REAL K3 SURFACES WITH NON-SYMPLECTIC INVOLUTIONS AND APPLICATIONS

VIACHESLAV V. NIKULIN AND SACHIKO SAITO

Abstract. Classification of real K3 surfaces X with a non-symplectic involution τ is considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected component of their moduli is defined by the isomorphism class of the action of τ and the anti-holomorphic involution φ in the homology lattice. (There are very few similar cases known.) For their classification we apply invariants of integral lattice involutions with conditions which were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves A ∈ |−2K_{V} | for the classical real surfaces: V = ℙ^{2}, hyperboloid, ellipsoid, 𝔽_{1}, 𝔽_{4}.

As an application, we describe all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls (surfaces V above). There are very few exceptions. For example, any non-singular real quartic in ℙ^{3} can be constructed in this way.

Contents

1. Introduction 1 2. Connected components of moduli of real K3 surfaces with non-symplectic

involutions and real DPN-pairs 4 2.1. Connected components of moduli of non-degenerate real K3 surfaces with

non-symplectic involutions 4

2.2. Connected components of moduli of non-degenerate real right DPN-pairs 11 2.3. General classification results about real K3 surfaces with non-symplectic

involutions and real right DPN-pairs 15

2.4. Possible types of (S, θ) with rk S ≤ 2 25 3. Connected components of moduli of real non-singular curves of degree 6 in ℙ^{2} 26 4. Connected components of moduli of real non-singular curves of bidegree (4, 4) on

hyperboloid 29 5. Connected components of moduli of real non-singular curves of bidegree (4, 4) on

ellipsoid 37 6. Connected components of moduli of real non-singular curves in | − 2K_{𝔽_{1}}| 42 7. Connected components of moduli of real non-singular curves in | − 2K_{𝔽_{4}}| 48 8. Application: Real polarized K3 surfaces as deformations of general K3 double

rational scrolls 50 References 54

1. Introduction

In this paper, a K3 surface is a simply-connected Kälerian compact complex surface with the trivial canonical class. Thus, there exists a 2-dimensional holomorphic differential form ω_{X} with the zero divisor. The form ω_{X} is unique up to λω_{X} where λ ∈ ℂ and λ ≠ 0.

A K3 surface may have two very different kinds of holomorphic involutions τ: symplectic when τ^{∗}(ω_{X}) = ω_{X} and non-symplectic when τ^{∗}(ω_{X}) = −ω_{X}.

There exists only one type of symplectic involutions [18] and many types of non-symplectic involutions [20], [22]. The types of non-symplectic involutions are classified by invariants which are all presented in Figure 1 below. They describe irreducible components of moduli of complex K3 surfaces X with non-symplectic involutions τ. See more delicate classification in [1] and [2].

In this paper we consider K3 surfaces with non-symplectic involutions (X, τ). The quotient surface Y = X/{1, τ} is either Enriques surface when τ has no fixed points, or a rational surface when τ has fixed points. The set of fixed points is a non-singular curve A = X^{τ}, and its image A ⊂ Y is an element of the linear system | − 2K_{Y} | where K_{Y} is the canonical system of Y. Then X is the double covering of Y ramified in A and τ the involution of the double covering. Pairs (A, Y) when Y is either Enriques or a rational surface (they are characterized by the condition q(Y) = 0 where q is the irregularity) and A ⊂ | − 2K_{Y} | is a non-singular curve are called right DPN-pairs. Classification of K3 surfaces with nonsymplectic involutions (X, τ) is equivalent to classification of right DPN-pairs A ⊂ Y.

In this paper we consider description of connected components of moduli of real K3 surfaces with non-symplectic involutions and real right DPN-pairs. Equivalently, we consider classification of triplets (X, τ, φ) where X is a K3 surface, τ its holomorphic non-symplectic involution and φ its anti-holomorphic involution. The involutions τ and φ must commute: φτ = τφ. Equivalently, we consider classification of real right DPN-pairs A ⊂ Y with an anti-holomorphic involution θ where θ(A) = A.

This classification depends on the definition of degeneration of (X, τ, φ). We choose one of the weakest types of degeneration (i. e. giving one of the reachest discriminants). Geometrically (over ℂ) it is equivalent to appearance of an exceptional curve F with square −2 on the quotient Y. Over ℝ this curve should be either real: θ(F) = F, or θ(F) · F ≤ 1. See Section 2.1.

We mention that our definition of degeneration is weaker (it gives a much reacher discriminant) then the one used in [4] for description of connected components of moduli of real Enriques surfaces. Thus, even for real Enriques surfaces our results are stronger (more exact). Difference of our results from [4] is in the appearance of the group G.

On the one hand, our type of degeneration gives a reach discriminant. For example, it is enough to describe connected components of moduli of real bidegree (4, 4) curves on the hyperboloid which was one of the main purposes of this paper. A degeneration of this case gives non-singular curves A ∈ |−2K_{𝔽_{2}}| which we must exclude. Here we denote as 𝔽_{n}, n ≥ 0 a relatively minimal rational surface with the section s having s^{2} = −n. See Proposition 4. On the other hand, using Global Torelli Theorem [24] and epimorphicity of Torelli map [13] for K3 surfaces we prove that connected components of moduli are described by sufficiently simple invariants which are isomorphism classes of integral involutions with conditions. They are triplets (L, φ, (S, θ)). Here L = H_{2}(X, ℤ) is the even unimodular homology lattice with the action of φ, S = L^{τ} and θ = φ|S. The pair (S, θ) determines the type of real K3 surfaces (X, τ, φ) with non-symplectic involution whose connected components of moduli we want to describe. See Theorems 1, 6, 7. All genus invariants of integral involutions with condition were classified in [21]. See Sect. 2.3 about these invariants. They give very universal invariants which work in all cases we consider. This paper can be considered as one of applications of this classification and the invariants which it uses.

We mention that similar arithmetic description of connected components of moduli had been obtained in [19] for real polarized K3 surfaces. Similar result was obtained in [4] for real Enriques surfaces with the much stronger (or with the much smaller discriminant) type of degeneration used in [4]. Thus, this paper adds some more cases to these beautiful series. It seems there are very few geometrically significant cases when similar arithmetic results are valid.

Our results are valid for any type (S, θ) of real K3 surfaces with non-symplectic involutions (or the real right DPN-pairs). In Sections 3 — 7 we specify them in the most simple and general cases (by the dimension 20 − rk S of moduli). We consider all cases when rk S ≤ 2. They give the following real DPN-pairs and the corresponding real K3 surfaces with nonsymplectic involutions as double coverings:

ℙ^{2}: Real non-singular curves A of degree 6 on the real projective plane ℙ^{2}; see Sect. 3.

ℍ (hyperboloid): Real non-singular curves A of bidegree (4, 4) on hyperboloid and on ℙ^{1}×ℙ^{1} with all possible spinor (i. e. with empty set of real points) real structures; see Sect. 4.

𝔼 (ellipsoid): Real non-singular curves A of bidegree (4, 4) on ellipsoid, see Sect. 5.

𝔽_{1}: Real non-singular curves A ∈ | − 2K_{𝔽_{1}}| on 𝔽_{1} with real structure; see Sect. 6.

𝔽_{4}: Real non-singular curves A ∈ | − 2K_{𝔽_{4}}| on 𝔽_{4} with a real structure; see Sect. 7.

In all these cases we get classification of connected components of moduli of the nonsingular real curves A on the real surfaces up to the action of the automorphism group of the real surfaces.

The case of ℙ^{2} had been considered in [19].

For hyperboloid ℍ, the isotopy classification had been obtained by Gudkov [7]. Zvonilov [35] added description of divideness by A(ℝ) of A(ℂ) and possible complex schemes (i. e. complex orientations of A(ℝ) which are induced by A(ℂ) in the dividing case). We show that these visual invariants: isotopy type, divideness and complex orientations (in some very few dividing cases), define the connected component of moduli up to the action of the automorphism group which has 8 connected components for hyperboloid.

For ellipsoid 𝔼, the isotopy classification had been obtained by Gudkov and Shustin [8]. Zvonilov [35] added description of divideness by A(ℝ) of A(ℂ) and possible complex schemes which are prescribed by the isotopy type in this case. We show that these visual invariants: isotopy type and divideness, define the connected component of moduli, up to the action of the automorphism group of ellipsoid which has 2 connected components. (Similar result is valid for ℙ^{2}.)

For 𝔽_{1} we show that also the connected component of moduli is defined by the isotopy type and by divideness. The related result had been obtained by Itenberg [9], [10] who gave the classification of connected components of moduli of real curves of degree 6 with one non-degenerate quadratic singular point. In principle, our classification should follow from Itenberg’s results, but it would be hard for us to deduce it from [9], [10].

For 𝔽_{4} we also show that the connected component of moduli is defined by the isotopy type and by divideness. This case corresponds to very important class of K3 surfaces which are elliptic K3 surfaces with section. The involution τ is the inverse map of the corresponding elliptic pencil.

In Sect. 8 we consider an important application of the above results to description of real polarized K3 surfaces which are deformations of general real K3 double rational scrolls. We mean the following. Let (X, P) be a polarized K3 surface with a primitive ample H where P^{2} = n. For a general K3 double rational scroll the complete linear system |P | or |2P | gives the double covering π : X → Y = X/{1, τ} where Y is one of surfaces Y = ℙ^{2}, ℙ^{1} ×ℙ^{1}, 𝔽_{1} or 𝔽_{4} above. Moreover, |kP | is the preimage |kP | = π^{∗}|P̃| of some standard linear system P̃ on Y. General K3 double rational scrolls give codimension 1 moduli subspace (if n ≥ 4) of the connected moduli space M_{n} of the polarized K3 surfaces. Thus, any polarized K3 surface (X, P) over ℂ is a deformation of some general K3 double rational scroll. It is interesting to ask, if the same is valid over ℝ.

Using classification results of Sects. 3 — 7 and the classification results about real polarizes K3 surfaces [19], in Sect. 8 we show that a real polarized K3 surface (X, P) of degree n = P^{2} ≥ 6 is not a deformation of a general real K3 double rational scroll, if the real part X(ℝ) either consists of 10 spheres or it consists of k spheres and X(ℝ) ∼ P mod 2 in H_{2}(X, ℤ). “All other” (in some sense) real polarized K3 surfaces (X, P) are deformations of general real K3 double rational scrolls. In [19] all possible genus invariants (for the action of the anti-holomorphic involution φ in homology)

(r, a, δ_{P} , δ_{φ}, δ_{φP})

of real polarized K3 surfaces were defined and classified. Excluding few (possibly extremely few) exceptions the genus invariants determine the connected component of moduli of real polarized K3 surfaces. “All other” above means all possible genus invariants different from genus invariants of the shown above exceptions. Additional considerations show that for P^{2} = 2 (double coverings of ℙ^{2}) and P^{2} = 4 (quartics in ℙ^{3}) any real K3 surface is a deformation of a general real K3 double rational scroll. For P^{2} = 2 it is obvious. For quartics, P^{2} = 4, we can additionally use double coverings of ellipsoid 𝔼 which one cannot use for P^{2} ≠ 4. This gives a very effective method of construction of all (i. e. representatives of all connected components of moduli) real quartic surfaces. Using results of Shah [26] (over ℂ), one can describe all these deformations very explicitly. It gives an effective method of construction of real quartic surfaces.

The first author wants to thank University of Liverpool, Steklov Mathematical Institute, Max-Planck Institut für Mathematik and Hakodate Campus of Hokkaido University of Education for hospitality.

2. Connected components of moduli of real K3 surfaces with non-symplectic involutions and real DPN-pairs

2.1. Connected components of moduli of non-degenerate real K3 surfaces with non-symplectic involutions. We remind that a K3 surface X is a non-singular projective algebraic surface over ℂ (more generally a compact Kählerian surface) such that it has a holomorphic 2-dimensional differential form 0 ≠ ω_{X} with zero divisor (or the canonical class K_{X} = 0) and X is simply-connected. The form ω_{X} is defined uniquely up to λω_{X}, λ ∈ ℂ^{∗}. E.g. see [29], [24] and [27] about K3 surfaces.

