IIa} ∐ P (Y)_{IIb} ∐ P (Y)_{III}.
P (X)_{+} = P (X)_{+I} ∐ P (X)_{+IIa} ∐ P (X)_{+IIb} ∐ P (X)_{+III}.

It turns out that P (X)_{+} and, respectively P (Y) admit a description analogous to the description of P (X) for the set of exceptional curves on the surface X. Since the lattice S = (S_{X})_{+} is hyperbolic, and S ⊂ S_{X}, we have embedding of cones

V (S) ⊂ V (S_{X}), V^{+}(S) ⊂ V^{+}(S_{X})

and hyperbolic spaces L(S) ⊂ L(X).

The set L(S) contains the polarization ray ℝ^{+}(h+θ^{∗}h) of surface X, therefore, M(S)_{+}^{(2,4)} = M(X)∩L(S) is non-degenerate (i.e. containing a nonempty open subset of L(S)) convex locally finite polytope in L(S). Since S is even, it is easy to see that P (X)_{+} is precisely the set of primitive elements of S, perpendicular to facets of M(X)_{+}^{(2,4)} and directed outward. One has

P^{(2)}(X)_{+} =^{Def} {f ∈ P (X)_{+} | f^{2} = −2}
= P (X)_{+I} ∐ P (X)_{+IIa} ∐ P (X)_{+IIb},
P^{(4)}(X)_{+} =^{Def} {f ∈ P (X)_{+} | f^{2} = −4} = P (X)_{+III}.

Moreover, M(X)_{+}, as well as M(X) admits a description in terms of groups.

Let

Δ_{±}^{(4)} = {f_{±} ∈ (S_{X})_{±} | f_{±}^{2} = −4, and ∃f_{∓} ∈ (S_{X})_{∓},
for which f_{∓}^{2} = −4 and (f_{±} + f_{∓})/2 ∈ S_{X}};
Δ_{+}^{(2)} = Δ^{(2)}(S) = {f_{+} ∈ S | f_{+}^{2} = −2};
Δ_{+t}^{(2)} = {f_{+} ∈ Δ^{(2)}(S) | ∃f_{−} ∈ (S_{X})_{−},
for which f_{−}^{2} = −6 and (f_{+} + f_{−})/2 ∈ S_{X}};
Δ_{−}^{(6)} = {f_{−} ∈ (S_{X})_{−} | f_{−}^{2} = −6 and ∃f_{+} ∈ Δ_{+t}^{(2)},
for which (f_{+} + f_{−})/2 ∈ S_{X}}.

If f_{±} ∈ Δ_{±}^{(4)}, then f_{±} ·(S_{X})_{±} ≡ 0 mod 2. Hence, f_{±} ∈ Δ_{±}^{(4)} are roots of (S_{X})_{±}, and there exists a reflection s_{f_{±}} ∈ O′((S_{X})_{±}) with respect to element f_{±},

s_{f_{±}}(x) = x + \frac{(x · f_{±})}{2}f_{±}, x ∈ (S_{X})_{±},

and one has the very important property

(12)  s_{f_{±}}(Δ_{±}^{(2)} ∪ Δ_{±}^{(4)}) = Δ_{±}^{(2)} ∪ Δ_{±}^{(4)} ∀f_{±} ∈ Δ_{±}^{(2)} ∪ Δ_{±}^{(4)}

where we formally put Δ_{−}^{(2)} = ∅ because the lattice (S_{X})_{−} has no elements with square (−2). It follows that reflections with respect to all elements Δ^{(2)}(S)∪Δ_{+}^{(4)}(S) = Δ_{+}^{(2,4)}(S) generate a group W_{+}^{(2,4)} ⊂ O′(S) which geometrically is a group generated by reflections in hyperplanes of L(S) which are orthogonal to Δ_{+}^{(2,4)}(S), any reflection in hyperplane of L(S) from this group is reflection in elements of Δ_{±}^{(2)} ∪Δ_{±}^{(4)}. It easily follows that M(S)_{+}^{(2,4)} is a chamber for W_{+}^{(2,4)} (see [Dol84, Nik84.1, Nam85]). In this way,

P (X)_{+} = P (M(S)_{+}^{(2,4)})

is the set of primitive elements of S, which are orthogonal to facets of M(S)_{+}^{(2,4)} and directed outward. Denote by

A(X, θ) = {ϕ ∈ O′(S) | ϕ(M_{+}) = M_{+}}

the group of automorphisms of M_{+} in O′(S) and by Aut(X, θ) – the normalizer of the involution θ in Aut X. The action of Aut(X, θ) on S_{X} and S defines a contravariant representation Aut(X, θ) → A(X, θ) with a finite kernel. Global Torelli theorem for K3 surfaces [PS-Sh71] implies [Dol84, Nik84.1, Nam85] that over the base field ℂ the image has finite index in A(X, θ). Therefore, there is an isomorphism up to finite groups Aut(X, θ) ≈ A(X, θ).

2.3.1. Computing P (X)_{+}. First, we consider the calculation of the fundamental chamber M ⊂ L(S) of W^{(2)}(S).

For that, it is important to consider a larger group W^{(2,4)}(S) generated by reflections in all elements Δ^{(2)}(S) with square (−2) of the lattice S and all elements of the set

Δ^{(4)}(S) = {f ∈ S | f^{2} = −4 and f · S ≡ 0 mod 2}

of all roots with square (−4) of the lattice S. Both of these sets Δ^{(2)}(S) and Δ^{(4)}(S) are invariant with respect to W^{(2,4)}(S). It follows that every reflection from W^{(2,4)}(S) gives a hyperplane H_{f} where f ∈ Δ^{(2)}(S) ∪ Δ^{(4)}(S). The subgroup W^{(2)}(S) ◁ W^{(2,4)}(S) is normal, and any reflection from W^{(2)}(S) is reflection in an element of Δ^{(2)}(S). Similarly, the subgroup W^{(4)}(S)◁W^{(2,4)}(S), generated by reflections in Δ^{(4)}(S), is normal and any reflection from W^{(4)}(S) is reflection in an element of Δ^{(4)}(S).

This implies the following description of a fundamental chamber M^{(2)} of W^{(2)}(S). Let M^{(2,4)} ⊂ L(S) be a fundamental chamber of W^{(2,4)}(S). Let P^{(2)}(M^{(2,4)}) and P^{(4)}(M^{(2,4)}) are directed outwards and orthogonal to M^{(2,4)} (i. e. to facets of M^{(2,4)}) elements from Δ^{(2)}(S) and Δ^{(4)}(S) respectively.

Proposition 2.2. Let W^{(4)}(M^{(2)}) be the group generated by reflections in all elements of P^{(4)}(M^{(2,4)}).

Then the fundamental chamber M^{(2)} of W^{(2)}(S) containing M^{(2,4)} is equal to

M^{(2)} = W^{(4)}(M^{(2)})(M^{(2,4)}),
P (M^{(2)}) = W^{(4)}(M^{(2)})(P^{(2)}(M^{(2,4)})).

Moreover,

W^{(4)}(M^{(2)}) = {w ∈ W^{(4)}(S) | w(M^{(2)}) = M^{(2)}}.

Reflections which are contained in W^{(4)}(M^{(2)}) are exactly the reflections in elements

Δ^{(4)}(M^{(2)}) = {f ∈ Δ^{(4)}(S) | H_{f} intersects the interior of M^{(2)}}.

Proof. This easily follows from the facts that W^{(2)}(S) ◁ W^{(2,4)}(S) and W^{(4)}(S) ◁ W^{(2,4)}(S) are normal subgroups, and Δ^{(2)}(S) ∩ Δ^{(4)}(S) = ∅. We leave details to the reader. □

Example 1. Let us consider the hyperbolic 2-elementary lattice S = ⟨2⟩ ⊕ 5⟨−2⟩ with the invariants (r, a, δ) = (6, 6, 1). Here and in what follows we denote by ⟨A⟩ the integral lattice given by an integral symmetric matrix A in some its basis. We denote by ⊕ the orthogonal sum of lattices.

The Dynkin diagram of W^{(2,4)}(S) (which is equivalent to the Gram matrix of all elements of P (M^{(2,4)})) is (see [Vin85])

Here black vertices correspond to elements from P^{(4)}(M^{(2,4)}) and white vertices correspond to elements from P^{(2)}(M^{(2,4)}) (see Sect. 3.1 below about edges). From the diagram, one can see that W^{(4)}(M^{(2)}) is the Weyl group of the root system D_{5}, the Δ^{(4)}(M^{(2)}) is the root system D_{5}, the set P^{(2)}(M^{(2)}) is the orbit of the Weyl group of D_{5} applied to the unique element of P^{(2)}(M^{(2,4)}) which corresponds to the white vertex. Calculations show that the set P (M^{(2)}) consists of 16 elements, and it is not easy to draw their Dynkin (or Gram) diagram, but it is completely defined by the diagram above which has only 6 vertices.

Now let us consider a subset Δ_{+}^{(4)} ⊂ Δ^{(4)}(S) and the subgroup W_{+}^{(2,4)} of reflections generated by this subset and by the set Δ^{(2)}(S). As in our case (12), we shall assume that the set Δ_{+}^{(4)} is W_{+}^{(2,4)}-invariant. Then each reflection from W_{+}^{(2,4)} is a reflection in a hyperplane H_{f}, f ∈ Δ^{(2)}(S)∪Δ_{+}^{(4)}. As before, W^{(2)}(S)◁W_{+}^{(2,4)} is a normal subgroup. We denote by W_{+}^{(4)} the group generated by reflections in Δ_{+}^{(4)}, it is normal in W_{+}^{(2,4)} as well. Thus, for a fundamental chamber M_{+}^{(2,4)} ⊂ M^{(2)} of W_{+}^{(2,4)} we can similarly define P^{(4)}(M_{+}^{(2,4)}), W_{+}^{(4)}(M^{(2)}), Δ_{+}^{(4)}(M^{(2)}) and prove similar statements to Proposition 2.2. We shall do this below.

First, let us show how one can calculate a fundamental chamber M_{+}^{(2,4)} of W_{+}^{(2,4)} contained in the fixed fundamental chamber M^{(2)} of W^{(2)}(S). Let

W_{+}^{(4)}(M^{(2)}) = {w ∈ W_{+}^{(4)} | w(M^{(2)}) = M^{(2)}}

and

Δ_{+}^{(4)}(M^{(2)}) = {f ∈ Δ_{+}^{(4)} | H_{f} intersects the interior of M^{(2)}}.

By the analogous statement of Proposition 2.2 for W_{+}^{(2,4)}, the group W_{+}^{(4)}(M^{(2)}) is generated by reflections in the elements of Δ_{+}^{(4)}(M^{(2)}), and every reflection from W_{+}^{(4)}(M^{(2)}) is a reflection in a hyperplane H_{f}, f ∈ Δ^{(4)}(M^{(2)}).

Obviously, the fundamental chamber M_{+}^{(2,4)} for W_{+}^{(2,4)} contained in M^{(2)} is the fundamental chamber of the group W_{+}^{(4)}(M^{(2)}) considered as a group acting on M^{(2)}. Let P^{(4)}(M_{+}^{(2,4)}) and P^{(2)}(M_{+}^{(2,4)}) are the sets of all elements in Δ_{+}^{(4)} and Δ^{(2)}(S) respectively which are orthogonal to facets of M_{+}^{(2,4)}.

Proposition 2.3. We have:

P^{(2)}(M^{(2)}) = W_{+}^{(4)}(M^{(2)})(P^{(2)}(M_{+}^{(2,4)}))

and

P^{(2)}(M_{+}^{(2,4)}) = {f ∈ P^{(2)}(M^{(2)}) | f · P^{(4)}(M_{+}^{(2,4)}) ≥ 0}.

Proof. The first statement is analogous to Proposition 2.2. We denote the right hand side of the proving second equality as P^{(2)}. Since P^{(2)} ⊂ P (M^{(2)}) and M^{(2)} has acute angles, f · f′ ≥ 0 for any two different elements f, f′ ∈ P^{(2)} ∪ P^{(4)}(M_{+}^{(2,4)}). It follows that P (M_{+}^{(2,4)}) ⊂ P^{(2)} ∪ P^{(4)}(M_{+}^{(2,4)}) because the fundamental chamber M_{+}^{(2,4)} must have acute angles. Then

⋂_{f∈P^{(2)}∪P^{(4)}(M_{+}^{(2,4)})} H_{f}^{+} ⊂ M_{+}^{(2,4)}

where the left hand side is not empty. Indeed, it contains the nonempty subset

⋂_{f∈P^{(2)}(M^{(2)})∪P^{(4)}(M_{+}^{(2,4)})} H_{f}^{+} = M^{(2)} ⋂ ( ⋂_{f∈P^{(4)}(M_{+}^{(2,4)})} H_{f}^{+}) ⊃ M_{+}^{(2,4)}.

It follows (see Proposition 3.1 in [Vin85]) that

P (M_{+}^{(2,4)}) = P^{(2)} ∪ P^{(4)}(M_{+}^{(2,4)})

because for all f ≠ f′ ∈ P^{(2)} ∪ P^{(4)}(M_{+}^{(2,4)}) we have f · f′ ≥ 0. □

Propositions 2.2 and 2.3 imply the result. It will be very important in further considerations.

Theorem 2.4. Let M^{(2,4)} be a fundamental chamber of W^{(2,4)}(S) in L(S), and W^{(4)}(M^{(2)}) the group generated by reflections in all elements of P^{(4)}(M^{(2,4)}), and Δ^{(4)}(M^{(2)}) = W^{(4)}(M^{(2)})(P^{(4)}(M^{(2,4)})).

Then

(1) Subsets Δ_{+}^{(4)} ⊂ Δ^{(4)}(S) which are invariant for the group W_{+}^{(2,4)} generated by reflections in all elements of Δ^{(2)}(S) ∪ Δ_{+}^{(4)} are in oneto-one correspondence with subsets Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) which are invariant for the group W_{+}^{(4)}(M^{(2)}) generated by reflections in all elements of Δ_{+}^{(4)}(M^{(2)}). This correspondence is given by

Δ_{+}^{(4)}(M^{(2)}) = Δ_{+}^{(4)} ∩ Δ^{(4)}(M^{(2)}); Δ_{+}^{(4)} = W^{(2)}(S)(Δ_{+}^{(4)}(M^{(2)})).

(2) The fundamental chamber M^{(2)} of W^{(2)}(S) containing M_{+}^{(2,4)} is M^{(2)} = W^{(4)}(M^{(2)})(M^{(2,4)}).

(3) Under the one-to-one correspondence in (1), any fundamental chamber M_{+}^{(2,4)} ⊂ M^{(2)} of W_{+}^{(2,4)} can be obtained as follows: Let M_{+}^{(4)}(M^{(2)}) be a fundamental chamber for W_{+}^{(4)}(M^{(2)}). Then

M_{+}^{(2,4)} = M^{(2)} ∩ M_{+}^{(4)}(M^{(2)}) and P^{(4)}(M_{+}^{(2,4)}) = P^{(4)}(M_{+}^{(4)}(M^{(2)})),

P^{(2)}(M_{+}^{(2,4)}) = {f ∈ W^{(4)}(M^{(2)})(P^{(2)}(M^{(2,4)})) | f · P^{(4)}(M_{+}^{(2,4)}) ≥ 0}.

Proof. Only the statement (1) requires some clarification. Assume that Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) is invariant with respect to the subgroup W_{+}^{(4)}(M^{(2)}) generated by reflections in Δ_{+}^{(4)}(M^{(2)}). Note that M^{(2)} is invariant with respect to W_{+}^{(4)}(M^{(2)}). It follows that the fundamental chamber M_{+}^{(2,4)} for W_{+}^{(4)}(M^{(2)}) acting in M^{(2)} will be the fundamental chamber for the group W_{+}^{(2,4)} generated by reflections in all elements of Δ^{(2)}(S) and Δ_{+}^{(4)}(M^{(2)}). It follows that Δ_{+}^{(4)} = W^{(2)}(S)(Δ_{+}^{(4)}(M^{(2)})) is invariant with respect to W_{+}^{(2,4)}. It follows that Δ_{+}^{(4)}(M^{(2)}) = Δ_{+}^{(4)} ∩ Δ^{(4)}(M^{(2)}).

The remaining statements are obvious. □ In Sect. 3 we apply this theorem to describe DPN surfaces of elliptic

type.

2.4. Root invariants of pairs (X, θ). To describe W_{+}^{(2,4)} and sets P (X)_{+III}, and P (X)_{+IIa}, P (X)_{+IIb}, one should add to the main invariants (r, a, δ) (equivalently (k, g, δ)) of (X, θ)) the so called root invariants. We describe them below. For the first time, root invariants for DPN surfaces had been considered and used in [Nik84.1] and [Nik87].

Let M be a lattice. Following [Nik80.2], M(k) denotes a lattice obtained from M by multiplication of the form of M by k ∈ ℚ.

Let K(2) be the sublattice of (S_{X})_{−} generated by Δ_{−}^{(4)} ⊂ (S_{X})_{−}. Since Δ_{−}^{(4)} · (S_{X})_{−} ≡ 0 mod 2, the lattice K = K(2)(1/2) is integral and is generated by its subset Δ_{−}^{(4)} ⊂ K of elements with square (−2) defining in K a root system, since reflections with respect to elements of Δ_{−}^{(4)} send Δ_{−}^{(4)} to itself. If follows that the lattice K is isomorphic to an orthogonal sum of root lattices A_{n}, D_{m} and E_{k} corresponding to the root systems 𝔸_{n}, 𝔻_{m}, 𝔼_{k} (or their Dynkin diagrams), and Δ_{−}^{(4)} = Δ^{(2)}(K) is the set of all elements of K with square (−2). Equivalently, Δ_{−}^{(4)} = Δ^{(4)}(K(2)) is the set of all elements with square (−4) of K(2). Moreover, we have a natural homomorphism of groups

(13)  ξ : Q = \frac{1}{2}K(2)/K(2) → 𝔄_{S} = S^{∗}/S

such that ξ(\frac{1}{2}f_{−} +K(2)) = \frac{1}{2}f_{+} +S, if f_{∓} ∈ Δ_{∓}^{(4)} and (f_{−} +f_{+})/2 ∈ S_{X}. This defines a homomorphism of finite quadratic forms ξ : q_{K(2)}|Q → −q_{S} with values in \frac{1}{2}ℤ/2ℤ ⊂ ℚ/2ℤ. (Here q_{M} : 𝔄_{M} = M^{∗}/M → ℚ/2ℤ denotes the discriminant quadratic form of an even lattice M.) The homomorphism ξ is equivalent to the homomorphism (which we denote by the same letter ξ) of the finite quadratic forms

(14)  ξ : K mod 2 → −q_{S},

by the natural isomorphism \frac{1}{2}K(2)/K(2) ≅ K/2K, where q_{K(2)}|Q is replaced by the finite quadratic form \frac{1}{2}x^{2} mod 2 for x ∈ K. The triplet (15)

R(X, θ) = (K(2), Δ_{−}^{(4)}, ξ) ≅ (K(2), K(2)^{(4)}, ξ) ≅ (K, Δ^{(2)}(K), ξ),

considered up to isomorphisms of lattices K and automorphisms of the lattice S is called the root invariant of the pair (X, θ) or the corresponding DPN pair (Y, C). (Clearly, similarly we can introduce abstract root invariants, without any relation to K3 surfaces with involutions and DPN pairs; we leave to a reader the corresponding definition).

We have the following statement from [Nik80.1]

Lemma 2.5. Let S be an even hyperbolic 2-elementary lattice. Then the natural homomorphism O(S) → O(q_{S}) is surjective.

Proof. We remind the proof from [Nik80.1]. If rk S ≥ 3, this follows from the general theorem 1.14.2 in [Nik80.2] (see also [Nik80.1]). If

rkS = 2, then S ≅ U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), U(2), ⟨2⟩ ⊕ ⟨−2⟩. If rk S = 1, then

S = ⟨2⟩. For all these lattice one can check the statement directly. □

Lemma 2.6. Let S be an even hyperbolic 2-elementary lattice and rk S ≥ 2. Then every x ∈ 𝔄_{S} with q_{S}(x) = n/2 mod 2, n ∈ ℤ, can be represented as x = u/2 mod S where u ∈ S and u^{2} = 2n.

Proof. If rk S = 2, then S ≡ U, U(2), ⟨2⟩⊕⟨−2⟩, and the statement can be checked directly. Assume that the statement is valid for rk S ≤ k where k ≥ 2. Let rk S = k + 1. By Theorem 1.12.2 from [Nik80.2] about existence of an even lattice with a given discriminant quadratic form, we get that S = S′ ⊕ T where S′ is a hyperbolic 2-elementary lattice of the rank rk S′ ≥ 2, and T is a negative definite 2-elementary lattice of the rk T ≥ 1. Let x = y ⊕ z, x ∈ 𝔄_{S}, y ∈ 𝔄_{T}, and assume z = u/2 mod T where u ∈ T and u^{2} = 2m, m ∈ ℤ . By the induction assumption, there exists v ∈ S′ with y = v/2 mod S′ and v^{2} = 2n−2m since q_{S′}(y) = q(x) − q(z) = (n − m)/2 mod 2. □

Lemma 2.7. Let q : A → ℚ/2ℤ be a non-degenerate quadratic form on a finite 2-elementary group A and ϕ : H_{1} ≅ H_{2} be an isomorphism of two subgroups in A which preserves q|H_{1} and q|H_{2}. Assume that the characteristic element a_{q} of q on A either does not belongs to both subgroups H_{1} and H_{2} or belongs to both of them. In the second case we additionally assume that ϕ(a_{q}) = a_{q}.

Then ϕ can be extended to an authomorphism of q.

Proof. See Proposition 1.9.1 in [Nik84.2]. We recall that a_{q} ∈ A is the characteristic element of q, if q(x) ≡ b_{q}(x, a_{q}) mod 1 for any x ∈ A. Here b_{q} is the bilinear form of q. This defines the characteristic element a_{q} uniquely. □

Lemmas 2.5 — 2.7 imply

Proposition 2.8. The root invariant R(X, θ) of (X, θ) (or (Y, C)) is equivalent to the triplet

R(X, θ) = (K(2); H; α, \overline{a}) ≅ (K; H; α, \overline{a}).

Here H = Ker ξ is an isotropic for q_{K(2)} subgroup in Q (equivalently in K mod 2); α = 0, if ξ(Q) = ξ(K mod 2) contains the characteristic element a_{q_{S}} of the quadratic form q_{S}, and α = 1 otherwise; if α = 0, the element \overline{a} = ξ^{−1}(a_{q_{S}}) + H ∈ Q/H ; if α = 1, the element \overline{a} is not defined.

The root invariant R(X, θ) is important because it defines

Δ_{+}^{(4)} = {f_{+} ∈ S | f_{+}^{2} = −4, f_{+}/2 mod S ∈
ξ(\frac{1}{2}Δ^{(4)}(K(2)) mod K(2)) = ξ(Δ^{(2)}(K) mod 2K)\},

and W_{+}^{(2,4)}, M(X)_{+}^{(2,4)}, P (M(X)_{+}^{(2,4)}) = P (X)_{+}. Moreover, for f_{+} ∈ P (X)_{+}^{(4)}, the root invariant defines the decomposition f_{+} = f + θ^{∗}(f), f ∈ P (X), up to the action of O(S). More precisely, we have the following. Let f_{−} ∈ Δ^{(4)}(K(2)) and ξ(f_{−}/2 mod K(2)) = f_{+}/2 mod S.

Then f = (f_{+} ±f_{−})/2, θ(f) = (f_{+} ∓f_{−})/2. Indeed, if f_{−}′ ∈ Δ^{(4)}(K(2)) satisfies the same conditions, then (f_{−} + f_{−}′)/2 ∈ K(2). In K(2), if f_{−}′ ≠ ±f_{−} then either f_{−} · f_{−}′ = 0 or f_{−} · f′_{−} = ±2. If f_{−} · f_{−}′ = 0 then ((f_{−} + f_{−}′)/2)^{2} = −2, and we get a contradiction since (S_{X})_{−} does not have elements with the square (−2). If f_{−} · f_{−}′ = ±2 then f_{−}·(f_{−}+f_{−}′)/2 = −2±1, and we get a contradiction since f_{−}·(S_{X})_{−} ≡ 0 mod 2. Thus, f_{−}′ = ±f_{−}, and the pair of elements f and θ^{∗}(f) is defined uniquely.

Similarly one can define a generalized root invariant

R_{gen}(X, θ) = (K_{gen}(2), Δ_{−}^{(4)} ∪ Δ_{−}^{(6)}, ξ_{gen}) ≅
(K_{gen}, Δ_{−}^{(2)} ∪ Δ_{−}^{(3)}, ξ_{gen}),

where for f_{−} ∈ Δ_{−}^{(6)} one has

ξ_{gen}(f_{−}/2 mod K_{gen}(2)) = f_{+}/2 mod S

where f_{+} ∈ Δ_{+t}^{(2)}(S) and (f_{+} + f_{−})/2 ∈ S_{X}. Here K_{gen}(2) ⊂ (S_{X})_{−} is generated by Δ_{−}^{(4)} ∪ Δ_{−}^{(6)}.

Using Lemmas 2.5 — 2.7, one can similarly prove that it is equivalent to the tuple

R_{gen}(X, θ) = (K_{gen}(2), Δ_{−}^{(4)} ∪ Δ_{−}^{(6)}; H_{gen}; α_{gen}, \overline{a}_{gen}) ≅
(K_{gen}, Δ_{−}^{(2)} ∪ Δ_{−}^{(3)}; H_{gen}; α_{gen}, \overline{a}_{gen}).

It is defined similarly to the root invariant.

Importance of the generalized root invariant is that it contains the root invariant R(X, θ). Thus, it defines W_{+}^{(2,4)}, M(X)_{+}^{(2,4)} and also P (M(X)_{+}^{(2,4)}) = P (X)_{+} up to the action of O(S). But, it also defines

Δ_{+t}^{(2)} = {f_{+} ∈ S | (f_{+})^{2} = −2, f_{+}/2 mod S ∈

ξ_{gen}(\frac{1}{2}Δ_{−}^{(6)} mod K_{gen}(2) = ξ(Δ_{−}^{(3)} mod 2K_{gen})\},

and then it defines

P (X)_{+IIb} = P^{(2)}(X)_{+t} = {f_{+} ∈ P (X)_{+} | f_{+} ∈ Δ_{+t}^{(2)}}.

Thus, using root invariants, we know how to find P (X)_{+III} = P^{(4)}(X)_{+}, P (X)_{+IIb} = P^{(2)}(X)_{+t}, and hence, we know P (X)_{+I} ∪ P (X)_{+IIa} = P^{(2)}(X)_{+} − P^{(2)}(X)_{+t}. To distinguish P (X)_{+I} and P (X)_{+IIa}, it is sufficient to know how one can find P (X)_{+I}.

2.5. Finding the locus X^{θ}. Here we show how one can find P (X)_{+I}. This is based on the following considerations (similar to [Nik83]):

1) Since W^{(2)}(S) ◁ W_{+}^{(2,4)} is a normal subgroup, the fundamental polyhedron M_{+}^{(2,4)} is contained in one fundamental chamber M^{(2)} of W^{(2)}(S); we have M_{+}^{(2,4)} ⊂ M^{(2)}. One can consider replacing M_{+}^{(2,4)} by M^{(2)} as a deformation of a pair (X, θ) to a general pair (X̃, θ̃) having SX̃ = S, M(X̃) = M^{(2)} and P (X̃) = P (M^{(2)}).

The divisor classes of fixed points of the involution do not change under this deformation, thus P (X)_{+I} = P (X̃)_{+I}. In particular, P (X)_{+I} does not change when a root invariant changes (with fixed main invariants (r, a, δ) equivalent to the lattice S).

2) Let δ_{1}, δ_{2} belong to P^{(2)}(X)_{+} and δ_{1} · δ_{2} = 1, i. e. the corresponding to them curves D_{1}, D_{2} intersect transversally. Then one of δ_{1}, δ_{2} belongs to P (X)_{+I}, and another to P (X)_{+II} = P (X)_{+IIa}∪P (X)_{+IIb} = P (X)_{+IIa} for the general case we consider. See the diagrams below where an element of P (X)_{+I} is denoted by a double transparent vertex, and an element of P (X)_{+II} by a single transparent vertex.

Indeed, the intersection point D_{1} ∩ D_{2} is a fixed point of θ, tangent directions of D_{1} and D_{2} at this point are eigenvectors of θ_{∗}. We know that they have eigenvalues +1 and −1.

3) If the Gram diagram of elements δ_{0}, δ_{1}, δ_{2} and δ_{3} from P^{(2)}(X)_{+} has the form as shown

then δ_{0} ∈ P (X)_{+I}, and δ_{1}, δ_{2}, δ_{3} ∈ P (X)_{+II} = P (X)_{+IIa} (for the general case we consider). Indeed, the rational curve corresponding to δ_{0} has three different fixed points of θ, and hence belongs to X^{θ}.

4) If δ ∈ P (X)_{+III} = P^{(4)}(X)_{+} and δ_{1} ∈ P (X)_{+I}, then δ_{1} · δ = 0. This is obvious from the definition of P (X)_{+III}.

Considering all possible lattices S, it is not difficult to see that statements 1) — 3) are sufficient for finding P (X)_{+I} and the divisor class of the irreducible component C_{g} of the curve X^{θ} of fixed points. The statement 4) simplifies these considerations, if some elements of P^{(4)}(X)_{+} are known.

2.6. Conditions for the existence of root invariants. Assume that the main invariants (r, a, δ) (equivalently (k, g, δ)) of (X, θ) are known and fixed. Here we want to give conditions which are necessary and sufficient for the existence of a pair (X, θ) with a given root or a generalized root invariant. We consider the case of a root invariant. Similarly, one can consider the case of a generalized root invariant.

For the existence of a root invariant, it is necessary

Condition 1. The lattice

K_{H} = [K; x/2 where x + 2K ∈ H]

does not have elements with the square (−1). Equivalently, the lattice

K_{H}(2) = [K(2); x/2where x/2 + K(2) ∈ H]

does not have elements with the square (−2).

Indeed, the lattice K_{H}(2) ⊂ (S_{X})_{−}, but the lattice (S_{X})_{−} does not have elements with the square (−2). If f ∈ (S_{X})_{−} has f^{2} = −2 then ±f is effective (by Riemann-Roch Theorem applied to the K3 surface X). On the other hand, θ^{∗}(±f) = ∓f is then also effective which is impossible.

Condition 2. If k = ℂ, then rk S + rk K = r + rk K ≤ 20.

Indeed, S ⊕ K ⊂ S_{X} and rk S_{X} ≤ 20.

A pair (X, θ) (or the corresponding pair (Y, C))is called standard, if K_{H}(2) is a primitive sublattice of (S_{X})_{−}, and the primitive sublattice [S ⊕ K(2)]_{pr} in S_{X} generated by S ⊕ K(2) is defined by the homomorphism ξ, i. e. it is equal to

M =[S ⊕ K(2); {a + b | ∀a ∈ S^{∗}, ∀b ∈ K(2)/2,
such that ξ(b + K(2)) = a + S}].

(16)

Clearly, M ⊂ [S ⊕ K(2)]_{pr} is always a sublattice of finite index.

Let l(𝔄) be the minimal number of generators of a finite Abelian group 𝔄. Let 𝔄_{M} = M^{∗}/M be the discriminant group of a lattice M.

Proposition 2.9. Assume that k = ℂ.

Then a standard pair (X, θ) with a given root invariant R(X, θ) satisfying Conditions 1 and 2 does exist, if additionally

(17)  r + a + 2l(H) < 22

and

(18)  r + rk K + l(𝔄_{K_{p}}) < 22

for any prime p > 2.

Proof. By Global Torelli Theorem [PS-Sh71] and Surjectivity of Torelli Map [Kul77] for K3, the pair (X, θ) does exist, if there exists a primitive embedding of the lattice M described in (16) into an even unimodular lattice L_{K3} ≅ H^{2}(X, ℤ) of the signature (3, 19) (see the proof of Proposition 2.10 below). By Corollary 1.12.3 in [Nik80.2], such a primitive embedding does exist, if rk M + l(𝔄_{M_{p}}) < 22 for all prime p ≥ 2.

If p > 2, then rk M + l(𝔄_{M_{p}}) = r + rkK + l(𝔄_{K_{p}}) < 22 by (18) (here we remember that the lattice S is 2-elementary).

Assume that p = 2. Let Γ_{ξ} be the graph of ξ. Then 𝔄_{M} = (Γ_{ξ})^{⊥}/Γ_{ξ} for the discriminant form q_{S} ⊕q_{k(2)}. Let Q = H ⊕Q′ (see (13)) where Q′ is a complimentary subgroup, and ξ′ = ξ|Q′. Then Γ_{ξ′} ⊂ Γ_{ξ}, moreover Γ_{ξ′} ⊂ 𝔄_{S} ⊕ 𝔄_{K(2)} = 𝔄 is a 2-elementary subgroup, and Γ_{ξ′} ∩ 2 𝔄 = {0} since 2 𝔄 = {0} ⊕ 2 𝔄_{K(2)} and ξ′ is injective. Let 𝔄^{(2)} be the kernel of multiplication by 2 in 𝔄, and q = (q_{S} ⊕q_{K(2)})| 𝔄^{(2)}. It is easy to see that the kernel Ker q = 𝔄^{(2)} ∩2 𝔄. Since Γ_{ξ′} ∩ 2 𝔄 = {0}, then Γ_{ξ′} ∩ Ker q = {0}. Let 𝔄_{1}^{(2)} be a complementary to Ker q subgroup in 𝔄^{(2)} which contains Γ_{ξ′}. Then q_{S} ⊕ q_{K(2)} = q_{1} ⊕ q_{2} where q_{1} = q_{S} ⊕ q_{K(2)}| 𝔄_{1}^{(2)} and q_{2} is the orthogonal complement to q_{1} (since q_{1} is non-degenerate). The subgroup Γ_{ξ′} is isotropic for the non-degenerate 2-elementary form q_{1} and has rank rk K − rk H. It follows that

l((Γ_{ξ′})_{q_{1}}^{⊥}/Γ_{ξ′}) = l(𝔄_{1}^{(2)}) − 2l(Γ_{ξ′}),

and then

l(𝔄_{M_{2}}) ≤ l(𝔄) − 2l(Γ_{ξ′}) = a + rk K − 2(rk K − l(H)).

This implies that

rk M + l(𝔄_{M_{2}}) ≤ r + a + 2l(H) < 22

by (17). □

Finally, in general, by Global Torelli Theorem [PS-Sh71] and Surjectivity of Torelli Map [Kul77] for K3 (actually, The Global Torelli Theorem and local surjectivity of Torelli Map for K3 are sufficient for the proof, see [Nik80.1]), we have the following necessary and sufficient conditions of existence of a pair (X, θ) with a root invariant (K(2), ξ). It even takes under consideration the more delicate invariant which is the isomorphism class of embedding of lattices M ⊂ H^{2}(X, ℤ) ≅ L_{K3}.

Proposition 2.10. (over ℂ) There exists a K3 pair (X, θ) with a root invariant (K(2), ξ) and the isomorphism class of embedding ϕ : M ⊂ L_{K3} of lattices (see (16)), if and only if

1) ϕ(S) ⊂ L_{K3} is a primitive sublattice;

2) the primitive sublattice ϕ(K(2))_{pr} ⊂ L_{K3} generated by ϕ(K(2)) in L_{K3} does not have elements with square (−2);

3) we have:

ϕ(Δ^{(4)}(K(2))) ={f_{−} ∈ ϕ(K(2))_{pr} | f_{−}^{2} = −4, and ∃f_{+} ∈ S
such that f_{+}^{2} = −4 and (ϕ(f_{+}) + f_{−})/2 ∈ L_{K3}}.