An involution τ of a K3 surface X can be symplectic, if τ^{∗}ω_{X} = ω_{X}, and non-symplectic, if τ^{∗}ω_{X} = −ω_{X}. Symplectic involutions on K3 surfaces were classified in [18], for this case it is natural to consider Kählerian K3 surfaces, and there exists only one type of such involutions. Non-symplectic involutions on K3 surfaces were classified in [20], [22]; all K3 surfaces with non-symplectic involutions are algebraic.

The main invariant of a K3 surface with a non-symplectic involution τ is the isomorphism class of the lattice S which is the fixed part H_{2}(X, ℤ)^{τ} = S. Here and in what follows we denote

M^{ξ} = {x ∈ M |ξ(x) = x},

and by

M_{ξ} = {x ∈ M |ξ(x) = −x}

for an action of an involution ξ on a module M. By the Hodge decomposition, the lattice S is hyperbolic (i. e. it has exactly one positive square), it is contained in the Picard lattice N(X) of X (which is generated by classes of all algebraic curves on X). All possible S are known (up to isomorphisms) and are described in [20] (see also [22]).

We can additionally fix a half-cone (the light-cone) V^{+}(S) of the cone

V (S) = {x ∈ S ⊗ ℝ | x^{2} > 0}.

We can also fix a fundamental chamber M ⊂ V^{+}(S) for the group W^{(−2)}(S) generated by reflections in all elements with square (−2) in S. This is equivalent to fixing a fundamental subdivision Δ(S) = Δ(S)_{+}∪−Δ(S)_{+} of all elements with square −2 in S. The M and Δ(S)_{+} define each other by the condition (M, Δ_{+}) ≥ 0. These additional structures M ⊂ V^{+}(S) of the hyperbolic lattice S are defined uniquely up to the action of the group {±1}W^{(−2)}(S).

We can restrict considering K3 surfaces X with a non-symplectic involution τ, i.e. pairs (X, τ), such that V^{+}(S) contains a hyperplane section of X and the set Δ(S)_{+} contains only classes of effective curves with square −2 of X. I. e. V^{+}(S) and the fundamental subdivision Δ(S)_{+} are prescribed by the geometry of the K3 surface X.

If a pair (X, τ) is general, then S = N(X) is the Picard lattice of X (remind that it is generated by classes of algebraic curves) and M gives the nef cone (or Kählerian cone) of X. The weakest condition of degeneration (i. e. giving the most reach discriminant) is the following condition:

(D): We say that (X, τ) of the type H_{2}(X, ℤ)^{τ} = S is degenerate if there exists h ∈ M such that h is not nef for X. I.e., there are elements of S which are nef in general (when S = N(X)), but they are not nef for X. It is not difficult to see [22], that this is equivalent to existence of an exceptional curve with square −2 on the quotient Y = X/{1, τ}. This is also equivalent to have an element δ ∈ N(X) with δ^{2} = −2 such that δ = (δ_{1} + δ_{2})/2 where δ_{1} ∈ S, δ_{2} ∈ S_{N(X)}^{⊥} and δ_{1}^{2} = δ_{2}^{2} = −4. Remark that (δ_{1}, S) ≡ 0 mod 2 and (δ_{2}, S^{⊥}) ≡ 0 mod 2. I.e. δ_{1} and δ_{2} are roots with square −4 for lattices S and S^{⊥} respectively.

Let us consider examples of lattices S of a small rank, when rk S ≤ 2. All these cases will be considered in details in Sect. 2.4 below. We use notations: ⟨A⟩ denote a lattice defined

by a symmetric integral matrix A. We denote U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) . For a lattice K, we denote by

K(m) a lattice obtained from the lattice K by multiplication of the form of K by m ∈ ℚ. By ⊕ we denote the orthogonal sum of lattices.

If rk S = 1, then S ≅ ⟨2⟩, Y = X/{1, τ} = ℙ^{2} and all X are non-degenerate, because S has no elements with square −4.

If rk S = 2, then the lattice S is isomorphic to one of three lattices: S ≅ U(2), ⟨2⟩ ⊕ ⟨−2⟩ or U.

If S ≅ U(2), then non-degenerate K3 surfaces (X, τ) give X/{1, τ} ≅ ℙ^{1} × ℙ^{1} = 𝔽_{0}, but degenerate K3 surfaces (X, τ) give X/{1, τ} ≅ 𝔽_{2} where we denote by 𝔽_{n} a relatively minimal rational surface with a section having the square −n, n ≥ 0.

If S ≅ ⟨2⟩ ⊕ ⟨−2⟩, then Y = X/{1, τ} = 𝔽_{1}. All X are non-degenerate because S has no elements with square −4.

If S ≅ U, then Y = X/{1, τ} = 𝔽_{4}. All X are non-degenerate because S has no elements δ_{1} with δ_{1}^{2} = −4 and (δ_{1}, S) ≡ 0 mod 2.

We shall consider all these cases in Sect 2.4 below.

Further we fix one of possible lattices S together with M ⊂ V^{+}(S) and consider K3 surfaces with non-symplectic involutions (X, τ) and a fixed identification S = H_{2}(X, ℤ)^{τ} such that M ⊂ V^{+}(S) are defined by the geometry of X. In complex case we can consider two pairs (X, τ) and (X′, τ′) to be isomorphic if there exists an isomorphism f : X → X′ of K3 surfaces which is identical on S. Then fτ = τ′f (it is sufficient to check this identity for the action on second homology) and f defines the isomorphism of K3 with involutions. The moduli space of pairs (X, τ) is connected, and its subspace of degenerate pairs (X, τ) has codimension one over ℂ (if it is not empty). Thus, the space of non-degenerate pairs is also connected. This follows from the fact that there exists a unique (up to automorphisms) primitive embedding S → L where L ≅ H_{2}(X, ℤ) is an unimodular even lattice of signature (3, 19).

For real K3 surfaces with non-symplectic involutions (X, τ), we should additionally fix the type θ of the action of the anti-holomorphic involution φ of X on S. The θ should satisfy the following properties: θ(V^{+}(S)) = −V^{+}(S) and θ(Δ(S)_{+}) = −Δ(S)_{+}. It follows that the lattice S_{+} = S^{θ} is negative definite and it has no elements with square −2. Moreover, the linear subspace S_{−} ⊗ ℝ where S_{−} = S_{θ} must intersect the interior of the nef cone M. For the fixed type (S, θ) we consider K3 surfaces X with holomorphic involutions τ of the type S and an anti-holomorphic involution φ, such that φ|S = θ, in particular, φ(S) = S. It follows that the anti-holomorphic involution φ commutes with τ. The φ defines then the real structure of the pair (X, τ). The triplets (X, τ, φ) of the type (S, θ) give real K3 surfaces with non-symplectic involutions of the type (S, θ).

We consider the following real analogy for real K3 surfaces with non-symplectic involutions of the type (S, θ) of the degeneration (D) above. An element h ∈ S is called real if θ(h) = −h, i.e. h ∈ S_{−}. For a general real X we have N(X) = S, and all real nef elements are elements of S_{−} ∩ M.

(Dℝ): A real K3 surface (X, τ, φ) with a non-symplectic involution of the type (S, θ) is called degenerate if there exists a real element h ∈ S_{−} ∩ M which is not nef for X. I. e. h is nef for general real K3 surfaces with non-symplectic involutions of the type (S, θ), but it is not nef for the triplet (X, τ, φ) itself. This is equivalent to have an element δ ∈ N(X) with δ^{2} = −2 such that δ = (δ_{1} + δ_{2})/2 where δ_{1} ∈ S, δ_{2} ∈ S_{N(X)}^{⊥} and δ_{1}^{2} = δ_{2}^{2} = −4 (i. e. (X, θ) is degenerate in the sense of (D) as a complex surface). Additionally, δ_{1} must be orthogonal to an element h ∈ S_{−} ∩ int(M) with h^{2} > 0. Here int(M) denote the interior part of M, i. e. the polyhedron M without its faces.

The condition (Dℝ) for (X, τ, φ) implies the condition D for (X, τ). Thus, condition (Dℝ) is stronger for (X, τ) than (D), and it is not difficult to give examples of possible (S, θ) when (D) does not imply (Dℝ).

On the other hand, for some types (S, θ) the conditions (D) implies (Dℝ). E. g. it is obviously true if S = S_{−}. It is easy to see that (D) implies (Dℝ) for all lattices S of rk S ≤ 2 above and all possible θ for these lattices. Only for S = U(2), it is possible to have S_{−} ≠ S (this is the case when Y = X/{1, θ} is an ellipsoid as a real surface). Then S_{−} = ⟨4⟩, S_{+} = ⟨−4⟩, and all elements of S with square −4 belong to S_{+}. Then (D) implies (Dℝ). But, to formulate a result about connected components of moduli of non-degenerate real K3 surfaces with non-symplectic involutions (Theorem 1) in general (for arbitrary S), we have to consider the type (Dℝ) of degeneration.

Let L ≅ H_{2}(X, ℤ) where H_{2}(X, ℤ) is the homology lattice of K3 with intersection pairing. Then L is an even unimodular lattice of signature (3, 19). Let us fix a primitive embedding S ⊂ L, it is unique up to automorphisms of L. Let Δ(S, L)^{(−4)} be the set of all elements δ_{1} in S such that δ_{1}^{2} = −4 and there exists δ_{2} ∈ S_{L}^{⊥} such that (δ_{2})^{2} = −4 and δ = (δ_{1} + δ_{2})/2 ∈ L. Then δ^{2} = −2. All elements δ_{1} ∈ Δ(S, L)^{(−4)} are roots of S since −4 = (δ_{1})^{2}|2(δ_{1}, S) because (δ_{1}, S) ⊂ 2ℤ. Let W^{(−4)}(S, L) ⊂ O(S) be the group generated by reflections in all roots from Δ(S, L)^{(−4)}, and W^{(−4)}(S, L)_{M} the stabilizer subgroup of M in W^{(−4)}(S, L). The set Δ(S, L)^{(−4)} is invariant with respect to W^{(−2)}(S). It follows that the W^{(−4)}(S, L)_{M} is generated by reflections s_{δ_{1}} in δ_{1} ∈ Δ(S, L)^{(−4)} such that the the hyperplane (δ_{1})_{S}^{⊥} ⊗ ℝ in S ⊗ ℝ intersects the interior of M.

The real analogy of the group W^{(−4)}(S, L)_{M} is the subgroup G ⊂ W^{(−4)}(S, L)_{M} which is generated by reflections s_{δ_{1}} in all elements δ_{1} ∈ Δ(S, L)^{(−4)} which are contained either in S_{+} or in S_{−} (i. e., s_{δ_{1}} should commute with θ) and such that s_{δ_{1}}(M) = M. Obviously, G ⊂ W^{(−4)}(S, L)_{M}.

We consider two real K3 surfaces (X, τ, φ) and (X′, τ′, φ′) with non-symplectic involutions of the type (S, θ) to be isomorphic with respect to the group G, if there exists an isomorphism f : X → X′ such that fτ = τ′f, fφ = φ′f and f|S ∈ G.

We correspond to a real K3 surface(X, τ, φ) with a non-symplectic involution of the type (S, θ) an integral involution (L, φ, S) with the condition (S, θ) on the sublattice S (see [21]). Here L = H_{2}(X; ℤ), φ is the action of φ on L such that φ|S = θ. The integral involution (L, φ, S) has the properties (e.g. see [12] or Sect. 3.10 in [21]):

(RSK3) The lattice L is even unimodular of signature (3, 19). The lattice L^{φ} is hyperbolic (it has the signature (1, t_{(−)})).

Clearly, we can consider abstract integral involutions (L, φ, S) of the type (S, θ) satisfying (RSK3). We consider two integral involutions (L, φ, S) and (L′, φ′, S) with the condition (S, θ) to be isomorphic for the group G if there exists an isomorphism ξ : L → L′ of lattices such that ξφ = φ′ξ and ξ|S belongs to the group G. We denote by In(S, θ, G) the set of isomorphism classes of integral involutions (L, φ, S) satisfying (RSK3) with the condition (S, θ) and the group G.

By monodromy consideration, two triplets (X, φ, τ) and (X′, φ′, τ′) of the type (S, θ) which belong to one connected component of moduli give isomorphic integral involutions with the condition (S, θ). Thus, we have the natural map from the set of connected components of moduli of triplets (X, φ, τ) to the set In(S, θ, G).