(19)

We remind that here Δ^{(4)}(K(2)) is the set of all elements in K(2) with the square (−4).

Proof. Let us consider an abstract root invariant (K(2), ξ). This means that K is a negative definite lattice generated by its elements with square (−2), and K(2) is obtained by multiplication of the form of K by 2. The map

ξ : q_{K(2)}|Q → −q_{S}, Q = \frac{1}{2}K(2)/K(2),

is a homomorphism of finite quadratic forms. We assume that for each f_{−} ∈ Δ^{(4)}(K(2)) there exists f_{+} ∈ Δ^{(4)}(S) such that ξ(f_{−}/2 + K(2)) = f_{+}/2 + S (by Lemma 2.4.2, this condition satisfies always).

Using (16), we construct the even lattice M which contains K(2) ⊕ S ⊂ M as a sublattice of finite index. It contains S ⊂ M as a primitive sublattice, and it’s primitive sublattice generated by K(2) is K(2)_{H} where H = Ker ξ.

Let ϕ : M → L_{K3} be an embedding of lattices. If ϕ corresponds to a K3 pair (X, θ), with the root invariant (K(2), ξ), then conditions 1), 2) and 3) must satisfy. Now we assume that they are valid for the abstract embedding ϕ : M → L_{K3} of lattices we consider.

Then ϕ(S) ⊂ L_{K3} is a primitive sublattice. To simplify notations, we identify S = ϕ(S) ⊂ L_{K3} and K(2) = ϕ(K(2)). Since S is 2elementary, there exists an involution α on L_{K3} with (L_{K3})_{+} = S and (L_{K3})_{−} = S^{⊥}. Then α = −id on K(2). We denote by M̃ the primitive sublattice in L_{K3} generated by ϕ(M) = M.

Assume that f ∈ M̃ satisfies f^{2} = −2, f = f_{−}^{∗} + f_{+}^{∗} where f_{−}^{∗} ∈ (K(2)_{pr})^{∗}, f_{+}^{∗} ∈ S^{∗} and f_{+}^{2} < 0. Since 2f_{−}^{∗} = f_{−} = f − α(f) ∈ K(2)_{pr}, 2f_{+}^{∗} = f_{+} = f + α(f) ∈ S, K(2) is negative definite and satisfies 2), it follows that either f = (f_{−} + f_{+})/2 where f_{−} = 0 and f = f_{+}/2 ∈ Δ^{(2)}(S), or f = (f_{−} + f_{+})/2 where f_{−} ∈ K(2)^{(4)}, f_{+} ∈ Δ^{(4)}(S), or f = (f_{−} + f_{+})/2 where (f_{−})^{2} = −6 and f_{+} ∈ Δ^{(2)}(S).

It follows that there exists h_{+} ∈ S with (h_{+})^{2} > 0 such that h_{+}·f ≠ 0 for any f ∈ Δ^{(2)}(M̃).

By surjectivity of Torelli map for K3 surfaces [Kul77], we can assume that there exists a K3 surface X with H^{2}(X, ℤ) = L_{K3}, S_{X} = M̃ and a polarization h_{+}. The involution α preserves periods of X. By Global Torelli Theorem for K3 [PS-Sh71], α = θ^{∗} corresponds to an automorphism θ of X. The automorphism θ is symplectic because H^{2}(X, ℤ)_{+} = (S_{X})_{+} = M̃_{+} = S is hyperbolic. By 3), the root invariant of (X, θ) is (K(2), ξ). □

This implies the important for us result.

Corollary 2.11. Let (K(2), ξ) be the root invariant of a pair (X, θ) and K′(2) ⊂ K(2) a primitive sublattice of K(2) generated by its elements Δ^{(4)}(K′(2)) with the square (−4).

Then the pair (K′(2), ξ′ = ξ|Q′ = \frac{1}{2}K′(2)/K′(2)) is also the root invariant of some pair (X′, θ′).

Corollary 2.11 shows that to describe all possible root invariants of pairs (X, θ), it is enough to describe all possible root invariants of extremal pairs. Here a pair (X′, θ′) is called extremal, if its root invariant R(X′, θ′) = (K′(2), ξ′) cannot be obtained using Corollary 2.11 from the root invariant R(X, θ) = (K(2), ξ) of any other pair (X, θ′) with rk K(2) > rk K′(2).

2.7. Three types of non-symplectic involutions of K3 surfaces and DPN surfaces. It is natural to divide non-symplectic involutions (X, θ) of K3 and the corresponding DPN surfaces in three types:

Elliptic type: X^{θ} ≅ C ≅ C_{g} +E_{1} +⋯+E_{k} where C_{g} is an irreducible curve of genus g ≥ 2 (equivalently, (C_{g})^{2} > 0), and E_{1}, …, E_{k} are nonsingular irreducible rational curves. By Sect. 2.2, this is equivalent to r + a ≤ 18 and (r, a, δ) ≠ (10, 8, 0). Then Aut(X, θ) is finite because (C_{g})^{2} > 0.

Parabolic type: Either X^{θ} ≅ C ≅ C_{1} + E_{1} + ⋯ + E_{k} (using the same notations), or X^{θ} ≅ C ≅ C_{1}^{(1)} + C_{1}^{(2)} is sum of two elliptic (i. e. of genus 1) curves. By Sect. 2.2, this is equivalent to either r + a = 20 and (r, a, δ) ≠ (10, 10, 0), or (r, a, δ) = (10, 8, 0). Then Aut(X, θ) is Abelian up to finite index and usually non-finite. Here (C_{1})^{2} = 0.

Hyperbolic type: Either X^{θ} ≅ C ≅ E_{0} + ⋯ + E_{k} is sum of nonsingular irreducible rational curves, or X^{θ} = ∅. By Sect. 2.2, this is equivalent to either r+a = 22, or (r, a, δ) = (10, 10, 0). Then Aut(X, θ) is usually non-Abelian up to finite index.

Thus, pairs (X, θ) of elliptic type are the most simple, and we describe them completely in Sect. 3. On the other hand, for classification of log del Pezzo surfaces of index ≤ 2 we need only these pairs.

3. Classification of DPN surfaces of elliptic type

3.1. Fundamental chambers of W^{(2,4)}(S) for elliptic type. The most important property of the lattices S for elliptic type is that the subgroup W^{(2)}(S) ⊂ O(S) has finite index. We remark that it is parallel to Lemma 1.4, and it is an important step to prove that log del Pezzo surfaces of index ≤ 2 are equivalent to DPN surfaces of elliptic type.

This finiteness was first observed and used for classification of all hyperbolic lattices M with finite index [O(M) : W^{(2)}(M)] in [Nik83]. We repeat arguments of [Nik83]. Let us take a general pair (X, θ) with (S_{X})_{+} = S. Then S_{X} = S, and the involution θ of X is unique by the condition that it is identical on S_{X} = S and is −1 on the orthogonal complement to S_{X} in H^{2}(X, ℤ). Thus, Aut X = Aut(X, θ). By Global Torelli Theorem for K3 (see [PS-Sh71]), the action of Aut X on S_{X} gives that Aut X and O(S_{X})/W^{(2)}(S_{X}) are isomorphic up to finite groups. In particular, they are finite simultaneously. Thus, [O(S) : W^{(2)}(S)] is finite, if and only if Aut(X, θ) is finite. If (X, θ) has elliptic type, then Aut(X, θ) preserves X^{θ} and its component C_{g} with (C_{g})^{2} > 0. Since S_{X} is hyperbolic, it follows that the action of Aut(X, θ) in S_{X} is finite. But it is known for K3 (see [PS-Sh71]) that the kernel of this action is also finite. It follows that Aut(X, θ) and [O(S) : W^{(2)}(S)] are finite.

Since O(S) is arithmetic, W^{(2)}(S) has a fundamental chamber M^{(2)} in L(S) of finite volume and with a finite number of faces (e. g. see [Vin85]). Since W^{(2)}(S) ⊂ W^{(2,4)}(S) ⊂ O(S), the same is valid for W^{(2,4)}(S).

Let M^{(2,4)} ⊂ L(S) be a fundamental chamber of W^{(2,4)}(S), and Γ(P (M^{(2,4)})) its Dynkin diagram (see [Vin85]). Vertices corresponding to different elements f_{1}, f_{2} ∈ P (M^{(2,4)}) are not connected by any edge, if f_{1} · f_{2} = 0. They are connected by a simple edge of the weight k (equivalently, by k − 2 simple edges, if k > 2 is small), if

\frac{2 f_{1} · f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}} = 2 cos \frac{π}{k} , k ∈ ℕ.

They are connected by a thick edge, if

\frac{2 f_{1} · f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}} = 2.

They are connected by a punctured edge of the weight t, if

\frac{2 f_{1} · f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}} = t > 2.

Moreover, a vertex corresponding to f ∈ P^{(4)}(M^{(2,4)}) is black. It is transparent, if f ∈ P^{(2)}(M^{(2,4)}). It is transparent and doubled, if f ∈ P (X)_{+I} (i. e. it corresponds to the class of a rational component of X^{θ}), otherwise, it is simple transparent. Of course, here we assume that M^{(2,4)} ⊂ M(X)_{+}^{(2,4)} for a K3 surface with involution (X, θ) and (S_{X})_{+} = S.

Classification of DPN surfaces of elliptic type is based on the purely arithmetic calculations of the fundamental chambers M^{(2,4)} (equivalently, of the graphs Γ(P (M^{(2,4)})) of the reflection groups W^{(2,4)}(S) of the lattices S of elliptic type. Since S is 2-elementary and even, W^{(2,4)}(S) = W (S) is the full reflection group of the lattice S, any root f ∈ S has f^{2} = −2 or −4. We have

Theorem 3.1. 2-elementary even hyperbolic lattices S of elliptic type have fundamental chambers M^{(2,4)} for their reflection groups W^{(2,4)}(S) (it is the full reflection group of S), equivalently the corresponding Dynkin diagrams Γ(P (M^{(2,4)})), which are given in Table 1 below, where the lattice S is defined by its invariants (r, a, δ), see Sect. 2.2.

DEL PEZZO SURFACES OF INDEX ≤ 2 37

Table 1. Fundamental chambers M^{(2,4)} of reflection groups W^{(2,4)}(S) for 2-elementary even hyperbolic lattices S of elliptic type.

N r a δ k g l(H)  Γ(P (M^{(2,4)}))
1 1 1 1 0 10 0
2 2 2 0 0 9 0
3 2 2 1 0 9 0
4 3 3 1 0 8 0
5 4 4 1 0 7 0
6 5 5 1 0 6 0
7 6 6 1 0 5 0
8 7 7 1 0 4 0
9 8 8 1 0 3 0
10 9 9 1 0 2 0
11 2 0 0 1 10 0
12 3 1 1 1 9 0
13 4 2 1 1 8 0
14 5 3 1 1 7 0
15 6 4 0 1 6 0
16 6 4 1 1 6 0
17 7 5 1 1 5 0
18 8 6 1 1 4 1

N r a δ k g l(H)  Γ(P (M^{(2,4)}))

19 9 7 1 1 3 1

20 10 8 1 1 2 1
21 6 2 0 2 7 0
22 7 3 1 2 6 0
23 8 4 1 2 5 0

24 9 5 1 2 4 0
25 10 6 0 2 3 1

26 10 6 1 2 3 1

27 11 7 1 2 2 1

28 8 2 1 3 6 0

29 9 3 1 3 5 0
30 10 4 0 3 4 0

31 10 4 1 3 4 0

32 11 5 1 3 3 0

33 12 6 1 3 2 1

N r a δ k g l(H)  Γ(P (M^{(2,4)}))

34 9 1 1 4 6 0

35 10 2 0 4 5 0

36 10 2 1 4 5 0

37 11 3 1 4 4 0

38 12 4 1 4 3 0

39 13 5 1 4 2 0

40 10 0 0 5 6 0
41 11 1 1 5 5 0
42 12 2 1 5 4 0

43 13 3 1 5 3 0

44 14 4 0 5 2 0

N r a δ k g l(H)  Γ(P (M^{(2,4)}))

45 14 4 1 5 2 0
46 14 2 0 6 3 0

47 15 3 1 6 2 0

48 16 2 1 7 2 0
49 17 1 1 8 2 0
50 18 0 0 9 2 0

Proof. When S is unimodular (i.e. a = 0) or r = a (then S(1/2) is unimodular), i. e. for cases 1—11, 40, 50, these calculations were done by Vinberg. In all other cases they can be done using Vinberg’s algorithm for calculation of the fundamental chamber of a hyperbolic reflection group. See [Vin72], [Vin85]. For lattices S ≅ U ⊕D_{2k} ⊕mA_{1}, k ≥ 3, we used a computer.

To describe elements of P (X)_{+I} (i. e. double transparent vertices), we use the results of Sect. 2.5 and the fact that that their number is known, by results of Sect. 2.2. □

Remark 3.2. Using diagrams of Theorem 3.1, one can easily find the class in S of the component C_{g}, g ≥ 2, of X^{θ} as an element C_{g} ∈ S such that C_{g} · x = 0, if x corresponds to a black or a double transparent vertex, and C_{g} · x = 2 − s if x corresponds to a simple transparent vertex which has s edges to double transparent vertices.

3.2. Root invariants, and subsystems of roots in Δ^{(4)}(M^{(2)}) for elliptic case. We use the notations and results of Sect. 2.3.1. Let M^{(2)} ⊃ M^{(2,4)} be the fundamental chamber of W^{(2)}(S) containing M^{(2,4)}. Dynkin diagram of P^{(4)}(M^{(2,4)}) (i. e. black vertices) consists of components of types 𝔸, 𝔻 or 𝔼 (see Table 1). Thus, the group W^{(4)}(M^{(2)}) generated by reflections in all elements of P^{(4)}(M^{(2,4)}) is a finite Weyl group. It has to because W^{(4)}(M^{(2)})(M^{(2,4)}) = M^{(2)} has finite volume, and M^{(2,4)} is the fundamental chamber for the action of W^{(4)}(M^{(2)}) in M^{(2)}. Thus,

Δ^{(4)}(M^{(2)}) = W^{(4)}(M^{(2)})P^{(4)}(M^{(2,4)})

is a finite root system of the corresponding type in a negative definite sublattice

R(2) = [P^{(4)}(M^{(2,4)})] ⊂ S.

Let (X, θ) be a K3 surface with a non-symplectic involution, and (S_{X})_{+} = S. Let Δ_{+}^{(4)} ⊂ Δ^{(4)}(S) be the defined by (X, θ) subset which is invariant with respect to W_{+}^{(2,4)}. By Theorem 2.4, Δ_{+}^{(4)} = W^{(2)}(S)Δ_{+}^{(4)}(M^{(2)}) where Δ_{+}^{(4)}(M^{(2)}) = Δ_{+}^{(4)} ∩ Δ^{(4)}(M^{(2)}) is a root subsystem in Δ^{(4)}(M^{(2)}). Let

(20)  K^{+}(2) = [Δ_{+}^{(4)}(M^{(2)})] ⊂ R(2) ⊂ S

be the corresponding negative definite sublattice in S, and

(21)  Q = \frac{1}{2}K^{+}(2)/K^{+}(2), ξ^{+} : q_{K^{+}(2)}|Q → q_{S}

a homomorphism such that ξ^{+}(x/2 + K^{+}(2)) = x/2 + S, x ∈ K^{+}(2). We obtained a pair (K^{+}(2), ξ^{+}) which is similar to a root invariant, and it is equivalent to the root invariant for elliptic type.

Proposition 3.3. Let (X, θ) be a K3 surface with a non-symplectic involution of elliptic type, and S = (S_{X})_{+}.

In this case, the root invariant R(X, θ) is equivalent to the root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}), considered up to the action of O(S) (i. e. two root subsystems Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) and Δ_{+}^{(4)}(M^{(2)})′ ⊂ Δ^{(4)}(M^{(2)}) are equivalent, if Δ_{+}^{(4)}(M^{(2)})′ = ϕ(Δ_{+}^{(4)}(M^{(2)})) for some ϕ ∈ O(S)):

The root invariant R(X, θ) ≅ (K^{+}(2), ξ^{+}) is defined by (20) and (21).

The fundamental chamber M(X)_{+}^{(2,4)} is defined by the root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) (up to above equivalence), by Theorem 2.4. Moreover, P^{(4)}(M(X)_{+}^{(2,4)}) coincides with a basis of the root subsystem Δ_{+}^{(4)}(M^{(2)}).

Proof. Let E_{i}, i ∈ I, are all non-singular rational curves on X such that E_{i} · θ(E_{i}) = 0, i. e.

cl(E) + cl(θ(E)) = δ ∈ P^{(4)}(M(X)_{+}^{(2,4)}) = P^{(4)}(X)_{+} = P (X)_{+III}.

Since E_{i} · C_{g} = 0 and C_{g}^{2} = 2g − 2 > 0, the curves E_{i}, i ∈ I, generate in S_{X} a negative definite sublattice. Thus, their components define a Dynkin diagram Γ which consists of several connected components 𝔸_{n}, 𝔻_{m} and 𝔼_{k}. The involution θ acts on these diagrams and corresponding curves without fixed points. Thus it necessarily changes connected components of Γ. Let Γ = Γ_{1} ∐ Γ_{2} where θ(Γ_{1}) = Γ_{2}, and I = I_{1} ∐ I_{2} the corresponding subdivision of vertices of Γ. Then

δ_{i}^{+} = cl(E_{i}) + cl(θ(E_{i})), i ∈ I_{1},

and

δ_{i}^{−} = cl(E_{i}) − cl(θ(E_{i})), i ∈ I_{1}

give bases of root systems Δ_{+}^{(4)}(M^{2}) and Δ_{−}^{(4)} = Δ^{(4)}(K(2)) respectively. The map

δ_{i}^{−} = cl(E_{i}) − cl(θ(E_{i})) ↦ δ_{+} = cl(E_{i}) + cl(θ(E_{i})), i ∈ I_{1},

defines an isomorphism Δ_{−}^{(4)} ≅ Δ_{+}^{(4)}(M^{(2)}) of root systems, since it evidently preserves the intersection pairing. The homomorphism ξ of the root invariant R(X, θ) = (K(2), ξ) of the pair (X, θ) then goes to ξ^{+}.

In the opposite direction, the root invariant R(X, θ) defines Δ_{+}^{(4)} and Δ_{+}^{(4)}(M^{(2)}) = Δ^{(4)}(M^{(2)}) ∩ Δ_{+}^{(4)}.

The last statement follows from Sect. 2.3.1. □

By Proposition 3.3 in the elliptic case instead of root invariants one can consider root subsystems Δ_{+}^{(4)}(M^{(2)}) (in Δ^{(4)}(M^{(2)})). We say that a root subsystem Δ_{+}^{(4)}(M^{(2)}) “is contained” (respectively “is primitively contained”) in a root subsystem Δ_{+}^{(4)}(M^{(2)})′, if ϕ(Δ_{+}^{(4)}(M^{(2)})) ⊂ Δ_{+}^{(4)}(M^{(2)})′ (respectively [ϕ(Δ_{+}^{(4)}(M^{(2)}))] ⊂ [Δ_{+}^{(4)}(M^{(2)})′] is a primitive embedding of lattices) for some ϕ ∈ O(S). By Corollary 2.11, we obtain

Proposition 3.4. If a root subsystem Δ_{+}^{(4)}(M^{(2)}) in Δ^{(4)}(M^{(2)}) corresponds to a K3 surface with non-symplectic involution (X, θ), then any primitive root subsystem in Δ_{+}^{(4)}(M^{(2)}) corresponds to a K3 surface with non-symplectic involution. Thus, it is enough to describe extremal pairs (X, θ) such that their root subsystems Δ_{+}^{(4)}(M^{(2)}) in Δ^{(4)}(M^{(2)}) are not contained as primitive root subsystems of strictly smaller rank in a root subsystem Δ_{+}^{(4)}(M^{(2)})′ in Δ^{(4)}(M^{(2)}) corresponding to another pair (X′, θ′).

3.3. Classification of non-symplectic involutions (X, θ) of elliptic type of K3 surfaces. We have

Theorem 3.5. Let (X, θ) and (X′, θ′) be two non-symplectic involutions of elliptic type of K3 surfaces.

Then the following three conditions are equivalent:

(i) Their main invariants (r, a, δ) (equivalently, (k, g, δ)) coincide, and their root invariants are isomorphic.

(ii) Their main invariants (r, a, δ) coincide, and the root subsystems Δ_{+}^{(4)}(M^{(2)}) are equivalent.

(iii) Dynkin diagrams Γ(P (X)_{+}) and Γ(P (X′)_{+}) of their exceptional curves are isomorphic, and additionally the genera g are equal, if these diagrams are empty. The diagram Γ(P (X)_{+}) is empty if and only if either (r, a, δ) = (1, 1, 1) (then g = 10), or (r, a, δ) = (2, 2, 0) (then g = 9) and the root invariant is zero. The corresponding DPN surfaces are ℙ^{2} or 𝔽_{0} respectively.

Proof. By Sections 3.2 and 2.4, the conditions (i) and (ii) are equivalent, and they imply (iii).

Let us show that (iii) implies (i).

Assume that r = rk S ≥ 3.

First, let us show that S is generated by Δ^{(2)}(S), if r = rk S ≥ 3. If r ≥ a + 2, then it is easy to see that either S ≅ U ⊕ T or S ≅ U(2) ⊕ T where T is orthogonal sum of A_{1}, D_{2m}, E_{7}, E_{8} (one can get all possible invariants (r, a, δ) of S taking these orthogonal sums). We have U = [c_{1}, c_{2}] where c_{1}^{2} = c_{2}^{2} = 0 and c_{1} · c_{2} = 1 (the same for U(2), only c_{1} · c_{2} = 2). Then S is generated by elements with square −2 which are

Δ^{(2)}(T) ∪ (c_{1} ⊕ Δ^{(2)}(T)) ∪ (c_{2} ⊕ Δ^{(2)}(T)).

If r = a then S ≅ ⟨2⟩ ⊕ tA_{1}. Let h, e_{1}, …, e_{t} be the corresponding orthogonal basis of S where h^{2} = 2 and e_{i}^{2} = −2, i = 1, …, t. Then S is generated by elements with square (−2) which are e_{1}, …, e_{t} and h − e_{1} − e_{2}.

Let us show that P (X)_{+} generates S. Indeed, every element of Δ^{(2)}(S)∪Δ_{+}^{(4)} can be obtained by composition of reflections in elements of P (X)_{+} from some element of P (X)_{+}. It follows, that it is an integral linear combination of elements of P (X)_{+}. Since we can get in this way all elements of Δ^{(2)}(S) and they generate S, it follows that P (X)_{+} generates S.

It follows that the lattice S with its elements P (X)_{+} is defined by the Dynkin diagram Γ(P (X)_{+}). From S, we can find invariants (r, a, δ) of S, and they define invariants (k, g, δ).

Let K^{+}(2) ⊂ S be a sublattice generated by P^{(4)}(X)_{+} (i. e. by the black vertices), and ξ^{+} : Q = (1/2)K^{+}(2)/K^{+}(2) → q_{S} the homomorphism with ξ^{+}(x/2 + K^{+}(2)) = x/2 + S. By Proposition 3.3, the pair (K^{+}(2), ξ^{+}) coincides with the root invariant R(X, θ).

Now assume that r = rk S = 1, 2 for the pair (X, θ). Then S ≅ ⟨2⟩, U(2), U or ⟨2⟩ ⊕ ⟨−2⟩.

In the first two cases Δ^{(2)}(S) = ∅ and then P^{(2)}(X)_{+} = ∅. In the last two cases Δ^{(2)}(S) and P^{(2)}(X)_{+} are not empty.

Thus, only the first two cases give an empty diagram P^{(2)}(X)_{+}. This distinguishes these two cases from all others. In the case S = ⟨2⟩, the invariant g = 10, and the root invariant is always zero because S has no elements with square −4. Thus, in this case, the diagram P (X)_{+} is always empty. This case gives Y = X/{1, θ} ≅ ℙ^{2}. In the case S = U(2), the diagram P^{(2)}(X)_{+} is empty, but P^{(4)}(X)_{+} = ∅, if the root invariant is zero, and P^{(4)}(X)_{+} consists of one black vertex, if the root invariant is not zero (see Table 1 for this case). First case gives Y = 𝔽_{0}. Second case gives Y = 𝔽_{2}. In both these cases g = 9. Thus difference between two cases when the diagram is empty (ℙ^{2} or 𝔽_{1}) is in genus: g = 10 for the first case, and g = 9 for the second.

The difference of S = U(2) with a non-empty diagram Γ(P (X)_{+}) from all other cases is that this diagram consists of only one black vertex. All cases with rk S ≥ 3 must have at least 3 different vertices to generate S. In cases S = U and S = ⟨2⟩⊕⟨−2⟩, the diagram Γ(P (X)_{+}) also consists of one vertex, but it is respectively double transparent and simple transparent (see Table 1). Moreover, this consideration also shows the difference between cases S = U and S = ⟨2⟩ ⊕ ⟨−2⟩ and with all other cases. □

Theorem 3.5 shows that to classify pairs (X, θ) of elliptic type, we can use any of the following invariants: either the root invariant, or the root subsystem (together with the main invariants (k, g, δ) or (r, a, δ)), or the Dynkin diagram of exceptional curves.

It seems that the most natural and geometric is the classification by the Dynkin diagram. Using this diagram, on the one hand, it easy to calculate all other invariants. On the other hand, considering the corresponding DPN surface, we get the Gram diagram of all exceptional curves on it and all possibilities to get the DPN surface by blow-ups from relatively minimal rational surfaces.

However, the statements (i) and (ii) of Theorem 3.5 are also very important since they give a simple way to find out if two pairs (X, θ) and (X′, θ′) (equivalently, the corresponding DPN surfaces) have isomorphic Dynkin diagrams of exceptional curves. Moreover, the classification in terms of root invariants and root subsystems is much more compact, since the full Gram diagram of exceptional curves can be very large (e. g. recall the classical non-singular del Pezzo surface corresponding to 𝔼_{8}).

We have the following

Theorem 3.6 (Classification Theorem in the extremal case of elliptic type). Over ℂ, a K3 surface with a non-symplectic involution (X, θ) of elliptic type is extremal, if and only if the number of its exceptional curves with the square (−4), i. e. #P^{(4)}(X)_{+}, is equal to #P^{(4)}(M^{(2,4)}) (see Theorem 3.1) where M^{(2,4)} is a fundamental chamber of W^{(2,4)}(S), S = (S_{X})_{+}. Equivalently, numbers of black vertices of Dynkin diagrams Γ(P (X)_{+}) and Γ(P (M^{(2,4)})) with the same invariants (r, a, δ) are equal.

Moreover, the diagram Γ(P (X)_{+}) is isomorphic to (i. e. coincides with) Γ(P (M^{(2,4)})) (see Table 1) in all cases of Theorem 3.1 except cases 7, 8, 9, 10 and 20 of Table 1. In the last five cases, all possible diagrams Γ(P (X)_{+}) are given in Table 2. All diagrams of Tables 1 and 2 correspond to some extremal K3 (X, θ) over ℂ.

Proof. See Sect. 3.4 below. □

Now let us consider a description of non-extremal pairs (X, θ). The worst way to describe them is using full diagrams Γ(P (X)_{+}), since the number of non-extremal pairs (X, θ) is very large and diagrams Γ(P (X)_{+}) can be huge. It is better to describe them using Proposition 3.4 and Theorem 3.5. It is better to describe them by primitive root subsystems Δ_{+}^{(4)}(M^{(2)})′ in the root subsystems Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) of extremal pairs (X̃, θ̃).

Let us choose M^{(2)} in such a way that M^{(2)} ⊃ M(X̃)_{+}^{(2,4)}. By Sect. 2.3.1, then Δ_{+}^{(4)}(M^{(2)}) = Δ^{(4)}([P^{(4)}(X̃)_{+}]) is the subsystem of roots with the basis P^{(4)}(X̃)_{+}, i. e. Δ_{+}^{(4)}(M^{(2)}) = Δ^{(4)}([P^{(4)}(X̃)_{+}]) is the set of all elements with the square (−4) in the sublattice [P^{(4)}(X̃)_{+}] generated by P^{(4)}(X̃)_{+} in S = (SX̃)_{+}. Equivalently, Δ^{(4)}([P^{(4)}(X̃)_{+}]) = W_{+}^{(4)}(X̃)(P^{(4)}(X̃)_{+}), where W_{+}^{(4)}(X̃) is the finite Weyl group generated by reflections in all elements of P^{(4)}(X̃)_{+}.

Replacing a primitive root subsystem Δ_{+}^{(4)}(M^{(2)})′ ⊂ Δ^{(4)}([P^{(4)}(X̃)_{+}]) for a non-extremal pair (X, θ) by an equivalent root subsystem ϕ(Δ_{+}^{(4)}(M^{(2)})′), ϕ ∈ W_{+}^{(4)}(X̃), we can assume (by primitivity) that a basis of Δ_{+}^{(4)}(M^{(2)})′ is a part of the basis P^{(4)}(X̃)_{+} of the root system Δ^{(4)}([P^{(4)}(X̃)_{+}]). Thus, we can assume that the root subsystem Δ_{+}^{(4)}(M^{(2)})′ is defined by a subdiagram

D ⊂ Γ(P^{(4)}(X̃)_{+})

where Γ(P^{(4)}(X̃)_{+}) is the subdiagram of the full diagram Γ(P (X̃)_{+}) generated by all its black vertices. The D is a basis of Δ_{+}^{(4)}(M^{(2)})′.

By Propositions 2.2, 2.3 and Theorem 2.4, the subdiagram D ⊂ Γ(P^{(4)}(X̃)_{+}) defines the full Dynkin diagram Γ(P (X)_{+}) of the pair (X, θ) with the root subsystem Δ_{+}^{(4)}(M^{(2)})′: We have

(22)  P^{(2)}(X)_{+} = {f ∈ W_{+}^{(4)}(X̃)(P^{(2)}(X̃)_{+}) | f · D ≥ 0}.

The subdiagram of Γ(P (X)_{+}) defined by all its black vertices coincides with D. It is called Du Val’s part of Γ(P (X)_{+}), and it is denoted by Duv Γ(P (X)_{+}). Thus,

Duv Γ(P (X)_{+}) = D ⊂ Duv Γ(P (X̃)_{+}).

Double transparent vertices of Γ(P (X)_{+}) identify with double transparent vertices of Γ(P (X̃)_{+}) (see Sect. 2.5), and single transparent vertices of P (X)_{+} which are connected by two edges with double transparent vertices of Γ(P (X)_{+}) identify with such vertices of Γ(P (X̃)_{+}). Indeed, they are orthogonal to the set P^{(4)}(X̃)_{+} which defines the reflection group W_{+}^{(4)}(X̃) as the group generated by reflections in all elements of P^{(4)}(X̃)_{+}. Thus, the group W_{+}^{(4)}(X̃) acts identically on all these vertices, and all of them satisfy (22). All double transparent vertices and all single transparent vertices connected by two edges with double transparent vertices of Γ(P (X)_{+}) define the logarithmic part of Γ(P (X)_{+}), and it is denoted by Log Γ(P (X)_{+}). Thus, we have

Log Γ(P (X)_{+}) = Log Γ(P (X̃)_{+}),

logarithmic parts of X and X̃ identify. Moreover, the Du Val part Duv Γ(P (X)_{+}) and the logarithmic part Log Γ(P (X)_{+}) are disjoint in Γ(P (X)_{+}) because they are orthogonal to each other. Thus, the logarithmic part of Γ(P (X)_{+}) is stable, it is the same for all pairs (X, θ) with the same main invariants (r, a, δ). On the Du Val part of Γ(P (X)_{+}) we have only a restriction: it is a subdiagram of Du Val part of one of extremal pairs (X̃, θ̃) described in Theorems 3.1 and 3.6 (with the same main invariants (r, a, δ)).

All vertices of Γ(P (X)_{+}) which do not belong to Duv Γ(P (X)_{+}) ∪ Log Γ(P (X)_{+}) define a subdiagram Var Γ(P (X)_{+}) which is called the varying part of Γ(P (X)_{+}). By (22), we have

Var P (X)_{+} = {f ∈ W_{+}^{(4)}(X̃)(Var P (X̃)_{+}) | f · D ≥ 0}

(we skip Γ when we consider only vertices). It describes Var Γ(P (X)_{+}) by the intersection pairing in S.

Of course, different Dynkin subdiagrams D ⊂ Γ(P^{(4)}(X̃)_{+}) and D ⊂ Γ(P^{(4)}(X̃′)_{+}) of extremal pairs (X̃, θ̃) and (X̃′, θ̃′) (with the same main invariants) can give isomorphic Dynkin diagrams Γ(P (X)_{+}) and Γ(P (X′)_{+}) for defining by them K3 pairs (X, θ) and (X′, θ′). To have that, it is necessary and sufficient that defining by them root invariants ([D], ξ^{+}) and ([D], (ξ′)^{+}) are isomorphic. We remind that they can be obtained by restriction on [D] of the root invariants of pairs (X̃, θ̃) and (X̃′, θ̃′) respectively, and they can be easily computed. We remind that to have ([D], ξ^{+}) and ([D], (ξ′)^{+}) isomorphic, there must exist an automorphism γ : D → D of diagrams and an automorphism \overline{ϕ} ∈ O(q_{S}) of the discriminant quadratic form of the lattice S which send ξ^{+} for (ξ′)^{+}. Sect. 2.4 gives the very simple and effective method for that. Thus, we have a very simple and effective method to find out when different subdiagrams D above give isomorphic K3 pairs.

Note that we have used all equivalent conditions (i), (ii) and (iii) of Theorem 3.5 which shows their importance. Finally, we get

Theorem 3.7 (Classification Theorem in the non-extremal, i. e. general, case of elliptic type). Dynkin diagrams Γ(P (X)_{+}) of exceptional curves of non-extremal (i. e. general) non-symplectic involutions (X, θ) of elliptic type of K3 surfaces are described by arbitrary (without restrictions) Dynkin subdiagrams D ⊂ Duv Γ(P (X̃)_{+}) of extremal pairs (X̃, θ̃) (see Theorem 3.6) with the same main invariants (r, a, δ) (equivalently (k, g, δ)). Moreover,

Duv Γ(P (X)_{+}) = D, Log Γ(P (X)_{+}) = Log Γ(P (X̃)_{+}),

and they are disjoint to each other,

Var P (X)_{+} = {f ∈ W_{+}^{(4)}(X̃)(Var P (X̃)_{+}) | f · D ≥ 0}

where W_{+}^{(4)}(X̃) is generated by reflections in all elements of Duv P (X̃)_{+} = P^{(4)}(X̃)_{+}.

Dynkin subdiagrams D ⊂ Duv Γ(P (X̃)_{+}) give pairs (X, θ) with isomorphic Dynkin diagrams Γ(P (X)_{+}), if and only if the defining by them root invariants ([D], ξ^{+}) are isomorphic.

Table 2. Diagrams Γ(P (X)_{+}) of extremal (X, θ) of elliptic type which are different from Table 1 (in (a) we repeat the corresponding case of Table 1)

N r a δ k g l(H)  Γ(P (X)_{+})
7 6 6 1 0 5

a  0
b  1
8 7 7 1 0 4

a  0

b  1

c  0
9 8 8 1 0 3

a  0

b  1

c  1

d  1

e  1

N r a δ k g l(H)  Γ(P (X)_{+})
9 8 8 1 0 3

f  1
10 9 9 1 0 2

a  0

b  0

c  1

d  1

e  0

N r a δ k g l(H)  Γ(P (X)_{+})
10 9 9 1 0 2

f  1

g  1

h  0

i  1

j  2

k  2

N r a δ k g l(H)  Γ(P (X)_{+})
10 9 9 1 0 2

l  2

m  0

N r a δ k g l(H)  Γ(P (X)_{+})
20 10 8 1 1 2

a  1

b  2

c  1

d  2

3.4. Proof of Theorem 3.6. Let (X, θ) be a non-symplectic involution of elliptic type of a K3 surface over ℂ with the main invariants (r, a, δ), and (X, θ) is an extremal pair.