Using Global Torelli Theorem for K3 surfaces [24] and epimorphicity of Torelli map for K3 surfaces [13], we can prove the main result which is similar to Theorem 3.10.1 from [19] about moduli of real polarized K3 surfaces. This type of statements is very interesting because it reduces the problem of description of connected components of moduli of real algebraic varieties to a purely arithmetic problem.

Theorem 1. The natural map above gives the one to one correspondence between connected components of moduli of (Dℝ)-non-degenerate real K3 surfaces (X, τ, φ) with non-symplectic involutions of the type (S, θ) and the set In(S, θ, G) of isomorphism classes for G of integral involutions (L, φ, S) with the condition (S, θ) satisfying (RSK3).

Proof. Let (L, S, φ) be an isomorphism class from In(S, θ, G). We have the orthogonal decomposition up to finite index

(2.1)  S_{+} ⊕ S_{−} ⊕ L_{τ}^{φ} ⊕ L_{τ,φ} ⊂ L

Here is the key statement (compare with conditions (D) and (Dℝ) above).

Lemma 2. Let δ ∈ L has δ^{2} = −2. Assume that δ = (δ_{1} + δ_{2})/2 where δ_{1} ∈ S, δ_{2} ∈ L_{τ} and (δ_{1})^{2} = (δ_{2})^{2} = −4.

If additionally δ_{2} ∈ L_{τ}^{φ} ∪ L_{τ,φ} and (δ_{1}, h) = 0 for h ∈ S_{−} with h^{2} > 0, then δ_{1} ∈ S_{+} ∪ S_{−}.

Proof. We have (δ_{1})^{∗} = δ_{1}/2 = (δ_{1})_{+}^{∗}+(δ_{1})_{−}^{∗} where (δ_{1})_{±}^{∗} ∈ (S_{±})^{∗}. Since (δ_{1}, h) = ((δ_{1})_{−}^{∗}, h) = 0 and the lattice S_{−} is hyperbolic, then ((δ_{1})_{−}^{∗})^{2} < 0 if (δ_{1})_{−}^{∗} ≠ 0.

Assume that δ_{2} ∈ L_{τ,φ}. Then (δ_{1})_{−}^{∗} + δ_{2}/2 ∈ (L_{φ})^{∗}. The lattice L_{φ} is 2-elementary (since L is unimodular) which means that 2L_{φ}^{∗} ⊂ L_{φ}. It follows, δ̃ = 2(δ_{1})_{−}^{∗} + δ_{2} ∈ L_{φ} and δ̃^{2} = (2(δ_{1})_{−}^{∗})^{2} − 4 < −4 if (δ_{1})_{−}^{∗} ≠ 0. Since L is even, it follows that δ̃^{2} ≤ −6. Thus, we have δ = (δ_{1})_{+}^{∗} + δ̃/2. We then get (δ_{1})_{+}^{∗} = (δ_{1})_{+}/2 where (δ_{1})_{+} ∈ S_{+}. Since S_{+} is negative definite, even and does not have elements with square −2, then ((δ_{1})_{+})^{2} ≤ −4 if (δ_{1})_{+}^{∗} ≠ 0. If both elements (δ_{1})_{±}^{∗} ≠ 0, we then get δ^{2} ≤ (−4 − 6)/2 < −2 which is a contradiction. It follows δ_{1} ∈ S_{+} ∪ S_{−}.

If δ_{2} ∈ L_{τ}^{φ}, one should replace φ by φ̃ = τφ and argue similarly. We have Lφ̃ ; τ = L_{τ,φ} and L_{τ,φ̃} = L_{τ}^{φ}. □

We shall use the lemma later. The rest considerations are similar to the proof of Theorem 3.10.1 in [19] and are now standard.

Both lattices L_{τ}^{φ} and L_{τ,φ} are hyperbolic and define hyperbolic spaces L_{τ}^{φ} and L_{τ,φ}. E.g. to define L_{τ}^{φ}, one should choose one of half-cones V^{+}(L_{τ}^{φ}) of the cone V (L_{τ}^{φ}) = {x ∈ L_{τ}^{φ} ⊗ ℝ | x^{2} > 0} and consider the set L_{τ}^{φ} = V^{+}(L_{τ}^{φ})/ℝ^{+} of rays in V^{+}(L_{τ}^{φ}). Similarly, L_{τ,φ} = V^{+}(L_{τ,φ})/ℝ^{+}.

By global Torelli Theorem [24], the triplets

(2.2)  (Dℝ)-non-degenerate triplets (X, τ, φ) of the type (S, θ)

are described by their periods. Since S ⊂ N(X), periods should be orthogonal to S. It follows that periods of data (2.2) are pairs

(2.3)  (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ L_{τ}^{φ} × L_{τ,φ}

(see Sect. 3.10 in [19] for details). (We identify cohomology with homology using Poincaré duality.)

Let O(L, S, G, φ) be the group of all automorphisms α ∈ O(L) such that α(S) = S, α|S ∈ G and αφ = φα. The group O(L, S, G, φ) acts in L_{τ}^{φ} ×L_{τ,φ} as follows. For its element β we set β(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) = (ℝ^{+}(±β(ω_{+})), ℝ^{+}(±β(ω_{−}))) where ±β(ω_{+}) and ±β(ω_{−}) are chosen to be in V^{+}(L_{τ}^{φ}) and V^{+}(L_{τ,φ}) respectively. By global Torelli Theorem [24], periods (2.3) define isomorphic data (2.2), if and only if they belong to one orbit of O(L, S, G, φ). These periods we shall call equivalent.

Let W^{(−2)}(L_{τ}^{φ}) and W^{(−2)}(L_{τ,φ}) be the groups generated by all reflections in elements with square −2 of lattices L_{τ}^{φ} and L_{τ,φ} respectively. They are subgroups of O(L, S, G, φ). Both these groups are discrete in hyperbolic spaces L_{τ}^{φ} and L_{τ,φ} and have fundamental chambers Ω_{τ}^{φ} ⊂ L_{τ}^{φ} and Ω_{τ,φ} ⊂ L_{τ,φ} respectively. They are connected. We get all periods of data (2.2) up to equivalence if consider periods

(2.4)  (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω_{τ}^{φ} × Ω_{τ,φ}.

Further we consider only periods (2.4).

Consider (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω_{θ}^{φ} × Ω_{τ,φ}. Consider the set

(2.5)  Δ^{(−2)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) = {δ ∈ L_{τ} | δ^{2} = −2, (δ, ω_{+}) = (δ, ω_{−}) = 0}.

A point (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) in Ω_{θ}^{φ} × Ω_{τ,φ} does not correspond to periods of data (2.2) if Δ(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ≠ ∅. Really, assume that δ ∈ Δ(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}). Conditions (δ, ω_{+}) = (δ, ω_{−}) = 0 mean that δ ∈ N(X). Since δ^{2} = −2, by Riemann-Roch Theorem for K3 surfaces δ or −δ is effective. Since δ ∈ L_{τ}, then τ(δ) = −δ which is impossible for an effective class ±δ and a holomorphic involution τ.

Thus, we should delete from the set Ω_{τ}^{φ} × Ω_{τ,φ} of possible periods of data (2.2) all points (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) where ω_{+} or ω_{−} belongs to a face of the polyhedron Ω_{τ}^{φ} or Ω_{τ,φ} since they are orthogonal to elements of the corresponding lattices with square −2 and define elements of Δ(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}).

Thus, all possible periods of data (2.2) are points

(2.6)  (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ (Ω_{τ}^{φ})′ × (Ω_{τ,φ}^{−})′

where ′ means that we deleted all faces of the fundamental chambers (we can call them as open fundamental chambers). The set is connected since it is the product of open convex sets. For points (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) from (Ω_{φ}^{+})′×(Ω_{τ,φ}^{−})′, elements δ ∈ Δ^{(−2)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) cannot belong to L^{φ} or L_{τ,φ}.

Any δ ∈ Δ^{(−2)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) has the property that δ = δ_{+} ⊕ δ_{−} where δ_{+} ∈ (L_{τ}^{φ})^{∗} and δ_{−} ∈ L_{τ,φ}^{∗}. Moreover (δ_{+})^{2} < 0 if δ_{+} ≠ 0, and (δ_{−})^{2} < 0 if δ_{−}^{2} < 0 since they are elements of hyperbolic lattices orthogonal to their elements ω_{+} and ω_{−} with positive square. Since for points (2.6) elements of Δ^{(−2)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) cannot belong to L_{τ,φ} or L^{φ}, we can also assume that δ_{+} ≠ 0 and δ_{−} ≠ 0.

Let

\begin{matrix}Δ^{(−2)} = & \{δ = δ_{+} ⊕ δ_{−} ∈ L_{τ} | δ_{+} ∈ (L_{τ}^{φ})^{∗}, δ_{−} ∈ (L_{τ,φ})^{∗}, \\ (δ_{+})^{2} < 0, (δ_{−})^{2} < 0, δ^{2} = (δ_{+})^{2} + (δ_{−})^{2} = −2\}. \end{matrix} ; (2.7)

Let δ = δ_{+} ⊕ δ_{−} ∈ Δ^{(−2)}. Let H_{δ_{+}} = {ℝ^{+}ω_{+} ∈ L_{τ}^{φ} | (ω_{+}, δ_{+}) = 0} be the hyperplane in L_{τ}^{φ} which is orthogonal to δ_{+}. Similarly, let H_{δ_{−}} = {ℝ^{+}ω_{−} ∈ L_{τ,φ} | (ω_{−}, δ_{−}) = 0} be the hyperplane in L_{τ,φ} which is orthogonal to δ_{−}. Points (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ H_{δ_{+}} ×H_{δ_{−}} are exactly the points such that Δ(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) contains δ, and they should be deleted from the set of possible periods of data (2.2). Thus, the set of possible periods of data (2.2) is

(2.8)  (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω_{1} = (Ω_{τ}^{φ})′ × (Ω_{τ,φ})′ − ( ⋃_{δ∈Δ^{(−2)}} H_{δ_{+}} × H_{δ_{−}} ) .

The space Ω_{1} is connected because H_{δ_{+}} × H_{δ_{−}} has codimension 2. Moreover, the set of hyperplanes H_{δ_{+}}, δ ∈ Δ^{(−2)}, is locally finite in L_{τ}^{φ} because all δ_{+} belong to a lattice (L_{τ}^{φ})^{∗} and 0 > (δ_{+})^{2} > −2. Hyperplanes H_{δ_{−}}, δ ∈ Δ^{(−2)}, have the same property in L_{τ,φ}.

The set (2.8) gives periods of all triplets (X, τ, φ) of the type (S, θ) without additional condition that they are (Dℝ)-non-degenerate (it follows that Theorem 1 is valid if we delete the words (Dℝ)-non-degenerate). Let us add the last condition.

Let us denote by

(2.9)  Δ̃^{(−4)} ⊂ L_{τ}

the set of all elements δ_{2} ∈ L_{τ} such that δ_{2}^{2} = −4 and there exists δ_{1} ∈ S with conditions: δ_{1}^{2} = −4, δ = (δ_{1} +δ_{2})/2 ∈ L, there exists an element h ∈ S_{−} ∩int(M) such that (δ_{1}, h) = 0.

For (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ L_{τ}^{φ} × L_{τ,φ}, we denote by Ω^{(−4)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) the set of all δ_{2} ∈ Δ̃^{(−4)} such that (ω_{+}, δ_{2}) = (ω_{−}, δ_{2}) = 0. A triplet (2.2) is (Dℝ)-degenerate, if and only if its periods (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) have the non-empty set Ω^{(−4)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}). Thus, we should delete from Ω_{1} in (2.8) all these points.