By Theorem 3.5, the Γ(P (X)_{+}) is defined by the corresponding to (X, θ) root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) where M^{(2)} is a fundamental chamber of W^{(2)}(S), and S = (S_{X})_{+} has the invariants (r, a, δ). We can assume that M^{(2)} ⊃ M(X)_{+}^{(2,4)} ⊃ M^{(2,4)} where M^{(2,4)} is a fundamental chamber of W^{(2,4)}(S) defined by a choice of a basis P^{(4)}(M^{(2,4)}) of the root system Δ^{(4)}(M^{(2)}) (see Sect. 2.3.1).

Let Γ(P^{(4)}(M^{(2,4)})) be the Dynkin diagram of the root system Δ^{(4)}(M^{(2)}) and W^{(4)}(M^{(2)}) the Weyl group of the root system Δ^{(4)}(M^{(2)}). We use the following description of a root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}).

3.4.1. Let T ⊂ R be a root subsystem of a root system R and all components of R have types 𝔸, 𝔻 or 𝔼. We consider two particular cases of root subsystems.

Let T ⊂ R be a primitive root subsystem. Then T can be replaced by an equivalent root subsystem ϕ(T), ϕ ∈ W (R), such that there exists a basis B of R such that a part of the basis B gives a basis of T (see [Bour68]). Thus (up to equivalence defined by the Weyl group W (R)), primitive root subsystems T ⊂ R can be described by Dynkin subdiagrams Γ ⊂ Γ(B).

Now let T ⊂ R be a root subsystem of a finite index. Let R_{i} be a component of R. Let r_{j}, j ∈ J, be a basis of R_{i}. Let r_{max} = ∑_{j∈J} k_{j}r_{j} be the maximal root of R_{i} corresponding to this basis. Dynkin diagram of the set of roots

{r_{j} | j ∈ J} ∪ {−r_{max}}

is an extended Dynkin diagram of the Dynkin diagram Γ({r_{j} | j ∈ J}). Let us replace the component R_{i} of the root system R by the root subsystem R′_{i} ⊂ R_{i} of finite index having by its basis the set ({r_{j} | j ∈ J} ∪ {−r_{max}}) − {r_{t}} where t ∈ J is some fixed element. We get a subsystem R′ ⊂ R or the finite index equals to k_{t}. It can be shown [Dyn] that iterations of this procedure give any root subsystem of finite index of R up to the action of W (R).

Description of an arbitrary root subsystem T ⊂ R can be reduced to these two particular cases, moreover it can be done in two ways.

Firstly, any root subsystem T ⊂ R is a subsystem of finite index T ⊂ T_{pr} where T_{pr} ⊂ R is a primitive root subsystem generated by T.

Secondly, any root subsystem T ⊂ R can be considered as a primitive root subsystem T ⊂ R_{1} where R_{1} ⊂ R is root subsystem of finite index. One can take R_{1} generated by T and by any u = rk R − rk T roots r_{1}, …, r_{u} such that rk[T, r_{1}, …, r_{u}] = rk R.

3.4.2. Here we show that the root subsystems Δ_{+}(M^{(2)}) which coincide with the full root system Δ^{(4)}(M^{(2)}) can be realized by K3 pairs (X, θ). Obviously, they are extremal, for them M(X)_{+}^{(2,4)} = M^{(2,4)}, and the Dynkin diagrams Γ(P (X)_{+}) = Γ(P (M^{(2,4)})) coincide. All these diagrams are described in Table 1 of Theorem 3.1. It is natural to call such pairs (X, θ) as super-extremal. Thus, a non-symplectic involution (X, θ) of elliptic type of K3 (equivalently, the corresponding DPN pair (Y, C) or DPN surface) is called super-extremal, if for the corresponding root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) we have Δ_{+}^{(4)}(M^{(2)}) = Δ^{(4)}(M^{(2)}) (equivalently, Δ_{+}^{(4)} = Δ^{(4)}(S)). We have

Proposition 3.8. For any possible elliptic triplet of main invariants (r, a, δ) there exists a super-extremal, i. e.

Γ(P (X)_{+}) = Γ(P (M^{(2,4)})),

and standard (see Sect. 2.6) K3 pair (X, θ).

See the description of their graphs Γ(P (X)_{+}) = Γ(P (M^{(2,4)})) in Table 1 of Theorem 3.1).

Proof. Let us consider an elliptic triplet of main invariants (r, a, δ) and the corresponding Dynkin diagram Γ(P (M^{(2,4)})) which is described in Theorem 3.1. Denote K^{+}(2) = [P^{(4)}(M^{(2,4)})], i. e. it is the sublattice generated by all black vertices of Γ(P (M^{(2,4)})). Consider the corresponding root invariant (K^{+}(2), ξ^{+}), see (20) and (21). Consider H = Ker ξ^{+}. By Propositions 3.3 and 2.9, there exists a super-extremal standard pair (X, θ), if the inequalities

r + rk K^{+} + l(𝔄_{(K^{+})_{p}}) < 22 for all prime p > 2,
r + a + 2l(H) < 22

are valid together with Conditions 1 and 2 from Sect. 2.6.

By trivial inspection of all cases in Table 1, we can see that first inequality is valid. To prove second inequality, it is enough to show that l(H) ≤ 1 since r +a ≤ 18 in elliptic case. The inequality l(H) ≤ 1 can be proved by direct calculation of l(H) in all cases of Table 1 of Theorem 3.1.1. These calculations simplify by the general statement.

Lemma 3.9. In elliptic super-extremal case,

l(H) = #P^{(4)}(M^{(2,4)}) − l(𝔄_{S}^{(1)})

where 𝔄_{S}^{(1)} ⊂ 𝔄_{S} is the subgroup generated by all elements x ∈ 𝔄_{S} such that q_{S}(x) = 1 mod 2. Moreover, we have:

If δ = 0 then l(𝔄_{S}^{(1)}) = a except (a = 2 and sign S = 2−r ≡ 0 mod 8). In the last case l(𝔄_{S}^{(1)}) = a − 1.

If δ = 1, then l(𝔄_{S}^{(1)}) = a − 1 except cases (a = 2 and sign S ≡ 0 mod 8), (a = 3 and sign S ≡ ±1 mod 8), and (a = 4 and sign S ≡ 0 mod 8). In these cases l(𝔄_{S}^{(1)}) = a − 2.

Proof. We know (see Sect. 2.3.1) that Δ^{(4)}(S) = W^{(2)}(S)(Δ^{(4)}(M^{(2,4)})). The group W^{(2)}(S) acts identically on 𝔄_{S}. Therefore,

Im ξ^{+} = [{ξ^{+}(f/2 + K^{+}(2)) | f ∈ Δ^{(4)}(M^{(2,4)})}] =
[{f/2 + S | f ∈ Δ^{(4)}(S)} = 𝔄_{S}^{(1)}.

In the last equality, we use Lemma 2.6. For Q = (K^{+}(2)/2)/K^{+}(2), we have l(Q) = rk K^{+} = #P^{(4)}(M^{(2,4)}). Thus, l(H) = l(Q)−l(𝔄_{S}^{(1)}) = #P^{(4)}(M^{(2,4)}) − l(𝔄_{S}^{(1)}).

The rest statements of Lemma can be proved by direct calculations using a decomposition of a 2-elementary non-degenerate finite quadratic form as sum of elementary ones: q_{±1}^{(2)}(2), u_{+}^{(2)}(2) and v_{+}^{(2)}(2) (in notations of [Nik80.2]). □

One can easily check Condition 2 of Sect. 2.6.

To check Condition 1 of Sect. 2.6, note that if the lattice K_{H}^{+}(2) has elements with the square (−2), than the sublattice [P^{(4)}(M^{(2,4)})]_{pr} of S also has elements with the square (−2). Let us show that this is not the case.

Let us consider the subspace

γ = ⋂_{f∈P^{(4)}(M^{(2,4)})} H_{f}

of L(S) which is orthogonal to [P^{(4)}(M^{(2,4)})] (equivalently, we consider the corresponding face γ ∩ M^{(2,4)} of M^{(2,4)}). If the sublattice [P^{(4)}(M^{(2,4)})]_{pr} ⊂ S has elements with square (−2), then some hyperplanes H_{e}, e ∈ Δ^{(2)}(S), also contain γ and give reflections from W^{(2,4)}(S). On the other hand (e. g. see [Vin85]), all hyperplanes of reflections from W^{(2,4)}(S) containing γ must be obtained from the hyperplanes H_{f}, f ∈ P^{(4)}(M^{(2,4)}), by the group generated by reflections in P^{(4)}(M^{(2,4)}). All these hyperplanes are then also orthogonal to elements with square (−4) from S. They cannot be orthogonal to elements with square (−2) from S too.

This finishes the proof of Proposition 3.8.

3.4.3. Let us prove Theorem 3.6 in all cases except 7 — 10 and 20 of Table 1. These cases (i. e. different from 7 — 10 and 20 of Table 1) characterize by the property that Dynkin diagram Γ(P^{(4)}(M^{(2,4)})) consists of components of type 𝔸 only. By Sect. 3.4.1, any primitive root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) is then primitive. In particular, any root subsystem Δ_{+}^{(4)}(M^{(2)}) ⊂ Δ^{(4)}(M^{(2)}) of finite index is Δ_{+}^{(4)}(M^{(2)}) = Δ^{(4)}(M^{(2)}). By Proposition 3.8, we then obtain

Proposition 3.10. For any elliptic triplet (r, a, δ) of main invariants which is different from (6,6,1), (7,7,1), (8,8,1), (9,9,1) and (10,8,1), any extremal K3 pair (X, θ) is super-extremal, i. e. Γ(P (X)_{+}) = Γ(P (M^{(2,4)})) (see their description in Table 1 of Theorem 3.1).

3.4.4. Now let us consider cases 7—10 and 20 of Table 1. In these cases, the root system R = Δ^{(4)}(M^{(2)}) is 𝔻_{5} in the case 7, 𝔼_{6} in the case 8, 𝔼_{7} in the case 9, 𝔼_{8} in the case 10, and 𝔻_{8} in the case 20.

We have

Lemma 3.11. If R is one of root systems of types 𝔻_{5}, 𝔼_{6}, 𝔼_{7}, 𝔼_{8} or 𝔻_{8}, then its root subsystem T ⊂ R of finite index is defined by the isomorphism type of the root system T itself, up to the action of W (R). Moreover, the type of T can be the following and only the following which is given in Table of Lemma 3.11 below (we identify the type with the isomorphism class of the corresponding root lattice).

Table of Lemma 3.11.

N R  T
7 D_{5} a) D_{5}, b) A_{3} ⊕ 2A_{1}
8 E_{6} a) E_{6}, b) A_{5} ⊕ A_{1}, c) 3A_{2}
a) E_{7}, b) A_{7}, c) A_{5} ⊕ A_{2}, d) 2A_{3} ⊕ A_{1}, e) D_{6} ⊕ A_{1},
9 E_{7} f) D_{4} ⊕ 3A_{1}, g) 7A_{1}
a) E_{8}, b) A_{8}, c) A_{7} ⊕ A_{1}, d) A_{5} ⊕ A_{2} ⊕ A_{1}, e) 2A_{4},

f) D_{8}, g) D_{5} ⊕ A_{3}, h) E_{6} ⊕ A_{2}, i) E_{7} ⊕ A_{1}, j) D_{6} ⊕ 2A_{1},

10 E_{8} k) 2D_{4}, l) 2A_{3} ⊕ 2A_{1}, m) 4A_{2}, n) D_{4} ⊕ 4A_{1}, o) 8A_{1}

a) D_{8}, b) D_{6} ⊕ 2A_{1}, c) D_{5} ⊕ A_{3}, d) 2D_{4}, e) 2A_{3} ⊕ 2A_{1},

20 D_{8} f) D_{4} ⊕ 4A_{1}, g) 8A_{1}

Proof. It is done by inspection using description of root subsystems of finite index in Sect. 3.4.1. □

Consider a root subsystem T ⊂ R of Lemma 3.11. By Theorem 2.4, the root subsystem T ⊂ R defines a subset Δ_{+}^{(4)}(S) ⊂ Δ^{(4)}(S), the corresponding reflection group W_{+}^{(2,4)}, and Dynkin diagram Γ(P (M_{+}^{(2,4)})) of its fundamental chamber M_{+}^{(2,4)}. Direct calculation of these diagrams using Theorem 2.4 gives diagrams of Table 2 of Theorem 3.6 (where Γ(P (M_{+}^{(2,4)})) is replaced by Γ(P (X)_{+})) in all cases 7a, b; 8a — c; 9a — f; 10a — m; 20a — d. In the rest cases 9g; 10n, o; 20e — g we get diagrams Γ(P (M_{+}^{(2,4)})) which we describe below.

In the Case 9g, it is better to describe Γ(P (M_{+}^{(2,4)})) indirectly. Its transparent vertices correspond to all points of ℙ^{2}(F_{2}) which is the projective plane over the field F_{2} with two elements. Its black vertices correspond to all lines in ℙ^{2}(F_{2}). Both sets have seven elements. Transparent vertices are disjoint; black vertices are also disjoint; a transparent vertex is connected with a black vertex by the double edge, if the corresponding point belongs to the corresponding line, otherwise, they are disjoint.

In the Case 10n, the diagram Γ(P (M_{+}^{(2,4)})) is given below in Figure 3. Since it is quite complicated, we divide it in three subdiagrams shown. The first one shows all its edges connected black and transparent vertices. The second one shows the edge connected the transparent vertices numerated by 1 and 2. The third one shows edges connected transparent vertices 3 — 12. Each edge of Γ(P (M_{+}^{(2,4)})) is shown in one of these diagrams. (All other our similar descriptions of diagrams

Figure 3. The diagram 10n

as unions of their subdiagrams have the same meaning. In particular, we have used it in some diagrams of Table 2.

In the Case 10o, we describe the diagram Γ(P (M_{+}^{(2,4)})) indirectly. Its black vertices f_{v}, v ∈ V, correspond to all points of a three-dimensional affine space V over F_{2}. Its transparent vertices are of two types. Vertices e_{v} of the first type also correspond to all points v ∈ V. Vertices e_{π} of the second type correspond to all (affine) planes π ⊂ V (there are 14 of them). Black vertices f_{v} are disjoint. A black vertex f_{v} is connected with a transparent vertex e_{v′}, if and only if v = v′; the edge has the weight \sqrt{8}. A black vertex f_{v} is connected with a transparent vertex e_{π}, if and only if v ∈ π; the edge is double. Transparent vertices e_{v}, e_{v′} are connected by a thick edge. A transparent vertex e_{v} is connected with a transparent vertex e_{π}, if and only if v ∉ π; the edge is thick.

Figure 4. The diagram 20e

Figure 5. The diagram 20f

Transparent vertices e_{π}, e_{π′} are connected by edge, if and only if π∥π′; the edge is thick.

In Cases 20e, 20f and 20g diagrams Γ(P (M_{+}^{(2,4)})) are shown in figures 4—6 below.

We remark that a calculation of Γ(P (M_{+}^{(2,4)})) in cases 7a,b, 8a— c, 9a—g and 10a—o can be obtained from results of [BBD84] where (in our notations) the dual diagram of all exceptional curves on the quotient Y = X/{1, θ} is calculated using completely different method (under the assumption that Y does exist). By Sect. 2.4, both diagrams can be easily obtained from one another (compare with Sect. 3.5 below). Therefore, we explain our method of calculation of Γ(P (M_{+}^{(2,4)}))

Figure 6. The diagram 20g

in more details than it has done in Sect. 2.3.1 only in the Case 20 (i. e. cases 20a—g).

In the Case 20, the lattice S has invariants (r, a, δ) = (10, 8, 1), and we can take in S ⊗ ℚ an orthogonal basis h, a, v_{1}, …, v_{8} with h^{2} = 2, a^{2} = v_{1}^{2} = ⋯ = v_{8}^{2} = −2. As P (M^{(2,4)}), we can take

\begin{matrix}P^{(4)}(M^{(2,4)}) = & \{f_{1} = v_{1} − v_{2}, f_{2} = v_{2} − v_{3}, f_{3} = v_{3} − v_{4}, f_{4} = v_{4} − v_{5}, \\ f_{5} = v_{5} − v_{6}, f_{6} = v_{6} − v_{7}, f_{7} = v_{7} − v_{8}, f_{8} = v_{7} + v_{8}, \} \end{matrix} (23)

and

P^{(2)}(M^{(2,4)}) = \{a, b = \frac{h}{2} − \frac{a}{2} − v_{1}, c = h − \frac{1}{2} (v_{1} + v_{2} + ⋯ + v_{8})\}.

These elements have Dynkin diagram

of the case 20a, and they generate and define S.

By Sect. 2.3.1, the set P^{(2)}(M^{(2)}), where M^{(2)} ⊃ M^{(2,4)}, is

W (M^{(2)})({a, b, c})

where W (M^{(2)}) is generated by reflections in f_{1}, …, f_{8}. It follows that

P (M^{(2)}) = P^{(2)}(M^{(2)}) = {a; b_{±i}; c_{i_{1}...i_{k}}}

where

b_{±i} = \frac{h}{2} − \frac{a}{2} ± v_{i}, i = 1, 2, …, 8;

c_{i_{1}...i_{k}} = h + \frac{1}{2} (v_{1} + v_{2} + ⋯ + v_{8}) − v_{i_{1}} − ⋯ − v_{i_{k}},

where 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ 8 and k ≡ 0 mod 2. Here all b_{±i} give the W (M^{(2)})-orbit of b, and all c_{i_{1}...i_{k}} give the W (M^{(2)})-orbit of c.