Like for Δ^{(−2)} above, the set of these points has codimension one only if there exists δ_{2} ∈ Ω^{(−4)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) such that δ_{2} ∈ L_{τ}^{φ} ∪ L_{τ,φ}. By Lemma 2, then the corresponding δ_{1} belongs to S_{−} ∪ S_{+} (see definition of Δ̃^{(−4)} in (2.9)). Let δ = (δ_{1} + δ_{2})/2 ∈ L and δ′ = (−δ_{1} + δ_{2})/2 ∈ L. Since δ^{2} = (δ′)^{2} = −2, there exist reflections s_{δ}, s_{δ′} in these elements which belong to O(L). Their composition s = s_{δ}s_{δ′} acts in S = L^{τ} as the reflection s_{δ_{1}} in δ_{1}, and it acts in L_{τ} as the reflection s_{δ_{2}} in δ_{2}. Moreover, the reflection s_{δ_{1}} ∈ G since δ_{1} ∈ S_{−} ∪ S_{+} and (h, δ_{1}) = 0 for h ∈ int(M). It follows that s ∈ O(L, S, G, φ).

Thus, we get all equivalent periods of data (2.2), if we replace Ω_{τ}^{φ}, (Ω_{τ}^{φ})′ and Ω_{τ,φ}, (Ω_{τ,φ})′ by the corresponding fundamental chambers (and open fundamental chambers) Ω̃_{τ}^{φ}, (Ω̃_{τ}^{φ})′ and Ω̃_{τ,φ}, (Ω̃_{τ,φ})′ for the groups W (L_{τ}^{φ}) and W (L_{τ,φ}) generated by reflections in all elements with square −2 and in all elements from Δ̃^{(−4)} which are contained in lattices L_{φ}^{τ} and L_{τ,φ} respectively.

Let Δ^{(−4)} = Δ̃^{(−4)} − (L_{τ}^{φ} ∪ L_{τ,φ}). If δ_{2} ∈ Δ^{(−4)}, then we write δ_{2} = (δ_{2})_{+} + (δ_{2})_{−} where (δ_{2})_{+} ∈ (L_{τ}^{φ})^{∗} and (δ_{2})_{−} ∈ (L_{τ,φ})^{∗}.

The set of all periods of data (2.2) is then

(2.10)  (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω = (Ω̃_{φ}^{τ})′ × (Ω̃_{τ,φ})′ − ( ⋃_{δ∈Δ∪Δ^{(−4)}} H_{δ_{+}} × H_{δ_{−}} )

where (Ω̃_{φ}^{τ})′ and (Ω̃_{τ,φ})′ are open locally-finite polyhedra in hyperbolic spaces; the products H_{δ_{+}} × H_{δ_{−}} have codimension two and their set is locally finite in (Ω̃_{φ}^{τ})′ × (Ω̃_{τ,φ})′. It follows that Ω is connected.

By epimorphicity of Torelli map for K3 surface [13] applied to real K3 surfaces, there exist data (2.2) with any periods from (2.10). Thus, by Global Torelli Theorem [24] and epimorphicity of Torelli map for K3 surfaces [13], the moduli space of data (2.2) is covered by the connected space Ω defined in (2.10). More exactly, the moduli space of data (2.2) is the quotient of Ω by the stabilizer subgroup of Ω in O(L, S, G, φ), and it is connected.

This finishes the proof of the Theorem.

Let us add the standard details to that. Assume (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω. We can choose ω_{+} and ω_{−} to satisfy conditions ω_{+}^{2} = ω_{−}^{2} > 0. Let ω = ω_{+} + iω_{−}. Then ω^{2} = 0 and (ω, \overline{ω}) > 0. We have τ(ω) = −ω and φ(ω) = \overline{ω}. Let H^{2,0} = ℂω ⊂ L ⊗ ℂ. We then have τ(H^{2,0}) = H^{2,0} and φ(H^{2,0}) = H^{0,2} = \overline{H^{2,0}}.

We choose h ∈ S_{−} such that h belongs to the interior of the fundamental chamber M. This is possible because of our conditions on θ.

Let N = {x ∈ L | (ω, x) = 0} = {x ∈ L | (ω_{+}, x) = (ω_{−}, x) = 0}. The lattice N is hyperbolic. Let δ ∈ N and δ^{2} = −2. We claim that (h, δ) ≠ 0.

We can write δ = (δ_{1} + δ_{2})/2 where δ_{1} ∈ S and δ_{2} ∈ L_{τ}. We have δ_{2}^{2} < 0 if δ_{2} ≠ 0 because its orthogonal complement in L_{τ} ⊗ ℝ contains, ω_{+} ⊥ ω_{−} with positive squares and L_{τ} has exactly two positive squares.

We have δ_{1} ≠ 0. Otherwise, δ ∈ L_{τ} and Δ^{(−2)}(ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ≠ ∅ which is not the case for points of Ω by our considerations above.

Assume δ_{1} ≠ 0 and δ_{1}^{2} ≥ 0. Then (h, δ) = (h, δ_{1}) ≠ 0 because S is hyperbolic and h^{2} > 0. Assume δ_{1} ≠ 0 and δ_{1}^{2} < 0. Since −2 = δ^{2} = (δ_{1}^{2} + δ_{2}^{2})/4 and L is even, then we

have three possibilities: (0) δ_{2} = 0 and δ = δ_{1}/2 ∈ S; (1) δ_{1}^{2} = −2 and δ_{2}^{2} = −6. (2) δ_{1}^{2} = δ_{2}^{2} = −4. Let us consider all of them.

(0) and (1). Since h is in interior of M and M is the fundamental chamber for the group generated by reflections in all elements of S with square −2, we then have (h, δ) ≠ 0 for (0), and (h, δ) = (h, δ_{1})/2 ≠ 0 for (1).

(2) If (h, δ_{1}) = 0, we get a contradiction with our construction of Ω in (2.10). Thus, (h, δ) = (h, δ_{1})/2 ≠ 0.

It proves the claim: (h, δ) ≠ 0 for any δ ∈ N with δ^{2} = −2.

By epimorphicity of the period map for K3 surfaces [13], there exists a complex K3 surface X and an isomorphism ξ : H_{2}(X, ℤ) → L such that ξ^{−1}(H^{2,0}) = H^{2,0}(X) and ξ^{−1}(h) is a polarization of X. By Global Torelli Theorem for K3 surfaces [24], the choice of (X, ξ) is unique up to isomorphisms. The pair (X, τξ) satisfies the same conditions. Thus, ξ^{−1}τξ is induced by a unique automorphism of X which we denote by the same letter τ (for simplicity). The τ acts as −1 on H^{2,0}(X) and as an involution in H_{2}(X, ℤ). It follows that τ is a non-symplectic involution of X of the type S.

The pair (\overline{X}, φξ) is another pair with the same properties as (X, ξ) where \overline{X} is the surface X with the conjugate complex structure. It follows that ξ^{−1}φξ is induced by an antiholomorphic involution of X which we denote by the same letter φ.

Finally we get a unique, up to isomorphisms, a triplet (X, τ, φ) from (2.2) with periods which are equivalent to (ℝ^{+}ω_{+}, ℝ^{+}ω_{−}) ∈ Ω.

□

2.2. Connected components of moduli of non-degenerate real right DPN-pairs. Below we consider a description of K3 surfaces with non-symplectic involutions using the corresponding quotients and double coverings. See [20], [22], [1], [2] and also [4] for details.

Let (X, τ) be a K3 surface with a non-symplectic involution τ and A = X^{τ} the set of fixed points of τ. Since τ^{∗}(ω_{X}) = −ω_{X} and ω_{X} has zero divisor, it follows that A is a nonsingular curve of X. It follows that the surface Y = X/{1, τ} is non-singular, the quotient map π : X → Y is a double covering ramified along a non-singular curve A ⊂ Y. Simple considerations show that Y is Enriques surface if A = ∅, and Y is a rational surface if A ≠ ∅. The non-singular curve A ∈ | − 2K_{Y} | and the irregularity q_{Y} = dim H^{0}(Y, Ω^{1}) = 0. The opposite statement is also valid. If Y is a non-singular surface, A ∈ | − 2K_{Y} | a non-singular curve, and q_{Y} = 0, then the double covering of Y ramified along A gives a K3 surface X with a non-symplectic involution of the double covering. A pair (Y, A) with these properties is called a right DPN pair. The surface Y is called a right DPN-surface. We follow the terminology of [22], [1] and [2].

We mention that a general (not necessarily right) DPN-pair is a pair (Y, A) where Y is a non-singular surface with q_{Y} = 0, A ∈ |−2K_{Y} | and A has only ADE-singularities; the surface Y is called a DPN-surface. Any DPN-pair gives a right one after a sequence of blow-ups in singular points of A. This sequence is canonical, and studying of general DPN-pairs can be reduced to studying right ones. Any general DPN-pair can be obtained from a right one by a sequence of contractions of exceptional curves of the 1st kind. Further we consider only right DPN-pairs and right DPN-surfaces.

Like K3 surfaces with non-symplectic involutions, we call a right DPN-pair (Y, A) to be (D)-degenerate if the corresponding K3 surface with non-symplectic involution (X, τ) is (D)-degenerate. Similarly, we call a real (i.e. with an anti holomorphic involution θ) DPNpair (Y, A) to be (Dℝ)-degenerate if the corresponding real K3 surface with non-symplectic involution (X, τ, φ) is (Dℝ)-degenerate. Below we discuss that. In particular, we shall see that the property to be degenerate depends only on the quotient Y = X/{1, τ}. It does not depend on A.

We follow notations from [1] which are a little different from [22]. We use the following facts about right DPN-pairs (Y, A) and the corresponding K3 surfaces with non-symplectic involutions (X, τ) of the type S.

Any exceptional curve F of Y (i. e. an irreducible algebraic curve F with negative square F^{2} < 0) is a non-singular rational curve with square −1, −2 or −4. If F^{2} = −4 (the type I), then F is a component of A and its preimage E = π^{−1}(F) is a component of X^{τ} with E^{2} = −2. If F^{2} = −1 (the type II), then (F, A) = 2. If A intersects F in two different points (the type IIa), then π^{−1}(F) = E where E is a non-singular rational curve with square −2. If F^{2} = −1 and A is tangent to F with multiplicity 2 (the type IIb), then π^{−1}(F) = E + τ(E) where E is a non-singular rational curve with square −2 and (E, τ(E)) = 1 (remark that (E + τ(E))^{2} = −2 like for E^{2} of the type IIa). If F^{2} = −2 (the type III), then π^{−1}(F) = E + τ(E) is disjoint union of two non-singular rational curves of X with square −2.

We have the following description of exceptional curves on Y in terms of X. Let Δ(S)^{(−2)} = {δ ∈ S | δ^{2} = −2}. Let Δ(S, X)^{(−4)} be the set of all δ_{1} ∈ S such that δ_{1}^{2} = −4 and there exists δ_{2} ∈ S_{N(X)}^{⊥} such that δ_{2}^{2} = −4 and δ = (δ_{1} + δ_{2})/2 ∈ N(X) (then δ^{2} = −2). All elements of Δ(S, X)^{(−4)} are roots of S with square −4. Let W (S, X) ⊂ O(S) be the group generated by reflections in all elements of Δ(S, X) = Δ(S)^{(−2)}∪Δ(S, X)^{(−4)}. The set Δ(S, X) is invariant with respect to W (S, X). The group W (S, X) is a discrete reflection group in V^{+}(S) and in the corresponding hyperbolic space V^{+}(S)/ℝ^{+}. All reflections from W (S, X) are reflections with respect to elements from Δ(S, X). All faces of a fundamental chamber of W (S, X) are orthogonal to Δ(S, X). Let K(X) ⊂ N(X)⊗ℝ be the Kählerian (or the nef cone) of X. The crucial fact is that M(S, X) = (S ⊗ ℝ) ∩ K(X) is a fundamental chamber for the group W (S, X). Its faces are orthogonal exactly to preimages π^{∗}(F) of exceptional curves F of Y.

Let us take a face of M(S, X). Then it is orthogonal to δ ∈ Δ(S, X). We have: δ = π^{∗}(F) where F^{2} = −1, if δ^{2} = −2 and δ is not the class of a component of X^{τ}; 2δ = π^{∗}(F) where F^{2} = −4, if δ^{2} = −2 and δ is the class of a component of X^{τ}. δ = π^{∗}(F) where F^{2} = −2, if δ^{2} = −4, i.e. δ ∈ Δ(S, X)^{(−4)}.

By the projection formula, we also have M(S, X) = π^{∗}(M(Y)) where M(Y) is the Kählerian cone of Y.