Elements f_{1}, …, f_{8} give a basis of the root system R of type 𝔻_{8}. If T ⊂ R is its subsystem of rank m, and t_{1}, …, t_{m} a basis of T, then defining by T and by its basis t_{1}, …, t_{m} fundamental chamber M_{+}^{(2,4)} ⊂ M^{(2)} has P (M_{+}^{(2,4)}) = P^{(4)}(M_{+}^{(2,4)}) ∪ P^{(2)}(M_{+}^{(2,4)}) where

\begin{matrix}P^{(4)}(M_{+}^{(2,4)}) = & \{t_{1}, …, t_{m}\}, \\ P^{(2)}(M_{+}^{(2,4)}) = & \{a\} ∪ \{b_{±i} | b_{±i} · t_{s} ≥ 0, 1 ≤ s ≤ m\} \\ ∪ & \{c_{i_{1}...i_{k}} | c_{i_{1}...i_{k}} · t_{s} ≥ 0, 1 ≤ s ≤ m\}. \end{matrix} ; (24)

This describes Γ(P (M_{+}^{(2,4)})) completely.

For example, assume that T ⊂ R has the type 2A_{1}⊕D_{6} with the basis f_{1}, f_{9} = −v_{1} − v_{2}, f_{3}, …, f_{8}. Then we get (after simple calculations)

P^{(2)}(M_{+}^{(2,4)}) = {a, b_{2}, b_{−3}, c_{345678}, c_{134567}},

and Γ(P (M_{+}^{(2,4)})) is

This gives Case 20b of Table 2.

Exactly the same calculations of Γ(P (M_{+}^{(2,4)})) can be done in all cases 20a—g, and cases 7a,b — 10a—o of Table of Lemma 3.11 as well. Details of these calculations require too much space, and we leave them to a reader. The final results have been presented.

3.4.5. Here we prove

Proposition 3.12. Cases 9g, 10n,o and 20 e — g of root subsystems T ⊂ R of Lemma 3.11 do not correspond to non-symplectic involutions (X, θ) of K3 (in characteristic 0 and even in characteristic ≥ 3).

Proof. Assume that a root subsystem T ⊂ R corresponds to a K3 pair (X, θ). Then the corresponding Dynkin diagram Γ(P (M_{+}^{(2,4)})) given in Sect. 3.4.4 coincides with Dynkin diagram Γ(P (X)_{+}) of exceptional curves of the pair (X, θ). It follows the dual diagram of exceptional curves Γ(P (Y)) on the corresponding DPN surface Y = X/{1, θ} (see Sect. 2.3). Using this diagram, it is easy to find a sequence of exceptional curves E_{1}, …, E_{k} on Y where k = r − 1 such that their contraction gives a morphism σ : Y → ℙ^{2}. Then other (different from E_{1}, …, E_{k}) exceptional curves on Y corresponding to Du Val and logarithmic part of Γ(P (Y)) give a configuration of rational curves on ℙ^{2} which cannot exist in characteristic 0 and even in characteristic ≥ 3 (but it exists in characteristic 2). In cases 9g; 10n,o; 20e,f we get Fano’s configuration of seven lines of the finite projective plane over F_{2} which can exist only in characteristic 2. In the case 9g one should contract exceptional curves corresponding to all transparent vertices. In the case 10n — corresponding to vertices 1, f, 3 — 8. In the case 10o — corresponding to vertices e_{π} where π contains a fixed point 0 ∈ V and e_{0}; then curves corresponding to f_{v}, v ≠ 0, give Fano’s configuration. In cases 20e,f — corresponding to vertices 1 — 9. In the case 20g — corresponding to vertices 1 — 9, then we get a conic (corresponding to the double transparent vertex) and four its tangent lines (corresponding to black vertices different from 5 — 8) passing through one point. It is possible only in characteristic 2. □

Another purely arithmetic proof of Proposition 3.12 can be obtained using Proposition 2.10. This proof is more complicated, but it can also be done. Here we preferred more short (if diagrams have calculated) the geometric considerations.

□

3.4.6. Here we prove

Proposition 3.13. Cases 7a,b; 8a—c, 9a—f; 10a—m and 20a—d of Table 2 of Theorem 3.6 correspond to standard extremal non-symplectic K3 involutions (X, θ) over ℂ.

Proof. Let us calculate corresponding to these cases root invariants (K^{+}(2), ξ^{+}).

Consider the sequence of embeddings of lattices

K^{+}(2) = [T] ⊂ [R] ⊂ S.

It defines the homomorphism

ξ^{+} : Q = \frac{1}{2} K^{+}(2)/K^{+}(2) → S^{∗}/S ⊂ \frac{1}{2}S/S

with the kernel H. It can be decomposed as

(25)  ξ^{+} : Q \underrightarrow{ξ̃^{+}} \frac{1}{2} [R]/[R] ⟶^{ξ_{R}^{+}} S^{∗}/S ⊂ \frac{1}{2} S/S.

Let H_{R} = Ker ξ_{R}^{+}. Then H = (ξ̃^{+})^{−1}(H_{R}). As we know (from our considerations in the super-extremal case), H_{R} = 0 in cases 7, 8, 9, 10. In the case 20, the H_{R} = ℤ/2ℤ is

H_{R} = [ \frac{1}{2} (f_{1} + f_{3} + f_{5} + f_{7}) + R]/[R]

(see Sect. 3.4.4 about this case). Thus, H can be identified with H = ( \frac{1}{2} [T] ∩ [R])/[T] in cases 7, 8, 9, 10, and with

H = ( \frac{1}{2} [T] ∩ [ \frac{1}{2} (f_{1} + f_{3} + f_{5} + f_{7}) + R])/[T]

in the case 20. Using Lemma 3.11 and the described in Sect. 3.4.1 method of construction of root subsystems T ⊂ R, the subgroups H can be easily calculated.

In Case 7a the H = 0.

In Case 7b with the basis (in T)

the H = [(f_{1} + f_{2} + f_{3} + f_{5})/2] mod [T].

In Case 8a the H = 0.

In Case 8b with the basis

the H = [(f_{1} + f_{2} + f_{4} + f_{6})/2] mod [T].

In Cases 8c and 9a the H = 0.

In Case 9b with the basis

the H = [(f_{1} + f_{3} + f_{5} + f_{7})/2] mod [T].

In Case 9c the H = 0.

In Case 9d with the basis

the H = [(f_{1} + f_{3} + f_{4} + f_{6})/2] mod [T].

In Case 9e with the basis

the H = [(f_{1} + f_{2} + f_{4} + f_{6})/2] mod [T]. In Case 9f with the basis

the H = [(f_{1} + f_{2} + f_{4} + f_{6})/2] mod [T]. In Cases 10a and 10b the H = 0. In Case 10c with the basis

the H = [(f_{2} + f_{4} + f_{6} + f_{8})/2] mod [T]. In Case 10d with the basis

the H = [(f_{1} + f_{4} + f_{6} + f_{8})/2] mod [T]. In Case 10e the H = 0. In Case 10f with the basis

the H = [(f_{1} + f_{3} + f_{5} + f_{7})/2] mod [T]. In Case 10g with the basis

the H = [(f_{1} + f_{3} + f_{7} + f_{8})/2] mod [T]. In Case 10h the H = 0. In Case 10i with the basis the H = [(f_{1} + f_{2} + f_{4} + f_{8})/2] mod [T].

In Case 10j with the basis

the H = [(f_{1} + f_{3} + f_{5} + f_{7})/2, (f_{2} + f_{3} + f_{5} + f_{8})/2] mod [T]. In Case 10k with the basis

the H = [(f_{1} + f_{2} + f_{5} + f_{6})/2, (f_{2} + f_{3} + f_{6} + f_{7})/2] mod [T]. In Case 10l with the basis

the H = [(f_{1} + f_{3} + f_{7} + f_{8})/2, (f_{4} + f_{6} + f_{7} + f_{8})/2] mod [T]. In Case 10m the H = 0. In Case 20a with some basis

the H = [(f_{1} + f_{3} + f_{5} + f_{7})/2] mod [T].

In Case 20b with the basis

the H = [(f_{1} + f_{3} + f_{5} + f_{7})/2, (f_{2} + f_{3} + f_{5} + f_{8})/2] mod [T]. In Case 20c with the basis the H = [(f_{1} + f_{3} + f_{7} + f_{8})/2] mod [T].

In Case 20d with the basis

the H = [(f_{1} + f_{2} + f_{5} + f_{6})/2, (f_{2} + f_{3} + f_{6} + f_{7})/2] mod [T].

¿From these calculations, we get values of l(H) given in Table 2 of Theorem 3.6.

As in Sect. 3.4.2, using Proposition 2.9, one can prove that all these cases when

(26)  r + a + 2l(H) < 22

correspond to standard extremal non-symplectic K3 involutions (X, θ) (always over ℂ). Thus, we only need to consider cases when the inequality (26) fails. There are exactly five such cases: 10j,k,l and 20b,d. Further we consider these cases only.

In cases 10j,k,l the discriminant form of S is q_{S} = q_{1}^{(2)}(2) ⊕ 4u_{+}^{(2)}(2) (in notations of [Nik80.2]). Here, the generator of the first summand q_{1}^{(2)}(2) gives the characteristic element a_{q_{S}} of the q_{S}, and the second summand 4u_{+}^{(2)}(2) gives the image of ξ_{R}^{+} from (25), by Lemma 3.9. Thus, the image of ξ^{+} belongs to 4u_{+}^{(2)}(2). The discriminant form of the lattice M (from 2.6) is obtained as follows (see [Nik80.2] about this technique). Let

Γ_{ξ^{+}} ⊂ Q ⊕ 𝔄_{S} ⊂ 𝔄_{K^{+}(2)} ⊕ 𝔄_{S}

be the graph of the homomorphism ξ^{+} in 𝔄_{K^{+}(2)} ⊕ 𝔄_{S}. Then

(27)  q_{M} = (q_{K^{+}(2)} ⊕ q_{S} | (Γ_{ξ^{+}})_{q_{K^{+}(2)}⊕q_{S}}^{⊥})/Γ_{ξ^{+}}

(here Γ_{ξ^{+}} is an isotropic subgroup).Therefore, q_{M} ≅ q_{1}^{(2)}(2) ⊕ q′ since the image of ξ^{+} belongs to the orthogonal complement of the summand q_{1}^{(2)}(2). Considerations in the proof of Proposition 2.9 show that

(28)  rk M + l(𝔄_{M_{2}}) ≤ 22

since r + a + 2l(H) = 22 in cases 10j,k,l. It is easy to see that

rk M + l(𝔄_{M_{p}}) < 22

for all prime p > 2. Then, by Theorem 1.12.2 in [Nik80.2], there exists a primitive embedding M ⊂ L_{K3} when either the inequality (28) is strict or q_{M_{2}} ≡ q_{±1}^{(2)}(2) ⊕ q′, if it gives the equality. Thus, it always does exist. It follows that all cases 10j,k,l correspond to standard extremal non-symplectic K3 involutions (X, θ) over ℂ by Proposition 2.10 (where we have used fundamental Global Torelli Theorem [PS-Sh71] and surjectivity of Torelli map [Kul77] for K3).

In cases 20b,d, the proof is exactly the same, but it is more difficult to prove that q_{M_{2}} ≅ q_{θ}^{(2)}(2) ⊕ q′ where θ = ±1. In these cases

q_{S} = 3u_{+}^{(2)}(2) ⊕ q_{1}^{(2)}(2) ⊕ q_{−1}^{(2)}(2).

If α_{1} and α_{2} are generators of the summands q_{1}^{(2)}(2) and q_{−1}^{(2)}(2) respectively, then α_{q_{S}} = α_{1} + α_{2} is the characteristic element of q_{S}, and the image of ξ^{+} belongs to 3u_{+}^{(2)}(2) ⊕ [α_{q_{S}}]. In these cases, the lattice K_{H}^{+}(2) (see Sect. 2.6) is isomorphic to E_{8}(2). For example, this is valid because the subgroups H are the same in cases 10j and 20b, and in cases 10k and 20d, besides, in cases 10j and 10 k we have E_{8}/K^{+} ≅ H. It follows that

q_{K_{H}^{+}(2)} = (q_{K^{+}(2)} | (H)_{q_{K^{+}(2)}}^{⊥})/H ≅ q_{E_{8}(2)} ≅ 4u_{+}^{(2)}(2).

We set \overline{Γ}_{ξ^{+}} = Γ_{ξ^{+}}/H. By (27)

q_{M} = (q_{K_{H}^{+}(2)} ⊕ q_{S} | (\overline{Γ}_{ξ^{+}})_{q_{K{+ \atop H} (2)}⊕q_{S}}^{⊥})/\overline{Γ}_{ξ^{+}}.

We have q_{K_{H}^{+}(2)} ⊕ q_{S} = 7u_{+}^{(2)}(2) ⊕ q_{1}^{(2)}(2) ⊕ q_{−1}^{(2)}(2). Since u_{+}^{(2)}(2) takes values in ℤ/2ℤ, the element α_{q_{S}} (more exactly, 0⊕α_{q_{S}}) is the characteristic element of q_{K_{H}^{+}(2)} ⊕ q_{S} again. Moreover, α_{q_{S}} ∉ \overline{Γ}_{ξ^{+}} since Γ_{ξ^{+}} is the graph of a homomorphism with the kernel H. Therefore (\overline{Γ}_{ξ^{+}})_{q_{K{+ \atop H} (2)}⊕q_{S}}^{⊥}

contains v which is not orthogonal to α_{q_{S}}. Then

(q_{K_{H}^{+}(2)} ⊕ q_{S})(v) = ±\frac{1}{2} mod 2

and

[v mod \overline{Γ}_{ξ^{+}}] ≅ q_{θ}^{(2)}(2), θ = ±1,

is the orthogonal summand of q_{M_{2}} we were looking for. □

Remark 3.14. We can give another proof of Proposition 3.13 which uses Theorem 1.5 and considerations which are inverse to the proof of the previous Proposition 3.12. Indeed, by Theorem 1.5, it is enough to prove existence of rational surfaces with Picard number r and configuration of rational curves defined by Dynkin diagram of Table 2 of Theorem 3.6 (assuming that these Dynkin diagrams correspond to K3 pairs (X, θ) and considering the quotient by θ). One can prove existence of these rational surfaces considering appropriate sequences of blow-ups of appropriate relatively minimal rational surfaces ℙ^{2}, 𝔽_{0}, 𝔽_{1}, 𝔽_{2}, 𝔽_{3} or 𝔽_{4} with appropriate configurations of rations curves defined by Dynkin diagrams of Table 2 of Theorem 3.6 (see the proof of Proposition 3.12). This proof does not use Global Torelli Theorem and surjectivity of Torelli map for K3. This gives a hope that results of Sect. 2 and Sect. 3 can be generalized to characteristic p > 0. Unfortunately, we have proved Theorem 1.5 in characteristic 0 only. Thus, we preferred the proof of Proposition 3.13 which is independent on the results of Sect. 1.

3.4.7. To finish the proof of Theorem 3.6, we need to prove only

Proposition 3.15. Let (X, θ) be a non-symplectic involution of K3 which corresponds to one of cases 7 — 10 or 20 of Table 1 of Theorem 3.1 and a root subsystem T ⊂ R = Δ^{(4)}(M^{(2)}).

If (X, θ) is extremal, then rk T = rk R

Proof. We can assume (see Sect. 3.4.1) that T has a basis which gives a part of a basis of a root subsystem T̃ ⊂ R = Δ^{(4)}(M^{(2)}) of the same rank rk T̃ = rk R. Then T̃ ⊂ R is one of root subsystems of Lemma 3.11. If the root subsystem T̃ ⊂ R corresponds to a non-symplectic involution of K3, i. e. T̃ gives cases 7a—b, 8a—c, 9a—f, 10a—m and 20a—d, then T is extremal, only if T = T̃ (by definition). Then rk T = rk T̃ = rk R as we want. Thus, it is enough to consider T̃ of cases 9g, 10n—o, 20e—g and T ⊂ T̃ to be a primitive root subsystem of a strictly smaller rank. Let us consider all these cases.

Case 9g. Then T̃ = 7A_{1}, and T = kA_{1}, k ≤ 6. The root subsystem T̃ = 7A_{1} is obtained from the root system D_{4} ⊕ 3A_{1} by the procedure described in Sect. 3.4.1. One can see that then T is a primitive root subsystem in D_{4} ⊕ 3A_{1}. Since D_{4} ⊕ 3A_{1} corresponds to K3, the T is not extremal.

Case 10n. Then T̃ = D_{4} ⊕ 4A_{1}. It is obtained from the root subsystems D_{6} ⊕ 2A_{1} or D_{4} ⊕ D_{4} by the procedure described in Sect. 3.4.1. It follows that T is a primitive root subsystem in D_{6} ⊕ 2A_{1}, D_{4} ⊕ D_{4} or 2A_{3} ⊕ 2A_{1} (and it is not then extremal) in all cases except when T = 7A_{1}.

Let us consider the last case T = 7A_{1} and show (as in Sect. 3.4.5) that it does not correspond to K3. As in Sect.3.4.4, one can calculate Dynkin diagram Γ = Γ(P (M_{+}^{(2,4)})). It is similar to the case 10o (see Sect. 3.4.4), but it is more complicated. We describe it indirectly. One can relate with this diagram a 3-dimensional linear vector space V over F_{2}.

Black vertices f_{v} of Γ correspond to v ∈ V − {0} (there are seven of them). Its transparent vertices (all of them are simple) are e_{v}, v ∈ V − {0}; e_{0}^{(+)}, e_{0}^{(−)}; e_{π}, π ⊂ V is any hypersurface in V which does not contain 0; e_{π}^{(+)}, e_{π}^{(−)}, π ⊂ V is any hyperspace (0 ∈ π) of V.