Using these results, we get Proposition 3. A right DPN-pair (Y, A) (or the corresponding non-symplectic involution (X, τ)) is (D)-degenerate, if and only if Y = X/{1, τ} has an exceptional curve with square −2. Proof. We have M(S, X) ⊂ M where M is the fundamental chamber for the group W^{(−2)}(S) generated by reflections in Δ^{(−2)}(S).

If X is (D)-degenerate, then M(S, X) ≠ M. It follows that at least one face of M(S, X) is not orthogonal to an element δ ∈ S with δ^{2} = −2. Then it is orthogonal to δ ∈ Δ(S, X)^{(−4)} where δ^{2} = −4. It follows that δ = π^{∗}(F) where F is an exceptional curve of Y with F^{2} = −2.

Let F be an exceptional curve on Y with F^{2} = −2. Then δ = π^{∗}(F)) ∈ S where δ^{2} = 2F^{2} = −4. It follows that at least one face of M(S, X) is orthogonal to an element of S with square −4. Then it cannot be orthogonal to an element of S with square −2. Otherwise these two elements are ℚ-proportional which is impossible for an integral lattice. It follows that M(S, X) ≠ M and Y is (D)-degenerate. □

As an example, let us apply cited above theory of DPN-surfaces to S = U(2). The lattice S has no elements with square −2. Then Y has no exceptional curves with square −1. Moreover, rk H^{2}(Y, ℝ) = rk S = 2. Since Y can be only a rational or Enriques surface, it follows that Y is a relatively minimal rational surface Y ≅ 𝔽_{n} where n = 0, 2 or 4 since a right DPN-surface may have only exceptional curves with squares −1, −2 or −4. For S = U(2), the set of fixed points X^{τ} = C_{9} ∈ | − K_{Y} | is a non-singular irreducible curve of genus 9 (see the formula (2.12) for X^{τ} below). This is only possible for 𝔽_{0} or 𝔽_{2} since any curve C ∈ | − 2K_{𝔽_{4}}| contains the exceptional section (rational) of 𝔽_{4} (see Sect. 7 below). The case Y = 𝔽_{2} gives (D)-degenerate case because the exceptional section of 𝔽_{2} gives an exceptional curve with square (−2). In the (D)-non-degenerate case, Y ≅ ℙ^{1} × ℙ^{1}. Thus, we get Proposition 4. For a non-symplectic involution τ of a K3 surface X with the lattice S = U(2) the surface Y = X/{1, τ} is Y ≅ ℙ^{1} × ℙ^{1} in the D-non-degenerate case, and Y ≅ 𝔽_{2} in the D-degenerate case.

We have a similar to Proposition 3 statement over ℝ.

Proposition 5. A right real DPN-pair (Y, A, θ) (or the corresponding real non-symplectic involution (X, τ, φ) where θ = φ mod {1, τ}) is (Dℝ)-degenerate, if and only if Y = X/{1, τ} has an exceptional curve F with square −2 such that (F, θ(F)) ≤ 1 (equivalently, either θ(F) = F or the intersection matrix of F and θ(F) is negative definite).

Proof. If \overline{Y} is the complex conjugate surface to Y, then \overline{Y} has the same exceptional curves as Y with the complex structures on them changed by the conjugate. It follows that for any exceptional curve F of Y we have θ(F) = F_{1} where F_{1} is another exceptional curve of Y (called θ-conjugate) and θ^{∗}([F]) = −[F_{1}] where [C] denotes the homology class (or cohomology class since we always identify homology and cohomology) of a holomorphic curve C.

The pair (Y, A, θ) is (Dℝ)-degenerate, if and only if the inclusion

M(S, X) ∩ (S_{−} ⊗ ℝ) ⊂ M ∩ (S_{−} ⊗ ℝ)

is strict. By considerations for the proof of Proposition 3, it is equivalent to existence of an exceptional curve F of Y with F^{2} = −4 such that π^{∗}[F] = δ ∈ Δ(S, X)^{(−4)} and the inequality (x, δ) ≥ 0 considered in M∩(S_{−} ⊗ℝ) gives its proper subset. Let δ = (δ_{−} +δ_{+})/2 where δ_{−} ∈ S_{−}^{∗} and δ_{+} ∈ S_{+}^{∗}.

The inequality (x, δ) ≥ 0 is equivalent to (x, δ_{−}) ≥ 0 for x ∈ S_{−} ⊗ ℝ. If δ_{−}^{2} ≥ 0, the inequality is trivial on M ∩ (S_{−} ⊗ ℝ) since S_{−} is hyperbolic and M has only elements with non-negative square.

Thus, we can assume that δ_{−}^{2} < 0 if the pair (Y, A, θ) is (Dℝ)-degenerate. Let F̃ be one of components of π^{−1}(F). We have

δ = [F̃] + [τ(F̃)],
δ_{−} = [F̃] + [φ(F̃)] + [τ(F̃)] + [τ(φ(F̃)],
δ_{+} = [F̃] − [φ(F̃)] + [τ(F̃)] − [τ(φ(F̃)].

It follows, 0 > δ_{−}^{2} = (π^{∗}(F) + π^{∗}(θ(F)))^{2} = 2(F + θ(F))^{2} = 2(F^{2} + θ(F)^{2} + 2(F, θ(F)) = 2(−4 + 2(F, θ(F)). Then (F, θ(F)) ≤ 1.

Let us prove the opposite statement. Let F be an exceptional curve of Y with F^{2} = −2 and (F, θ(F)) ≤ 1. If (F, θ(F)) < 0, then θ(F) = F because both curves F and θ(F) are irreducible. If (F, θ(F)) = 0 or 1, then θ(F) ≠ F, and the intersection matrix of the curves F, θ(F) is negative definite. It follows that F and θ(F) (where θ(F) = F for the first case) define a face of the nef cone M(Y) of Y: there exists a nef element H of Y with H^{2} > 0 such that (H, F) = (H, θ(F)) = 0 and (H, E) > 0 for any other exceptional curve E of Y which is different from F and θ(F). The elements (−θ^{∗}(H)) and h = H − θ^{∗}(H) are also nef. We have θ^{∗}(h) = −h, thus h is a real nef element. Since θ^{∗}[F] = −[θ(F)] , it follows that (h, F) = (H, F)−(θ^{∗}(H), F) = (H, F)−(H, θ^{∗}[F]) = (H, F)+(H, θ(F)) = 0. Similarly, (h, θ(F)) = 0. Moreover, (h, E) = (H, E) + (−θ^{∗}(H), E) ≥ (H, E) > 0 for any exceptional curve E of Y which is different from the curves F and θ(F). It follows that the real nef element h belongs to a face of M(Y) which is orthogonal to F and θ(F), and h does not belong to a face of M(Y) which is orthogonal to any other exceptional curve E of Y different from F and θ(F).

We have M(X, S) = π^{∗}(M(Y)) and π^{∗}(h) ∈ M(X, S) is a real nef element which belongs to one or two faces of M(X, S) which are orthogonal to elements π^{∗}[F] and π^{∗}[θ(F)] with square −4 of S, and π^{∗}(h) does not belong to any other face of M(X, S). It follows that π^{∗}(h) is inside of M because faces of M are orthogonal to elements with square −2. It finishes the proof of the statement.

□

Let (Y, θ, A) be a real right DPN-pair, i. e. (Y, θ) is a non-singular projective algebraic surface over ℂ with an anti-holomorphic involution θ (i. e. a real projective algebraic surface Y/ℝ), and A ∈ | − 2K_{Y} | is a non-singular curve such that θ(A) = (A) (i. e. a real curve A/ℝ). Later we shall also use notation (Y, A)/ℝ (or just (Y, A)) for a pair defined over ℝ. It is natural to consider two such pairs (Y, A, θ) and (Y′, A′, θ′) to be isomorphic if there exists an isomorphism f : Y → Y′ of algebraic surfaces over ℂ such that f(A) = A′ and fθ = θ′f. The corresponding moduli space should parameterize isomorphism classes of such pairs. But we consider a more delicate isomorphism relation between real right DPN-pairs. Studying connected components of moduli for this more delicate isomorphism relation gives an additional information about monodromy.

There exist two real double coverings of Y ramified along A which give two real K3 surfaces with non-symplectic involutions (X, τ, φ) and (X, τ, φ̃) where φ̃ = τφ. They define the same real right DPN-pair (Y = X/{1, τ}, A = X^{τ}, θ = φ mod {1, τ}). The real K3 surfaces with non-symplectic involutions (X, τ, φ) and (X, τ, φ̃ = τφ) are called related. A real right DPN-pair (Y, A, θ) together with a choice of one, between two, real double coverings of Y ramified along A (equivalently together with a choice of one, between two, real K3 surfaces with non-symplectic involutions (X, τ, φ) such that its quotient by τ gives (Y, θ, A)) is called a positive real right DPN-pair. We shall denote a positive DPN-pair as (Y, θ, A)^{+} (or (Y, A)^{+} for (Y, A)/ℝ). Then a related DPN-pair will be denoted by (Y, θ, A)^{−}. If (Y, θ, A)^{+} is given by (X, τ, φ), then (Y, θ, A)^{−} is given by (X, τ, φ̃ = τφ). Obviously, considering of real right DPN-pairs (Y, θ, A) is equivalent to considering of sets of related positive real right DPN-pairs {(Y, θ, A)^{+}, (Y, θ, A)^{−}}. An isomorphism of two positive real right DPN-pairs is equivalent to an isomorphism of the corresponding real K3 surfaces with non-symplectic involutions. Thus, we can define positive real right DPN-pairs of the type (S, θ) and isomorphism of real right DPN-pairs with respect to the group G defined by (S, θ). Obviously, an isomorphism of real non-symplectic involutions (X, τ, φ) and (X′, τ′, φ′) defines the corresponding isomorphism of the related non-symplectic involutions (X, τ, φ̃) and (X′, τ′, φ̃′). Moreover, the type (S, θ) and the group G don’t change for related real non-symplectic involutions. Moreover, by Proposition 5 related pairs are (Dℝ)-degenerate simultaneously.

Since positive real right DPN-pairs are equivalent to real non-symplectic involutions of K3, we get from Theorem 1 an equivalent

Theorem 6. The natural map gives the one to one correspondence between connected components of moduli of (Dℝ)-non-degenerate positive real right DPN-pairs (Y, θ, A)^{+} of the type (S, θ) and the set In(S, θ, G) of isomorphism classes for G of integral involutions (L, φ, S) with the condition (S, θ) satisfying (RSK3).

Integral involutions (L, φ, S) and (L, φ̃, S) corresponding to related pairs (Y, θ, A)^{+} and (Y, θ, A)^{−} are related as φ̃ = τφ. We remind that τ acts as +1 on S and as −1 on S_{L}^{⊥}. Naturally, integral involutions (L, φ, S) and (L, φ̃, S) are called related. We denote the involution (L, φ, S) → (L, φ̃, S) in In(S, θ, G) as τ too. From Theorem 6 we get

Theorem 7. The natural map gives the one to one correspondence between connected components of moduli of (Dℝ)-non-degenerate real right DPN-pairs (Y, θ, A) of the type (S, θ) and

[figure: means δ = 0, and; means δ = 1.); 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]
Figure 1. All possible (r, a, δ) = (r(S), a(S), δ(S)) = (r(τ), a(τ), δ(τ))

the set In(S, θ, G)/{1, τ} of isomorphism classes for G of pairs {(L, φ, S), (L, φ̃ = τφ, S)} of related integral involutions with the condition (S, θ) satisfying (RSK3).