Edges which connect f_{v}, e_{v}, e_{0}^{(+)}, e_{π}, e_{π}^{(+)} are the same as for the diagram 10o (forget about (+)). The same is valid for f_{v}, e_{v}, e_{0}^{(−)}, e_{π}, e_{π}^{(−)} (forget about (−)). Vertices e_{0}^{(+)} and e_{0}^{(−)} are connected by the punctured edge of the weight 6. Vertices e_{0}^{(+)} and e_{π}^{(−)} (and e_{0}^{(−)}, e_{π}^{(+)} as well) are connected by the punctured edge of the weight 4. This gives all edges of Γ.

Assume that Γ corresponds to a K3 pair (X, θ). Consider the corresponding DPN surface and contract exceptional curves corresponding to e_{π}^{(+)} and e_{0}^{(+)}. Then exceptional curves of f_{v}, v ∈ V − {0}, give Fano’s configuration on ℙ^{2} which exists only in characteristic 2. We get a contradiction.

Case 20e. Then T̃ = 2A_{3} ⊕ 2A_{1}. It is obtained from the root subsystems D_{6} ⊕ 2A_{1} or D_{5} ⊕ A_{3} by the procedure described in Sect. 3.4.1. It follows that T is a primitive root subsystem in D_{6} ⊕ 2A_{1} or D_{5} ⊕ A_{3} (and it is not then extremal) in all cases except when T = A_{3} ⊕ 4A_{1}.

Let us consider the last case T = A_{3} ⊕ 4A_{1} and show (as in Sect. 3.4.5) that it does not correspond to K3. As in Sect.3.4.4, one can calculate Dynkin diagram Γ = Γ(P (M_{+}^{(2,4)})). It has exactly one transparent double vertex a and eight simple transparent vertices c_{v}, v ∈ V (K), where V (K) is the set of vertices of a 3-dimensional cube K with distinguished two opposite 2-dimensional faces α, α′ ∈ γ(K) where γ(K) is the set of all 2-dimensional faces of K. Black vertices of Γ are f_{γ}, γ ∈ γ(K), and one more black vertex f_{0}. Simple transparent vertices of Γ which are connected by a simple edge with a are either b\overline{γ}, \overline{γ} ∈ \overline{γ(K)}, where \overline{γ(K)} is the set of pairs of opposite 2-dimensional faces of K, or b_{t}, t ∈ \overline{V (K)}. Here \overline{V (K)} consists of two elements corresponding to a choice of one vertex from each pair of opposite vertices of K in such a way that neither of three of them are contained in a 2-dimensional face γ ∈ γ(K) (they define a regular tetrahedron with edges which are diagonals of 2-dimensional faces of K).

Let us describe edges of Γ different from above. Thick edges connect c_{v} corresponding to opposite vertices v ∈ V (K), vertices b_{t_{1}} and b_{t_{2}} where \{t_{1}, t_{2}\} = \overline{V (K)}, vertices b_{t} and c_{v} where v ∈ t. Simple edges connect f_{0} with f_{α} and f_{α′}. Double simple edges connect c_{v} with f_{γ}, if v ∈ γ, and b\overline{γ} with f_{γ}, if γ ∈ \overline{γ} − \{α, α′\}, and the vertex b\overline{α} with f_{0}.

Assume that Γ corresponds to a K3 pair (X, θ). On its DPN surface, let us contract exceptional curves corresponding to c_{v}, v ∈ t; b\overline{γ}, \overline{γ} ∈ \overline{γ(K)}; f_{0} and b_{t′}, t′ ≠ t (here t ∈ \overline{V (K)} is fixed). Then curves corresponding to f_{v}, v ∈ V (K), and the vertex a define Fano’s configuration of lines in ℙ^{2} which can exist only in characteristic 2.

Cases 20f,g. In these cases, T̃ = D_{4} ⊕ 4A_{1} or T̃ = 8A_{1}. As for analogous cases 10, everything is reduced to prove that T = 7A_{1} does not correspond to a K3 pair (X, θ).

In this case, Γ = Γ(P (M_{+}^{(2,4)})) is as follows. Let I = {1, 2, 3, 4} and J = {1, 2}. The Γ has: exactly one double transparent vertex a; black vertices f_{ij}, i ∈ I, j ∈ J, and (i, j) ≠ (4, 2); simple transparent vertices b_{i}, i = 1, 2, 3, and b_{4(+)}, b_{4(−)} which are connected by a simple edge with a; simple transparent vertices c_{j_{1}j_{2}j_{3}j_{4}} where j_{1}, j_{2}, j_{3} ∈ J, j_{4} ∈ {1, −2, +2} and j_{1} + j_{2} + j_{3} + j_{4} ≡ 0 mod 2 which are disjoint to a.

Edges of Γ which are different from above, are as follows.

Double edges connect b_{i} with f_{ij}, if i = 1, 2, 3, and b_{4(+)}, b_{4(−)} with f_{41}, and c_{j_{1}j_{2}j_{3}j_{4}} with f_{1j_{1}}, f_{2j_{2}}, f_{3j_{3}}, and c_{j_{1}j_{2}j_{3}1} with f_{41}.

Thick edges connect b_{4(±)} with c_{j_{1}j_{2}j_{3}(∓2)}, and c_{j_{1}j_{2}j_{3}j_{4}} with c_{j′_{1}j′_{2}j′_{3}j_{4}′}, if j_{1} ≠ j′_{1}, j_{2} ≠ j′_{2}, j_{3} ≠ j′_{3}, |j_{4}| ≠ |j′_{4}|, and c_{j_{1}j_{2}j_{3}(+2)} with c_{j′_{1}j′_{2}j′_{3}(−2)}, if (j_{1}, j_{2}, j_{3}) ≠ (j′_{1}, j′_{2}, j′_{3}).

Assume that Γ corresponds to a K3 pair (X, θ). On its DPN surface, let us contract exceptional curves corresponding to b_{1}, b_{2}, b_{3}, b_{4(+)}, f_{11}, f_{21}, f_{31}, f_{41}, c_{222(+2)}. The curve corresponding to a gives a conic in ℙ^{2}. Corresponding to f_{12}, f_{22}, f_{32} curves give lines touching to the conic and having a common point. This is possible in characteristic 2 only. □

This finishes the proof of Theorem 3.6

3.5. Classification of DPN surfaces of elliptic type. Each nonsymplectic involution of elliptic type (X, θ) of K3 gives rise to the “good” DPN pair (Y, C) where

(29)  Y = X/{1, θ}, C = π(X^{θ}) ∈ | − 2K_{Y} |,

π : X → Y the quotient morphism; and vice versa. ¿From Theorems 3.6, 3.7, we then get classification of “good” DPN pairs (Y, C) and DPN surfaces Y of elliptic type. See Sect. 2 and especially Sects 2.1 and 2.7. It is obtained by the reformulation of Theorems 3.6 and 3.7 and by redrawing of the diagrams. But, for readers convenience, we do it below.

Theorem 3.16 (Classification Theorem for “good” DPN surfaces of elliptic type in the extremal case). Over ℂ, a “good” DPN surface Y of elliptic type is called extremal, if the number of its exceptional curves with the square (−2) (i. e. the number of black vertices in the diagram Γ below) is maximal for the fixed main invariants (r, a, δ) (equivalently, (k, g, δ)).

Moreover, the dual diagram Γ = Γ(Y) of all exceptional curves on Y is one of diagrams given in Table 3, and vice versa any diagram of Table 3 corresponds to some of the Y.

In the diagrams Γ, simple transparent vertices correspond to curves of the 1st kind (i. e. to non-singular rational irreducible curves with the square (−1)), double transparent vertices correspond to non-singular rational irreducible curves with the square (−4), black vertices correspond to non-singular rational irreducible curves with the square (−2), a k-multiple edge (or an edge with the weight k when k is large) means the intersection index k for the corresponding curves. Any exceptional curve on Y is one of these curves.

For a not necessarily extremal “good” DPN surface Y of elliptic type the dual diagram Γ = Γ(Y) of all exceptional curves on Y also consists of simple transparent, double transparent and black vertices which have exactly the same meaning as in Theorem 3.16 above. All black vertices of Γ define theDu Val part Duv Γ of Γ. All double transparent vertices of Γ, and all simple transparent vertices of Γ which are connected by two edges with double transparent vertices of Γ (there are always two of these double transparent vertices) define the logarithmic part Log Γ of Γ. The rest (vertices different from vertices of Duv Γ and Log Γ) define the varying part Var Γ of Γ. In Theorem below we identify vertices of Γ(Y) with elements of Picard lattice Pic Ỹ, then weights of edges are equal to the corresponding intersection pairing in this lattice which makes sense to the descriptions of the graphs Var Γ(Y) and Γ(Y).

Theorem 3.17 (Classification Theorem for DPN surfaces in the nonextremal, i. e. general, case of elliptic type). Over ℂ, dual diagrams Γ(Y) of all exceptional curves of not necessarily extremal “good” DPN surfaces Y of elliptic type are described by arbitrary (without any restrictions) subdiagrams D ⊂ Duv Γ of extremal DPN surfaces described in Theorem 3.16 above with the same main invariants (r, a, δ) (equivalently (k, g, δ)).

Moreover, Duv Γ(Y) = D, Log Γ(Y) = Log Γ, and these subdiagrams are disjoint to each other;

Var Γ(Y) = {f ∈ W (Var Γ) | f · D ≥ 0}

where W is the subgroup of automorphisms of the Picard lattice of the extremal DPN surface (the Picard lattice is defined by the diagram Γ), generated by reflections in elements with square −2 corresponding to all vertices of Duv Γ.

Two such subdiagrams D ⊂ Duv Γ and D′ ⊂ Duv Γ′ (with the same main invariants) give DPN surfaces Y with isomorphic diagrams Γ(Y), if and only if they have isomorphic root invariants ([D]^{+}, ξ^{+}) and ([D′]^{+}, ξ′^{+}) (see Theorem 3.7).

In Theorem 3.17, to calculate the root invariant ([D]^{+}, ξ^{+}) of a DPN surface, one has to go back from the graph Γ of Table 3 to the corresponding graph of Tables 1 or 2.

72 VALERY ALEXEEV AND VIACHESLAV V. NIKULIN

Table 3. Dual diagrams Γ of all exceptional curves of extremal “good” DPN surfaces (over ℂ) of elliptic type

N r a δ k g r̃  Γ
1 1 1 1 0 10 1  Γ = ∅, ℙ^{2}
2 2 2 0 0 9 1  𝔽_{0} or 𝔽_{2}
3 2 2 1 0 9 2  𝔽_{1}
4 3 3 1 0 8 2
5 4 4 1 0 7 1
6 5 5 1 0 6 1
7 6 6 1 0 5 1

a b

8 7 7 1 0 4 1

a b

c

9 8 8 1 0 3 1

a b

DEL PEZZO SURFACES OF INDEX ≤ 2 73

N r a δ k g r̃  Γ
9 8 8 1 0 3 1

c d e f

10 9 9 1 0 2 1

a b

74 VALERY ALEXEEV AND VIACHESLAV V. NIKULIN

N r a δ k g r̃  Γ
10 9 9 1 0 2 1

d e f g h

DEL PEZZO SURFACES OF INDEX ≤ 2 75

N r a δ k g r̃  Γ
10 9 9 1 0 2 1

i j k

N r a δ k g r̃  Γ
10 9 9 1 0 2 1

m

11 2 0 0 1 10 1  𝔽_{4}
12 3 1 1 1 9 2
13 4 2 1 1 8 2
14 5 3 1 1 7 2
15 6 4 0 1 6 1
16 6 4 1 1 6 2
17 7 5 1 1 5 2
18 8 6 1 1 4 1

19 9 7 1 1 3 1

DEL PEZZO SURFACES OF INDEX ≤ 2 77

N r a δ k g r̃  Γ
20 10 8 1 1 2 1

a b c d

21 6 2 0 2 7 1
22 7 3 1 2 6 2
23 8 4 1 2 5 2
24 9 5 1 2 4 2
25 10 6 0 2 3 1
26 10 6 1 2 3 1
27 11 7 1 2 2 1
28 8 2 1 3 6 2
29 9 3 1 3 5 3
30 10 4 0 3 4 1

N r a δ k g r̃  Γ

31 10 4 1 3 4 3

32 11 5 1 3 3 2

33 12 6 1 3 2 1

34 9 1 1 4 6 2

35 10 2 0 4 5 2

36 10 2 1 4 5 3

37 11 3 1 4 4 3

38 12 4 1 4 3 2

39 13 5 1 4 2 2

40 10 0 0 5 6 1
41 11 1 1 5 5 2

N r a δ k g r̃  Γ

42 12 2 1 5 4 2

43 13 3 1 5 3 2

44 14 4 0 5 2 1

45 14 4 1 5 2 2
46 14 2 0 6 3 1

47 15 3 1 6 2 2

48 16 2 1 7 2 2
49 17 1 1 8 2 2
50 18 0 0 9 2 1

3.6. Application: On classification of plane sextics with simple singularities. Let Y be a “good” DPN surface of elliptic type which were classified in Theorems 3.16 and 3.17. Let Γ(Y) be the dual diagram of all exceptional curves on Y. By definition of “good” DPN surfaces, there exists a non-singular curve

(30)  C = C_{g} + E_{a_{1}} + ⋯ + E_{a_{k}} ∈ | − 2K_{Y} |

where E_{a_{1}}, …, E_{a_{k}} are exceptional curves with square (−4) corresponding to all double transparent vertices a_{1}, …, a_{k} of Γ(Y) and g > 1 the genus of the irreducible non-singular curve C_{g}. Here (k, g, δ) (equivalent to (r, a, δ)) are the main invariants of Y.

We denote by E_{v} the exceptional curve on Y corresponding to a vertex v ∈ V (Γ(Y)). If v is black, then C · E_{v} = C_{g} · E_{v} = 0. If v is simple transparent, then C · E_{v} = 2.

If v is simple transparent and v is not connected by any edge with double transparent vertices of Γ(Y) (i. e. E_{v} · E_{a_{i}} = 0, i = 1, …, k) then C_{g} · E_{v} = 2. This intersection index can be obtained in two ways:

(31)  C_{g} intersects E_{v} transversally in two points;
(32)  C_{g} simply touches E_{v} in one point.

(For example, in Case 47 of Table 3 we have two such vertices v.)

Up to this ambiguity, we know (from the diagram Γ(Y)) how components of C intersect exceptional curves. Which of possibilities (31) or (32) does take place is defined by the generalized root invariant which we don’t consider in this work.

Let t_{1}, …, t_{r−1} ∈ V (Γ(Y)) be a sequence of vertices such that the contraction of exceptional curves E_{t_{1}}, …, E_{t_{r−1}} gives a morphism σ : Y → ℙ^{2} which is a sequence of contractions of curves of the 1st kind. By Sect. 2.1, the image D = σ(C) ⊂ ℙ^{2} is then a sextic (it belongs to |−2K_{ℙ^{2}}|) with simple singularities. What components and what singularities the curve D does have is defined by the subgraph Γ(t_{1}, …, t_{r−1}) generated by vertices t_{1}, …, t_{r−1} in Γ(Y). We formalize that below.

Let

D̃ = C_{g} + ∑_{v_{i}∈{a_{1},...,a_{k}}−{t_{1},...,t_{r−1}}} E_{v_{i}}

be the curve of components of C which are not contracted by σ. Then σ : D̃ → D is the normalization of D. In pictures, we denote D̃ (or D) by the symbol ⊗ and evidently denote the intersection of this curve and its local branches at the corresponding singular point with the components E_{t_{j}} which are contracted to this point. For connected components of Γ(t_{1}, …, t_{r−1}) we then have possibilities presented in Table 4 below depending on types of the corresponding singular points of D.

By Table 4, the ambiguity (31) or (32) takes place only for singularities of the types A_{2k−1} or A_{2k}. Thus, we have to introduce the notation 𝔄_{2k−1} for the singularity of the type A_{2k−1} or A_{2k} of the component σ(C_{g}) of D of the geometric genus g > 1.

In the “good” column of Table 4, we denote by A_{n}, D_{n}, E_{n} connected components of graphs Γ(t_{1}, …, t_{r−1}) corresponding to singularities A_{n}, D_{n} and E_{n} of the curve D respectively. Obviously, finding of all possible contractions σ : Y → ℙ^{2} reduces to enumeration of all subgraphs Γ ⊂ Γ(Y) with the connected components A_{n}, D_{n}, E_{n} and with the common number r −1 of vertices. A choice of such a subgraph Γ ⊂ Γ(Y) defines the sextic D with the corresponding irreducible components and simple singularities, and the related configuration of rational curves

(33) σ(E_{v}), v ∈ V (Γ(Y)) − ({a_{1}, …, a_{k}} ∪ {t_{1}, …, t_{r−1}}) ,

which one can call as the exceptional curves of a sextic D with simple singularities.

Thus, the classification in Theorems 3.16 and 3.17 of DPN surfaces of elliptic type implies a quite delicate classification of sextics D having an irreducible component of the geometric genus g ≥ 2. For this classification, we correspond to a sextic D ⊂ ℙ^{2} a subgraph Γ ⊂ Γ(Y) up to isomorphisms of graphs Γ(Y) which send the subgraphs Γ to one another. The analogous classification can be repeated for to classify curves with simple singularities in | − 2K_{𝔽_{n}}|, n = 0, …, 4. One should only replace r − 1 by r − 2. We also note that a choice of different subgraphs Γ ⊂ Γ(Y) for the same curve C defines birational transformations of the corresponding rational surfaces (ℙ^{2} or 𝔽_{n}) which transform the curves D to one another. Thus, the graph Γ(Y) itself classifies the corresponding curves D up to some their birational equivalence.

A complete enumeration of all cases has no principal difficulties, and it is only related to a long enumeration using Theorem 3.17 of all possible diagrams Γ(Y) and their subdiagrams Γ. Unfortunately, it seems, number of cases is enormous. But the complete enumeration can be important in some problems of real algebraic geometry and singularity theory. For example, it could be important for classification of irreducible quartics in ℙ^{3} with double rational singularities by the method of projection from a singular point. To remove the ambiguity (31) or (32), one has to perform similar (to ours) classification of generalized root invariants. Table 4. Correspondence between connected components of Γ and singularities of D = σ(D̃).