2.3. General classification results about real K3 surfaces with non-symplectic involutions and real right DPN-pairs. The first, the most important invariant of a K3 surface with a non-symplectic involution (X, τ) is the isomorphism class of the lattice S = H_{2}(X, ℤ)^{τ}. The lattice S defines the type of (X, τ) and of the corresponding right DPNpair (Y = X/{1, τ}, A = X^{τ}). The lattice S is any hyperbolic (i. e. of signature (1, t_{(−)}(S)) and of the rank r(S) = 1 + t_{(−)}(S)) even lattice having a primitive embedding S ⊂ L to the even unimodular lattice L ≅ H_{2}(X, ℤ) of signature (3, 19) and such that there exists an involution τ of L with L^{τ} = S. The last property is equivalent for S to be 2-elementary, i. e. S^{∗}/S ≅ (ℤ/2ℤ)^{a(S)} where a(S) ≥ 0 is an invariant of S. Another invariant δ(S) of S is equal to 0 or 1. The δ(S) = 0 if and only if (x^{∗})^{2} ∈ ℤ for any x^{∗} ∈ S^{∗}; equivalently, (z, τ(z)) ≡ 0 mod 2 for any z ∈ L. (Namely, δ(S) means δ(τ) like δ(φ) below.) The triplet (r(S), a(S), δ(S)) is the complete list of invariants of the isomorphism class of S. All possible triplets (r, a, δ) = (r(S), a(S), δ(S)) were classified in [19], see also [20] and [22]. They are presented in Figure 1 (it is the same as Figure 2 in [22]).

We have

\begin{matrix}X^{τ} = & ∅, if (r(S), a(S), δ(S)) = (10, 10, 0); \\ X^{τ} = & C_{1} + C_{1}′ if (r(S), a(S), δ(S)) = (10, 8, 0); \\ X^{τ} = & C_{g(S)} + E_{1} + ⋯ + E_{k(S)}, g(S) = (22 − r(S) − a(S))/2, k(S) = (r(S) − a(S))/2. \end{matrix} (2.11)

Here C_{g} denote a curve of genus g and E_{i} ≅ ℙ^{1}. We have

(2.12)  X^{τ} ∼ 0 mod 2 in H_{2}(X, ℤ)

if and only if δ(S) = 0. The dimension of moduli of pairs (X, τ) and the corresponding DPN-pairs (Y, A) is equal to 20 − r(S). See [20] and [22].

The type (S, θ) of a real K3 surface with a non-symplectic involution (X, τ, φ) is defined by the isomorphism class θ of the action of φ on S. Any θ with the conditions S^{θ} is negative definite and does not have elements x ∈ S^{θ} with x^{2} = −2 can be taken as a type. Equivalently, (−θ) ∈ O^{+}(S) and (−θ) gives an involution of a fundamental chamber M of W^{(−2)}(S). Finding of all possible types (S, θ) is a purely arithmetic and not difficult problem related with the automorphism group of the hyperbolic lattice S and its subgroup W^{(−2)}(S). In this paper, we shall consider lattices S with r(S) ≤ 2 when the problem of finding of possible types (S, θ) is very simple. It is why we don’t consider this problem in this paper any more. See [4] about some results in this direction.

Assume that the type (S, θ) is fixed. We denote S_{+} = S^{θ} and S_{−} = S_{θ}. We shall use invariants

(2.13)  s = rk S, p = rk S_{+}

Then S has the signature (s_{(+)}, s_{(−)}) = (1, s−1) and S_{+} has the signature (p_{(+)}, p_{(−)}) = (0, p).

First we calculate the group G which we had uses in Theorems 1, 6 and 7. Let α ∈ S and α^{2} = −4. If α ∈ Δ(S, L)^{(−4)}, then α is a root of S, i. e. (α, S) ≡ 0 mod 2. Let ⟨−4⟩ be a lattice generated by β with β^{2} = −4. Let S ⊕ ⟨−4⟩ be the orthogonal sum of S and ⟨−4⟩. Let S ⊕ ⟨−4⟩ ⊂ S(α) be its overlattice of index two which is generated by (α + β)/2.

Lemma 8. The element α ∈ Δ(S, L)^{(−4)} if and only if there exists a primitive embedding S(α) ⊂ L.

In particular, this is true if r(S) + a(S) < 20. It is also valid if r(S) < 10. Thus under one of these conditions, Δ(S, L)^{(−4)} consists of all roots α ∈ S with α^{2} = −4.

Proof. The first statement is trivial.

We have rk S(α) = r(S) + 1, the minimal number l(A_{S(α)}) of generators of A_{S(α)} = S(α)^{∗}/S(α) is not more than a(S) + 1. Thus, rk S(α) + l(A_{S(α)}) ≤ r(S) + 1 + a(S) + 1 < 22 = rk L. By Corollary 1.12.3 in [19], there exists a primitive embedding S(α) ⊂ L.

The last statement follows from the obviouse inequality a(S) ≤ r(S).

□ Using results of [19] more heavily, one can give a necessary and sufficient condition of

α ∈ Δ(S, L)^{(−4)}. We have restricted here the simple sufficient condition which is enough for this paper.

Let us introduce

Δ(S_{±}, S)^{(−4)} = {α ∈ S_{±} | α^{2} = −4, (α, S) ≡ 0 mod 2}.

Let W^{(−4)}(S_{±}, S) be the group generated by reflections in all elements of Δ(S_{±}, S)^{(−4)}. From Lemma 8, we get

Lemma 9. If a(S) + r(S) < 20 (in particurlar, if r(S) < 10), then the group G = W^{(−4)}(S_{+}, S) × W^{(−4)}(S_{−}, S).

In general, the group G is a subgroup of W^{(−4)}(S_{+}, S) × W^{(−4)}(S_{−}, S). In Theorems 1, 6 and 7 one can replace G by the bigger group W^{(−4)}(S_{+}, S) × W^{(−4)}(S_{−}, S) to simplify considerations. Of course, the results can be weaker then.

We say that two integral involutions (L, φ, S) and (L′, φ′, S) of the type (S, θ) have the same genus with respect to the group G of (S, θ) if there exists an automorphism ξ : S → S from G which can be continued to an isomorphism (L, φ, S) ⊗ ℝ → (L′, φ′, S) ⊗ ℝ over ℝ, and an isomorphism (L, φ, S) ⊗ ℤ_{p} → (L′, φ′, S) ⊗ ℤ_{p} over the ring ℤ_{p} of p-adic integers for any prime p. Other speaking, the integral involutions (L, φ, S) and (L′, φ′, S) cannot be distinguished considering over ℝ and modulo any integer N using isomorphisms on S which belong to the group G. All the genus invariants (for an arbitrary even lattice S with an involution θ) were found in [21] together with necessary and sufficient conditions of existence. In many cases the genus has only one isomorphism class. Then the genus invariants give complete invariants of the isomorphism class. Below we remind these results adapting them to our case.

We assume that the integral involution (L, φ, S) satisfies the condition (RSK3). I. e. the lattice L is even unimodular of signature (l_{(+)}, l_{(−)}) = (3, 19). The lattice L^{φ} is hyperbolic of the signature (t_{(+)}, t_{(−)}) = (1, t_{(−)}). Then the only real invariant of (L, φ, S) is

(2.14)  r = rk L^{φ} = 1 + t_{(−)}.

Below we describe the genus invariants of the integral evolution (L, φ, S) of the type (S, θ). To simplify notations, we temporarily denote L_{+} = L^{φ} and L_{−} = L_{φ}.

Since L is unimodular, we have

(2.15)  A_{L_{±}} = L_{±}^{∗}/L_{±} ≅ L/(L_{+} ⊕ L_{−}) ≅ (ℤ/2ℤ)^{a}

where a ≥ 0 is an integer. It is one of the most important genus invariants.

We also have genus invariants

(2.16)  δ_{φ} = \{ \begin{matrix}0 & if (x, φ(x)) ≡ 0 mod 2 ∀x ∈ L \\ 1 & otherwise \end{matrix} ,

and

(2.17)  δ_{φS} = \{ \begin{matrix}0 & if (x, φ(x)) ≡ (x, s_{φ}) mod 2 ∀x ∈ L \\ & for some element s_{φ} in S \\ 1 & otherwise \end{matrix} .

If δ_{φS} = 0, then the element s_{φ} occurring in the definition of δ_{φS} is uniquely defined modulo 2S. It is called the characteristic element of the involution φ.

Depending on these invariants, we divide (L, φ, S) into the following 3 types. Type 0: δ_{φS} = 0 and δ_{φ} = 0;

Type Ia: δ_{φS} = 0 and δ_{φ} = 1;
Type Ib: δ_{φS} = 1.

For x_{±} ∈ S_{±} we put ,

(2.18)  δ_{x_{±}} = \{ \begin{matrix}0 & if (x_{±}, L_{±}) ≡ 0 mod 2, \\ 1 & otherwise. \end{matrix}

Equivalently, δ_{x_{±}} = 0 iff \frac{1}{2}x_{±} ∈ 2L_{±}^{∗}.

Since L is unimodular,

(2.19)  δ_{x_{±}} = 0 ⇔ ∃x′_{±} ∈ L_{∓} : \frac{1}{2}(x_{±} + x′_{±}) ∈ L.

The elements x′_{±} are defined by the elements x_{±} uniquely modulo 2L_{∓}; this enables us, for elements x_{+} ∈ S_{+} and x_{−} ∈ S_{−} for which δ_{x_{+}} = δ_{x_{−}} = 0, to define the invariant

ρ_{x_{+}x_{−}} = \frac{1}{2}(x_{+}, x′_{−}) mod 2 = −\frac{1}{2}(x′_{+}, x_{−}) mod 2 ∈ ℤ/2ℤ.

Thus, we get a list of genus invariants

(2.20)  (r, a; δ_{x_{+}}, δ_{x_{−}}, ρ_{x_{+}x_{−}}; δ_{φ}, δ_{φS}, s_{φ})

where they are defined. E. g. the element s_{φ} ∈ S mod 2S is defined only if δ_{φS} = 0; the invariant ρ_{x_{+}x_{−}} is defined only if δ_{x_{+}} = δ_{x_{−}} = 0. From [21], it follows that:

Two integral involutions with the condition (S, θ) have the same genus with respect to G, if and only if the invariants (2.20) are conjugate by the group G of the condition (S, θ).

To formulate conditions of existence of an integral involution with the genus invariants (2.20), we need to reformulate the invariants (2.20).

We have the function δ_{±} : S_{±} → ℤ/2ℤ where x_{±} ↦ δ_{x_{±}}. We set

H_{±} = δ_{±}^{−1}(0)/2S_{±} ⊂ S_{±}/2S_{±}.

The subgroups are equivalent to the invariants δ_{x_{±}}. We have a more exact range for the subgroups H_{±}:

Γ_{±} = (2S)_{±}/2S_{±} ⊂ H_{±} ⊂ 2(S_{±}^{∗} ∩ (\frac{1}{2}S_{±}))/2S_{±} ≅ (S_{±}^{∗} ∩ (\frac{1}{2}S_{±}))/S_{±} = A_{S_{±}}^{(2)} ⊂ A_{S_{±}}.

Here (2S)_{±} are the orthogonal projections of 2S ⊂ S_{+} ⊕ S_{−} to S_{±} respectively. This projections also give the graph Γ of the isomorphism γ of the groups Γ_{+} and Γ_{−}. The A_{S_{±}}^{(2)} denote the subgroup of A_{S_{±}} generated by all elements of order two. Let

H = H_{+} ⊕_{γ} H_{−} = (H_{+} ⊕ H_{−})/Γ.

For simplicity we identify H_{±} = H_{±} mod Γ ⊂ H. We can define a finite quadratic form q_{ρ} : H → \frac{1}{2}ℤ/2ℤ as follows

q_{ρ}|H_{+} = q_{S_{+}}|H_{+}; q_{ρ}|H_{−} = −q_{S_{−}}|H_{−}; q_{ρ}(x_{+}, x_{−}) = \frac{ρ_{x_{+}x_{−}}}{2} mod ℤ for x_{±} ∈ H_{±}.

The finite quadratic form q_{ρ} is defined correctly and is equivalent to the invariants ρ_{x_{+}x_{−}}. In H_{+} × Γ_{−} and Γ_{+} × H_{−} the invariant ρ_{x_{+}x_{−}} is defined by the discriminant quadratic forms q_{S_{+}} and q_{S_{−}} of lattices S_{+} and S_{−} respectively. We shall also denote by

ρ : H_{+} × H_{−} → ℤ/2ℤ

the pairing ρ(x_{+}, x_{−}) = ρ_{x_{+}x_{−}}. Thus q_{ρ}(x_{+}, x_{−}) = ρ(x_{+}, x_{−})/2 mod 1.

Moreover (see [21]), the characteristic element

v = s_{φ} ∈ H = H_{+} ⊕_{γ} H_{−} ⊂ (S_{+} ⊕ S_{−})/2S, if δ_{φS} = 0.

In particular, v = 0 if δ_{φ} = 0. The element v should be characteristic for the quadratic form q_{ρ} on H which means that q_{ρ}(x, v) ≡ q_{ρ}(x, x) (mod 1) for any x ∈ H. The element v is zero, if δ_{φ} = 0, and v is not zero, if δ_{φ} = 1.

Thus, the genus invariants (2.20) are equivalent to the described above genus invariants

(2.21)  (r, a; H_{+}, H_{−}, q_{ρ}; δ_{φ}, δ_{φS}, v).

Now we introduce some numerical invariants of the data (2.21) which are important for their existence.

Let a_{M} be the rank over (ℤ/2ℤ) of a 2-elementary group M. Thus, we have the numerical invariants a_{H_{±}}, a_{Γ_{±}} = a_{Γ_{+}} = a_{Γ_{−}} and a_{H} = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}}.

Any finite 2-elementary quadratic form is an orthogonal sum ⊕ of elementary forms. The elementary forms are: z on ℤ/2ℤξ with z(ξ) = 0 mod 2; w on ℤ/2ℤξ with w(ξ) = 1 mod 2; u_{+}(2) on ℤ/2ξ_{1} + ℤ/2ℤξ_{2} with u_{+}(ξ_{1}) = u_{+}(ξ_{2}) = 0 mod 2, u_{+}(ξ_{1}, ξ_{2}) = \frac{1}{2} mod 1, σ(u_{+}(2)) ≡ 0 mod 8; v_{+}(2) on ℤ/2ξ_{1} + ℤ/2ℤξ_{2} with v_{+}(ξ_{1}) = v_{+}(ξ_{2}) = 1 mod 2, v_{+}(ξ_{1}, ξ_{2}) = \frac{1}{2} mod 1; σ(v_{+}(2)) ≡ 4 mod 8; q_{α}(2), α = ±1 mod 4, on ℤξ with q_{α}(2)(ξ) = α/2 mod 2. By definition, σ(q_{α}(2)) ≡ ±1 mod 8.

Any finite 2-elementary quadratic form f with a non-degenerate bilinear form is an orthogonal sum of u_{+}(2), v_{+}(2) and q_{α}(2) and has the invariant σ mod 8 which is equal to the sum of σ for all its summands. It has an additional invariant δ. The δ = 0 if f is orthogonal sum of only u_{+}(2) and v_{+}(2) (equivalently, f takes values in ℤ). Otherwise, δ = 1. Up to isomorphisms, the f is defined by its rank k, σ mod 8 and δ ∈ {0, 1}. Then we denote f as f = q(k, δ, σ).

The quadratic form q_{ρ} on H is one of forms of the rank a_{H} = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} with the invariants δ_{H} = 0 or 1; μ_{ρ} = 0 or 1:

δ_{H} = 0 : q_{ρ} = z^{k_{ρ}−μ_{r}} ⊕ w^{μ_{ρ}} ⊕ v_{+}(2)^{σ_{ρ}/4} ⊕ u_{+}(2)^{(a_{H}−k_{ρ})/2−σ_{ρ}/4},

where μ_{ρ} = 0 or 1, σ_{ρ} ≡ 0 or 4 mod 8 and σ_{ρ} ≥ 0, and μ_{ρ} + σ_{ρ}/4 ≤ 1;

δ_{H} = 1, μ_{ρ} = 0 : q_{ρ} = z^{k_{ρ}} ⊕ q(a_{H} − k_{ρ}, 1, σ_{ρ}),
δ_{H} = 1, μ_{ρ} = 1 : q_{ρ} = z^{k_{ρ}−1} ⊕ w ⊕ q_{1}(2)^{a_{H}−k_{ρ}}

where a_{H} > k_{ρ}.

The invariant δ_{H} = 0 iff q_{ρ}(x) ∈ ℤ for any x ∈ H. We similarly introduce the invariants δ_{H_{±}} for the form q_{ρ}|H_{±}. We have

δ_{H} = max{δ_{H_{+}}, δ_{H_{−}}}.

If δ_{φS} = 0 and v = s_{φ} ∈ H is the characteristic element of φ, we get the invariant

c_{v} mod 4 where c_{v}/2 = q_{ρ}(v) mod 2.

Thus, we get invariants

(2.22)  (a_{H_{±}}, a_{Γ_{±}}, δ_{H_{±}}; k_{ρ}, μ_{ρ}, σ_{ρ}, c_{v})

of the data (2.21).

We need some more invariants of data (2.21). An element ṽ ∈ A_{S_{±}}^{(2)} is called characteristic if q_{S_{±}}(x) ≡ q_{S_{±}}(ṽ, x) mod 1 for any x ∈ A_{S_{±}}^{(2)}. We remind that A_{S_{±}}^{(2)} is generated by all elements of order 2 in A_{S_{±}}.

We say that δ_{φS_{±}} = 0 if δ_{φS} = 0, the element v = s_{φ} ∈ H_{±} and v is equal to a characteristic element of A_{S_{±}}^{(2)}. Otherwise δ_{φS_{±}} = 1.

Assume that δ_{φS_{±}} = 0. We consider a non-degenerate finite quadratic form

γ_{±} = \{ \begin{matrix}0, & if δ_{φ} = 0, equivalently v = 0, \\ q_{1}(2) ⊕ q_{−1}(2), & if δ_{φ} = 1 and c_{v} ≡ 0 mod 4, \\ q_{1}(2)^{2}, & if c_{v} ≡ 2 mod 4, \\ q_{±1}(2), & if c_{v} ≡ ±1 mod 4. \end{matrix}

Let v_{γ_{±}} be the characteristic element of γ_{±} (it is unique). We have γ_{±}(v_{γ_{±}}) = c_{v}/2 mod 2. We denote (q_{S_{±}})_{v} = (v ⊕ v_{γ_{±}})_{q_{S_{±}}⊕(∓γ_{±})}^{⊥}/[v ⊕ v_{γ_{±}}]. There exists a unique even 2-adic lattice K((q_{S_{±}})_{v}) having the discriminant quadratic form (q_{S_{±}})_{v} and the same rank as the form (q_{S_{±}})_{v} (i. e. the minimal number of generators of the 2-group where the form is defined). We denote by A_{q} the group where a finite form q is defined and by |A_{q}| its order. The invariant

ε_{v_{±}} ∈ (ℤ/2ℤ)

is defined by

5^{ε_{v_{±}}} ≡ \frac{det K((q_{S_{±}})_{v})}{|A_{S_{±}}| · |A_{γ_{±}}|} (mod ± ℚ_{2}^{∗}^{2})

where ℚ_{p} is the field of p-adic integers.

Finally we get invariants

(2.23)  (δ_{φS_{+}}, δ_{φS_{−}}, ε_{v_{+}}, ε_{v_{−}})

By our construction, the finite quadratic form q_{ρ} on H has an embedding to the discriminant quadratic form q_{L_{+}} which is 2-elementary, and the characteristic element v (if δ_{φS} = 0) goes to the characteristic element of q_{L_{+}}. The following conditions from [21] are sufficient and necessary for its existence (we numerate these condition as in [21]). They use invariants (2.22) and invariants r, a, δ_{φS}, and δ_{φ}.

CONDITIONS 1.8.1 (from [21]). Type 0 (δ_{φ} = δ_{φS} = 0):

Conditions on H_{±}. δ_{H_{+}} = δ_{H_{−}} = 0. Inequality. a ≥ a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ}. Congruences. 1) a + r ≡ 0 (mod 2); 2) 2 − r ≡ 0 (mod 4). Boundary condition. If (a = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ}, μ_{ρ} = 0), then 2 − r ≡ σ_{ρ} (mod 8). Type Ia (δ_{φ} = 1, δ_{φS} = 0): Conditions on H, ρ and c_{v}. 1) c_{v} ≡ a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} − k_{ρ} (mod 2). 2) If μ_{ρ} = 0, then c_{v} ≡ σ_{ρ} (mod 4). 3) a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} > 0. 4) If δ_{H_{+}} = δ_{H_{−}} = 0, then k_{ρ} ≥ 1. 5) If (δ_{H_{+}} = δ_{H_{−}} = 0, k_{ρ} = μ_{ρ} = 1), then c_{v} ≡ 2 (mod 4). Inequality. a ≥ a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ}. Congruences. 1) a + r ≡ 0 (mod 2); 2) 2 − r ≡ c_{v} (mod 4). Boundary condition. If (a = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ}, μ_{ρ} = 0), then 2 − r ≡ σ_{ρ} (mod 8).

Type Ib (δ_{φ} = 1, δ_{φS} = 1): Inequality. a ≥ a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ} + 1. Congruences. a + r ≡ 0 (mod 2). Boundary condition. 1) If (a = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ} + 1, μ_{ρ} = 0), then 2 − r ≡ σ_{ρ} ± 1 (mod 8). 2) If (a = a_{H_{+}} + a_{H_{−}} − a_{Γ_{±}} + k_{ρ} + 2, μ_{ρ} = 0), then 2 − r ≢ σ_{ρ} + 4 (mod 8).

The following conditions from [21] are necessary and sufficient for existence of the lattices K_{±} = (S_{±})_{L_{±}}^{⊥} with the local invariants prescribed by the data (2.20) or (2.21). They also use the invariants (2.13) and (2.23). The l(A) denotes the minimal number of generators of a finite Abelian group A. We numerate the conditions as in [21] and drop unnecessary (for our case) conditions.

CONDITION 1.8.2 (from [21]). Conditions on the invariants δ_{φS_{±}}. If δ_{φS} = 1, then δ_{φS_{+}} = δ_{φS_{−}} = 1. Inequalities. 1) p + 1 ≤ r ≤ 21 − s + p; 2) r − a ≥ −2a_{H_{+}} + p + l(A_{S_{+}}); 3) r + a ≤ 2a_{H_{−}} + p − s − l(A_{S_{−}}) + 22.

Boundary conditions. 1) If δ_{φS_{+}} = 0 and r − a = −2a_{H_{+}} + p + l(A_{S_{+}}), then 2 − r ≡ 4ε_{v_{+}} + c_{v} (mod 8).

2) If δ_{φS_{−}} = 0 and r + a = 2a_{H_{−}} + p − s − l(A_{S_{−}}) + 22, then 2 − r ≡ 4ε_{v_{−}} + c_{v} (mod 8).

By [21] both these conditions are necessary and sufficient for existence of the integral involution. Thus, finally we get

Theorem 10. ([21]). The invariants (2.20) or (2.21) give the complete genus invariant of integral involutions (L, φ, S) of the type (S, θ) from the set In(S, θ, G). The conditions 1.8.1 and 1.8.2 above are necessary and sufficient for existence of the involution from In(S, θ, G) with the genus invariants.

Obviously, the genus of the integral involution (L, φ, S) defines the genus of the related involution (L, φ̃ = τφ, S). Thus, invariants (2.20) or (2.21) of (L, φ, S) define similar invariants of the related involution. We denote by

(2.24)  (r(φ), a(φ); H(φ)_{+}, H(φ)_{−}, q_{ρ(φ)}; δ_{φ}, δ_{φS}, v(φ))

and

(2.25)  (r(τφ), a(τφ); H(τφ)_{+}, H(τφ)_{−}, q_{ρ(τφ)}; δ_{τφ}, δ_{τφS}, v(τφ))

the genus invariants (2.21) for φ and φ̃ = τφ. Similarly we mark by φ or τφ the equivalent genus invariants (2.20).

Theorem 11. We have the following relations between genus invariants of the related involutions (L, φ, S) and (L, φ̃ = τφ, S):

(2.26)  r(φ) + r(τφ) = 22 − s + 2p,

(2.27)  a(τφ) − a(φ) = a(S) − 2a_{H(φ)} + 2 rk ρ(φ),

(2.28)  s_{φ} + s_{τφ} ≡ s_{θ} ∈ S mod 2L,

where rk ρ means the rank of the matrix which gives ρ in some bases of H_{+} and H_{−} over the field ℤ/2ℤ, and s_{θ} ∈ S is the characteristic element of (S, θ) defined by the property 2(x, θ(x)) ≡ (x, s_{θ}) mod 2 for any x ∈ S^{∗} (the s_{θ} is defined mod 2S). In particular,

(2.29)  δ_{φS} = δ_{(τφ)S}.