Type of  Equations of  Connected Components of Γ,
Singular  the Singularity  and Curve D̃ (denoted by ⊗)
point of D and its Branches

y^{2} − x^{2k} = 0
A_{2k−1} =  I: y − x^{k} = 0
𝔄_{2k−1}  II: y + x^{k} = 0

A_{2k} = 𝔄_{2k−1} y^{2} − x^{2k+1} = 0

D_{2k}  xy^{2} − x^{2k−1} = 0
I: x = 0
II: y − x^{k−1} = 0
III: y + x^{k−1} = 0

D_{2k+1}  xy^{2} − x^{2k} = 0
I: x = 0
II: y^{2} − x^{2k−1} = 0

E_{6}  y^{3} − yx^{4} = 0

E_{7}  y^{3} − yx^{3} = 0
I: y = 0
II: y^{2} − x^{3} = 0

E_{8}  y^{3} − x^{5} = 0

4. Classification of log del Pezzo surfaces of index ≤ 2 and applications

4.1. Classificaion of log del Pezzo surfaces of index ≤ 2. From results of Sects 1 — 3 we obtain

Theorem 4.1. (over ℂ) For any log del Pezzo surface Z of index ≤ 2 there exists and a unique resolution of singularities σ : Y → Z (it is called “good”) such that Y is a “good” DPN surface of elliptic type and σ contracts exactly all exceptional curves of the Du Val and the logarithmic part of Γ(Y). Vice versa, if Y is a “good” DPN surface of elliptic type, then there exists and a unique morphism σ : Y → Z of contraction of all exceptional curves corresponding to the Du Val and the logarithmic part of Γ(Y) which gives resolution of singularities of log del Pezzo surface Z of index ≤ 2.

Thus, classifications of log del Pezzo surfaces of index ≤ 2 and “good” DPN surfaces of elliptic type are equivalent, and they are given by Theorems 3.16 and 3.17.

Proof. Let Z be a log del Pezzo surface of index ≤ 2. In Sect. 1, a “canonical” (i. e. uniquely defined) resolution of singularities σ : Y → Z had been suggested such that Y is a “good” DPN surface of elliptic type. First, a minimal resolution of singularities σ_{1} : Y′ → Z is taken, and second, the blow-up of all intersection points of components of curves in preimages of non Du Val singularities K_{n} of Z is taken. Let us show that σ contracts exactly exceptional curves of Duv Γ(Y) and Log Γ(Y).

Let E be an exceptiona curve of Y corresponding to a vertex of Duv Γ(Y) of Log Γ(Y). Let C̃ ∈ | − 2K_{Z}| be a non-singular curve of Z which does not contain singular points of Z (it does exist by Theorem 1.5), and C_{g} = σ^{−1}(C̃_{g}). Then (see Sects 1.5 and 2) C_{g} +E_{1}+⋯+E_{k} ∈ |−2K_{Y} | where E_{i} are all exceptional curves on Y with the square (−4) and C_{g} a non-singular irreducible curve of genus g ≥ 2. By Sect. 2, then E · C_{g} = 0. If σ does not contract E, then for the curve σ(E) on Z we have σ(E) · C̃_{g} = σ(E) · (−2K_{Z}) = 0. Then −K_{Z} is not ample. We get a contradiction. Vice versa, by construction, σ contracts only curves from Duv Γ(Y) and Log Γ(Y). This shows that σ is a “good” resolution.

Now, let Y be a “good” DPN surface of elliptic type and σ : Y → Z a contraction of all exceptional curves corresponding to vertices Duv Γ(Y) and Log Γ(Y) (it does exist analytically because Duv Γ(Y)∪ Log Γ(Y) is negative, and we show that it does exist algebraically by the direct construction below). To prove Theorem, we should prove that Z is a log del Pezzo surface of index ≤ 2, and σ the “good” resolution of singularities of Z.

Second statement becomes obvious if one decomposes σ as the composition of contractions of all exceptional curves of 1st kind from Log Γ(Y) (they don’t intersect each other) and further contraction of the remaining exceptional curves.

To prove the first statement, one can use the double covering π : X → Y with the involution θ (see Sect. 2), the relation between exceptional curves of Y and (X, θ) (see Sect. 2), and that the contraction of 𝔸, 𝔻 and 𝔼 configurations of (−2) curves on X does exit and gives the corresponding quotient singularities ℂ/G_{i} where G_{i} ⊂ SL(2, ℂ) are finite subgroups. Using these considerations (i. e. first we consider the corresponding contraction, and second the quotient by involution), and Brieskorn’s results [Bri68], we obtain that all non Du Val singularities of Z are ℂ/G̃_{i} where G̃_{i} ⊂ GL(2, ℂ) and G̃_{i}∩SL(2, ℂ) = G_{i} have index 2 in G̃_{i}, and G̃_{i}/G_{i} = {1, θ}. It follows that Z is a complete algebraic surface with log-terminal singularities of index ≤ 2.

Let us show that −K_{Z} is ample. By Kleiman’s criterion, it is enough to show that (−K_{Z})^{2} > 0 and (−K_{Z}) · D > 0 for any curve D on Z. We have (see Sect. 1.5)

4(−K_{Z})^{2} = (−2K_{Z})^{2} = (σ^{∗}(−2K_{Z}))^{2} = (C_{g})^{2} > 0.

since Y is a DPN surface of elliptic type. Moreover,

−2K_{Z} · D = −2σ^{∗}K_{Z} · σ^{∗}D = C_{g} · σ^{∗}D ≥ 0

because C_{g} is irreducible with (C_{g})^{2} > 0 and σ^{∗}D is effective. Moreover, we get here zero, only if the effective divisor σ^{∗}D consists of exceptional curves F on Y with C_{g}·F = 0. But such curves F correspond to vertices of the logarithmic or the Du Val part of Γ(Y). They are contracted by σ into points of Z which is impossible for the divisor σ^{∗}D. □

Using Theorem 4.1, we can transfer to log del Pezzo surfaces Z of index ≤ 2 the main invariants (r, a, δ), equivalently (k, g, δ), the root invariant, the root subsystem, the exceptional curves which are defined for the surface Y of the “good” resolution σ : Y → Z. In particular, the Picard number of Z is

(34)  r̃ = rk Pic Z = r − #V (Duv Γ(Y)) − #V (Log Γ(Y)).

In Theorem 3.16 we have shown the Picard number r̃ in the extremal (for Y) case. Obviously, surfaces Z with the extremal Y are distinguished by the minimal Picard number r̃ for the fixed main invariants (r, a, δ) (equivalently, (k, g, δ)). Since the Log Γ(Y) is prescribed by the main invariants and is then fixed, this is equivalent to have the maximal rank (i. e. the number of −2 curves for the minimal resolution of singularities) for Du Val singularities of Z.

In Mori Theory, see [Mor82] and [Rei83], log del Pezzo surfaces Z with rk Pic Z = 1 are especially important. They give relatively minimal models (different from ℙ^{1}×ℙ^{1}) in the class of rational surfaces with log-terminal singularities: any rational surface with log-terminal singularities has a contraction morphism on such a model. ¿From Theorems 4.1 and 3.6, we obtain classification of such models with log-terminal singularities of index ≤ 2. By Theorem 3.16, they correspond to exactly extremal DPN surfaces of elliptic type with

r̃ = r − #V (Duv Γ(Y)) − #V (Log Γ(Y)) = 1,

and Theorem 3.16 gives their classification.

Theorem 4.2. (over ℂ) There exist, up to isomorphism, exactly 46 relatively minimal and different from ℙ^{1} × ℙ^{1}, i. e. with Pic Z = ℤ, log del Pezzo surfaces Z of index ≤ 2. The DPN surfaces Y of their “good” resolution of singularities are extremal and correspond to exactly the following cases of Theorem 3.16 where we show in brackets the type of their singularities.

\begin{matrix}1(Y = ℙ^{2}), 2(A_{1}), 5(A_{1}A_{2}), 6(A_{4}), 7a(D_{5}), 7b(2A_{1}A_{3}), 8a(E_{6}), \\ 8b(A_{1}A_{5}), 8c(3A_{2}), 9a(E_{7}), 9b(A_{7}), 9c(A_{2}A_{5}), 9d(A_{1}2A_{3}), \\ 9e(A_{1}D_{6}), 9f(3A_{1}D_{4}), 10a(E_{8}), 10b(A_{8}), 10c(A_{1}A_{7}), \\ 10d(A_{1}A_{2}A_{5}), 10e(2A_{4}), 10f(D_{8}), 10g(A_{3}D_{5}), 10h(A_{2}E_{6}), \\ 10i(A_{1}E_{7}), 10j(2A_{1}D_{6}), 10k(2D_{4}), 10l(2A_{1}2A_{3}), 10m(4A_{2}); \\ 11(K_{1}), 15(K_{1}A_{4}), 18(K_{1}A_{1}A_{5}), 19(K_{1}A_{7}), 20a(K_{1}D_{8}), \\ 20b(K_{1}2A_{1}D_{6}), 20c(K_{1}A_{3}D_{5}), 20d(K_{1}2D_{4}); 21(K_{2}A_{2}), \\ 25(2K_{1}A_{7}), 26(K_{2}2A_{3}), 27(K_{2}A_{7}), 30(K_{3}2A_{2}); 33(K_{3}A_{1}A_{5}); \\ 40(K_{5}), 44(K_{5}A_{4}); 46(K_{6}A_{2}); 50(K_{9}). \end{matrix} (35)

In particular, the isomorphism class of Z is defined by types of its singularities. The number of non-Du Val singularities is at most one except when the singularities are 2K_{1}A_{7}.

Proof. We need only to show that the corresponding to all these cases surfaces are unique up to isomorphisms. As in the proof of Proposition 3.12, we can choose an appropriate sequence of vertices of Γ(Y) such that the contraction of the corresponding sequence of the curves of the 1st kind gives a morphism of Y onto ℙ^{2} or 𝔽_{n}, n = 0, 2, 3, 4.

Considering images of other exceptional curves of Y, one can see (by direct inspection in each case) that the contracting exceptional curves give a rigid configuration of points on the corresponding surface. □

Remark 4.3. In the Gorenstein case (i. e. for index ≤ 1, when there are no singularities K_{n}) Theorem 4.2 had been proved by completely different method in [BBD84]. This case gives the first 28 cases 1 — 10m of Theorem 4.2, the remaining 18 of its cases 11 — 50 are non-Gorenstein.

If we consider log del Pezzo surfaces Z of index ≤ 2 and without Du Val singularities, we get an opposite case to the previous one. Thus, any singularity of Z must have index 2. This case includes and is surprisingly similar to the classical case of non-singular del Pezzo surfaces when there are no singularities at all. Applying Theorem 4.1 we get the following

Theorem 4.4. (over ℂ) Up to deformation, there exist exactly 50 types of log del Pezzo surfaces Z with singularities of index exactly 2 (if a singularity does exist). The DPN surfaces Y of their “good” resolution of singularities have empty Du Val part Duv Γ(Y), zero root invariants, and are defined by their main invariants (r, a, δ) (equivalently (k, g, δ)), up to deformation. The diagram Γ(Y) can be obtained from the diagram Γ of cases 1 — 50 of Table 3 (with the same main invariants) as follows: Γ(Y) consists of Log Γ(Y) = Log Γ and

(36)  Var Γ(Y) = W (Var Γ)

where W is generated by reflections in all vertices of Duv Γ (i. e. one should take D = ∅ in Theorem 3.17). In cases 7, 8, 9, 10, 20 one can consider only diagrams Γ of cases 7a, 8a, 9a, 10a and 20a (diagrams 7b, 8b,c, 9b—f, 10b—m, 20b—d give the same).

Proof. This case corresponds to Y with empty D ⊂ Duv Γ of Theorem 3.17. Then the root invariant is 0. Thus, all cases 7, 8, 9, 10 or 20 give the isomorphic root invariants and the same diagrams, and we can consider only the corresponding cases 7a, 8a, 9a, 10a and 20a to calculate the diagrams.

Let us show that moduli spaces of DPN surfaces Y with the same main invariants (r, a, δ) and zero root invariant (i. e. D = 0) are connected.

It is enough to show connectedness of the corresponding “good” DPN pairs (Y, C) where C ∈ | − 2K_{Y} | is non-singular. Taking double covering π : X → Y ramified in C, it is enough to prove connectedness of moduli of K3 surfaces with non-symplectic involutions (X, θ) and (S_{X})_{+} = S where S has invariants (r, a, δ). General such pairs have zero root invariant, as we want, because general (X, θ) have S_{X} = S. Using Global Torelli Theorem [PS-Sh71] and surjectivity of Torelli map [Kul77] for K3, the connectedness of moduli reduces to the uniqueness, up to automorphisms, of the primitive embedding S ⊂ L_{K3} where L_{K3} is the cohomology lattice of K3. This uniqueness had been proved in [Nik80.2]. □

We remark that equivalent to Theorem 4.4 results had been first obtained in [Nik83].

We remark that cases 1 — 10 of Theorem 4.4 give classical nonsingular del Pezzo surfaces. Therefore, Theorem 4.4 and all results of this work show that log del Pezzo surfaces of index ≤ 2 are very similar to classical non-singular del Pezzo surfaces.

4.2. Application: Minimal projective compactifications of affine surfaces in ℙ^{2} by relatively minimal log del Pezzo surfaces of index ≤ 2. Let us consider one of the 45 relatively minimal surfaces of Theorem 4.2 which are different from ℙ^{2} (i. e. except the case 1). Let σ : Y → Z be its minimal resolution of singularities, and v_{1}, …, v_{r−1} a sequence of vertices of Γ(Y) such that the corresponding exceptional curves on Y give a contraction of the sequence of curves of the 1st kind τ : Y → ℙ^{2}. Let C ⊂ ℙ^{2} be the union of images by τ of all exceptional curves E_{v}, v ∈ (V (Duv(Γ(Y))) ∪ V (Log(Γ(Y)))). Then, the embedding f = (στ^{−1}) : V → Z gives a compactification of the affine surface V = ℙ^{2} − C of ℙ^{2}. The morphism f is minimal in the sense that f cannot be extended through components of C (see [BBD84] for details). The description of all such affine surfaces V and compactifications is then reduced to the description of subdiagrams of Γ(Y) (defined by v_{1}, …, v_{r−1})) which were described by their connected components in Sect. 3.6.

This is similar to [BBD84] in the Gorenstein case.

4.3. Some questions. Here we discuss several questions of general interest.

4.3.1. Dimension of the moduli space. What is the dimension of the moduli space M_{Z} of log del Pezzo surfaces Z of index ≤ 2 and with the fixed invariants

(37)  (k, g, δ, root invariant) ,

equivalently,

(38) (r, a, δ, the dual diagram of exceptional curves Γ(Y))?

We have the obvious formula (where we identify M_{Z} with the moduli M_{Y} of the corresponding “good” DPN surfaces Y of the “good” resolution of singularities of Z):

(39)  dim M_{Y} + dim | − 2K_{Y} | − dim Aut Y = dim M_{(Y,C)}

where M_{(Y,C)} is the moduli space of “good” DPN pairs (Y, C) where C ∈ | − 2K_{Y} | is non-singular. Here we used the fact that the group Aut(Y, C) (equivalently Aut(X, θ) for the double covering) is finite. We remind that Aut X is 0-dimensional and discrete for a K3 surface X. The local Torelli Theorem and the local surjectivity of Torelli map for K3 surfaces then give

(40) dim M_{(Y,C)} = 20 − r − #V (Duv(Γ)) = 9 + g − k − #V (Duv(Γ)).

Moreover,

(41)  dim | − 2K_{Y} | = dim |C_{g}| = 3g − 3

(see Sects 1.4 and 1.5). It follows that

dim M_{Z} = dim M_{Y} =12 − 2g − k − #V (Duv(Γ(Y)) + dim Aut Y =
\frac{r + 3a}{2} − 10 − #V (Duv(Γ(Y)) + dim Aut Y

(42)

where we can assume that Y is general. The group Aut(Y) can be easily computed as follows: We can choose an appropriate sequence v_{1}, …, v_{k} ∈ Γ(Y) such that the contraction of the corresponding curves of the 1st kind gives a morphism τ : Y → V where V is one of relatively minimal rational surfaces ℙ^{2} or 𝔽_{n}, n = 0, 2, 3 or 4. Then the connected component of identity (Aut Y)_{0} gives a subgroup of Aut V which preserves the configuration K of images of the the exceptional curves E_{v_{1}}, …, E_{v_{k}} and corresponding to Duv Γ and Log Γ. It can be easily computed in concrete cases. Finally we obtain dim M_{Y} = dim M_{Z} using (42). Clearly, dim Aut Y must be zero in all interesting cases.

Is there a different from (42) formula for dim M_{Z} where dim Aut Y is computed?

4.3.2. Description of moduli spaces. It would be very interesting to describe the moduli spaces M_{Z} in the form it is known for the ordinary non-singular del Pezzo surfaces. In the notations above, this reduces to the explicit description of moduli spaces of all possible configurations K.

4.3.3. Finite characteristic. It would be very interesting to generalize results of this work to finite characteristic. As we had mentioned in Remark 3.14, it seems, the main problem is to generalize Theorem 1.5. We think that our results are valid in characteristic ≥ 3. As we have seen, in characteristic 2 the number of cases increases.

4.3.4. There are many results (e. g. see [Man86], [MT86] and [CT88]) where arithmetics of classical non-singular del Pezzo surfaces is studied. What is the arithmetics of log del Pezzo surfaces of index ≤ 2 and equivalent DPN surfaces of elliptic type?

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University of Georgia, Athens, GA 30605, USA

E-mail address: valerymath.uga.edu

Dept. of Pure Math., Univer. of Liverpool, Liverpool L69 3BX, UK; Steklov Math. Inst., ul. Gubkina 8, Moscow 117966, GSP-1, Russia

E-mail address: vnikulin@liv.ac.uk, vvnikulin@list.ru