The H(φ)_{±} and H(τφ)_{±} are orthogonal with respect to the discriminant bilinear form b_{S_{±}} on A_{S_{±}} respectively.

Moreover,

(2.30)  l(A_{S_{+}}) + a(φ) − 2a_{H(φ)_{+}} = l(A_{S_{−}}) + a(τφ) − 2a_{H(τφ)_{−}}.

Proof. We have the orthogonal decomposition up to a finite index

(2.31)  S_{+} ⊕ S_{−} ⊕ L_{τ}^{φ} ⊕ L_{τ,φ} ⊂ L.

We have r(τφ) = rk S_{+} + rk L_{τ,φ} = rk S_{+} + (rk L − rk L^{φ} − rk S_{−}) = p + 22 − r(φ) − (s − p). It follows the first relation (2.26).

The proof of second relation (2.27) follows the proof of Lemma 3.3.2 in [23]. There was shown that

2^{a(τφ)} = #A_{L_{φ}^{τ}} · #A_{L^{τ,φ}} · #A_{L_{φ}}/(2^{2 rk H(φ)_{+}} · 2^{2 rk (H(φ)_{+}^{⊥}∩H(φ)_{−})})

where H(φ)_{+}^{⊥} ⊂ H(φ)_{−} is the orthogonal complement to H(φ)_{+} with respect to the pairing ρ(φ). We have rk (H(φ)_{+}^{⊥} ∩ H(φ)_{−}) = rk H(φ)_{−} − rk ρ(φ). Moreover, #A_{L_{φ}} = 2^{a(φ)}, #A_{L_{φ}^{τ}} · #A_{L^{τ,φ}} = #A_{L^{τ}} · 2^{rk Γ_{±}} = 2^{a(S)+2 rk Γ_{±}}. Thus, we get

a(τφ) = a(φ) + a(S) + 2 rk Γ_{±} − 2 rk H(φ)_{+} − 2 rk H(φ)_{−} + 2 rk ρ(φ).

It follows (2.27).

We can write up x ∈ L as x = s_{+}^{∗} ⊕ s_{−}^{∗} ⊕ x_{τ}^{φ}^{∗} ⊕ x_{τ,φ}^{∗} where s_{+}^{∗} ∈ S_{+}^{∗}, s_{−}^{∗} ∈ S_{−}^{∗}, x_{τ}^{φ}^{∗} ∈ L_{τ}^{φ}^{∗}, x_{τ,φ}^{∗} ∈ L_{τ,φ}^{∗}. We have

(x, φ(x)) = (s_{+}^{∗})^{2} − (s_{−}^{∗})^{2} + (x_{τ}^{φ}^{∗})^{2} − (x_{τ,φ}^{∗})^{2} ≡ (x, s_{φ}) mod 2;
(x, τφ(x)) = (s_{+}^{∗})^{2} − (s_{−}^{∗})^{2} − (x_{τ}^{φ}^{∗})^{2} + (x_{τ,φ}^{∗})^{2} ≡ (x, s_{τφ}) mod 2.

Taking sum, we get

(x, s_{φ}) + (x, s_{τφ}) ≡ 2 ((s_{+}^{∗})^{2} − (s_{−}^{∗})^{2}) ≡ (p_{S}(x), s_{θ}) mod 2.

where p : L → S^{∗} the orthogonal projection. Here p(L) = S^{∗} since L is unimodular. Since L is unimodular, it follows that s_{φ} + s_{τφ} ≡ s_{θ} mod 2L. It proves the third relation (2.28).

Let u ∈ S_{−} has the invariant δ_{u} = 0 for φ and v ∈ S_{−} has the invariant δ_{v} = 0 for τφ. Then (see [21], (7.6)) there exist u_{1} ∈ L^{φ} and v_{1} ∈ L^{τφ} = L_{τ,φ} such that u/2 + u_{1}/2 ∈ L and v/2 + v_{1}/2 ∈ L. We then have (u/2, v/2) = (u/2 + u_{1}/2, v/2 + v_{1}/2) ∈ ℤ. Thus u/2 ∈ A_{S_{−}} and v/2 ∈ A_{S_{−}} are orthogonal with respect to the discriminant bilinear form b_{S_{−}} on A_{S_{−}}. It proves that the groups H(φ)_{−} and H(τφ)_{−} are orthogonal to each other. The proof for H_{+} is similar.

By the relation (8.9) from [21], we have

l(A_{L_{τ}^{φ}}) = l(A_{S_{+}}) + a(φ) − 2a_{H(φ)_{+}};
l(A_{L_{τ,φ}}) = l(A_{S_{−}}) + a(φ) − 2a_{H(φ)_{−}}.

Similar relation for τφ gives

l(A_{L_{τ}^{φ}}) = l(A_{S_{−}}) + a(τφ) − 2a_{H(τφ)_{−}};
l(A_{L_{τ,φ}}) = l(A_{S_{+}}) + a(τφ) − 2a_{H(τφ)_{+}}.

It follows the last relation (2.30) of the theorem. □

Theorem 11 permits to choose one from two related involutions by some conditions on its invariants. It helps to classify the pairs of related involutions which is important for a classification of the corresponding DPN-pairs.

One should follow [21] to give the exact construction of the genus invariants (2.25) of the related involution τφ. The idea is as follows. Using construction of [21], we can calculate the genus of the integral involution from the invariants (2.24) of φ. From the genus, we can find the invariants (2.25) of τφ. We don’t know a simpler procedure in general. Below we give this construction in details.

Let (A_{q}, q) = (A_{q(φ)}, q(φ)) be the discriminant quadratic form of the lattice L^{φ}. It is defined by the invariants (r(φ), a(φ), δ(φ)). Let ξ^{(+)} : H = H_{+}⊕_{γ} H_{−} → A_{q} be an embedding of (H, q_{ρ}) to (A_{q}, q) such that ξ^{(+)}(v(φ)) is the characteristic element of q, if δ_{φS} = 0, and ξ^{(+)}(H) does not contain a characteristic element of q if δ_{φS} ≠ 0. Such an embedding is defined uniquely up to automorphisms of q.

We denote by (A_{−q}, −q) the form (A_{q}, q) with the the opposite sign and by Δ_{q} ⊂ A_{q} ⊕A_{−q} the corresponding isotropic diagonal for q ⊕(−q). The ξ^{(+)} gives ξ^{(−)} : (H, −q_{ρ}) → (A_{−q}, −q) where ξ^{(−)} = ξ^{(+)} on the groups. Let Γ(H_{+}) ⊂ A_{S_{+}} ⊕ A_{−q} be the graph of ξ^{(−)}|H_{+}. We then get the non-degenerate quadratic form

(2.32)  (A_{k_{+}} = Γ(H_{+})_{q_{S_{+}}⊕(−q)}^{⊥}/Γ(H_{+}), k_{+} = −q_{S_{+}} ⊕ q|A_{k_{+}})

with the corresponding projection π_{k_{+}} : Γ(H_{+})_{q_{S_{+}}⊕(−q)}^{⊥} → A_{k_{+}} and the graph Γ_{π_{k_{+}}} ⊂ A_{S_{+}} ⊕ A_{−q} ⊕ A_{k_{+}}. The form k_{+} can be identified with the discriminant form of the lattice L_{τ}^{φ}. As a result, we get the quadratic form

(A_{S_{+}} ⊕ A_{−q} ⊕ A_{k_{+}}, q_{S_{+}} ⊕ A_{−q} ⊕ k_{+})

with the isotropic subgroup Γ_{(S_{+},−q,k_{+})} = Γ(H_{+}) ⊕ {0} + Γ_{π_{k_{+}}}. Further we identify A_{S_{+}}, A_{−q}, A_{k_{+}} with the corresponding subgroups in A_{S_{+}} ⊕ A_{−q} ⊕ A_{k_{+}}. By the construction, Γ_{(S_{+},−q,k_{+})}^{⊥} = Γ_{(S_{+},−q,k_{+})} is the maximal isotropic subgroup. Moreover, Γ_{(S_{+},−q,k_{+})} ∩ A_{S_{+}} = Γ_{(S_{+},−q,k_{+})} ∩ A_{−q} = Γ_{(S_{+},−q,k_{+})} ∩ A_{k_{+}} = {0}, Γ_{(S_{+},−q)} = Γ_{(S_{+},−q,k_{+})} ∩ (A_{S_{+}} ⊕ A_{−q}) = Γ(H_{+}). Let Γ_{(S_{+},k_{+})} = Γ_{(S_{+},−q,k_{+})} ∩ (A_{S_{+}} ⊕ A_{k_{+}}). By our construction, we then have the natural identification

(2.33)  (A_{q} = (Γ_{(S_{+},k_{+})})_{q_{S_{+}}⊕k_{+}}^{⊥}/Γ_{(S_{+},k_{+})}, q = q_{S_{+}} ⊕ k_{+}|A_{q})

where Γ_{(S_{+},−q,k_{+})}/Γ_{(S_{+},k_{+})} gives the graph of Δ_{q} ⊂ A_{q} ⊕ A_{−q} above. Let π_{q} : (Γ_{(S_{+},k_{+})})_{q_{S_{+}}⊕k_{+}}^{⊥} → A_{q} be the corresponding projection. Let Γ(H_{−}) ⊂ A_{q} ⊕ A_{S_{−}} be the

graph of ξ^{(+)}|H_{−} and Γ_{(S_{+},S_{−},k_{+})} = { \atop Γ(H_{−})} = (π_{q} ⊕ id(A_{S_{−}}))^{−1}(Γ(H_{−})) ⊂ A_{S_{+}} ⊕ A_{S_{−}} ⊕ A_{k_{+}}.

Let (2.34)

(A_{k_{−}} = (Γ_{(S_{+},S_{−},k_{+})})_{q_{S_{+}}⊕q_{S_{−}}⊕q_{k_{+}}}^{⊥}/Γ_{(S_{+},S_{−},k_{+})}, k_{−} = (−q_{S_{+}}) ⊕ (−q_{S_{−}}) ⊕ (−q_{k_{+}})|A_{k_{−}}) .

The form k_{−} can be identified with the discriminant form of the lattice L_{τ,φ}. We remark that one gets the same form as

(2.35)  (A_{k_{−}} = Γ(H_{−})_{q_{S_{−}}⊕q}^{⊥}/Γ(H_{−}), k_{−} = −q_{S_{−}} ⊕ (−q)|A_{k_{−}}).

Thus, finally we get the form

(2.36)  (A_{S_{+}} ⊕ A_{S_{−}} ⊕ A_{k_{+}} ⊕ A_{k_{−}}, q_{S_{+}} ⊕ q_{S_{−}} ⊕ k_{+} ⊕ k_{−})

which corresponds to the orthogonal decomposition (2.31). Further we use this form. Let

(2.37)  Γ = Γ_{(S_{+},S_{−},k_{+},k_{−})} ⊂ A_{S_{+}} ⊕ A_{S_{−}} ⊕ A_{k_{+}} ⊕ A_{k_{−}}

be the graph of the projection π_{k_{−}} : Γ_{(S_{+},S_{−},k_{+})})_{q_{S_{+}}⊕q_{S_{−}}⊕q_{k_{+}}}^{⊥} → A_{k_{−}}. By our construction, the group Γ ⊂ A_{S_{+}} ⊕ A_{S_{−}} ⊕ A_{k_{+}} ⊕ A_{k_{−}} is the maximal isotropic subgroup: Γ^{⊥} = Γ. We have

Γ = L/(S_{+} ⊕ S_{−} ⊕ L_{τ}^{φ} ⊕ L_{τ,φ})

for the orthogonal decomposition (2.31). The equality Γ^{⊥} = Γ is equivalent for the lattice L to be unimodular.

The orthogonal decomposition (2.36) and the isotropic subgroup Γ (together with the corresponding real invariants) are equivalent to the genus of the integral involution. Thus, we have calculated the genus from the invariants (2.24) of φ. Using this genus, now we can calculate the invariants (2.25) of τφ. For φ̃ = τφ, the lattices L_{τ}^{φ} and L_{τ,φ} change places:

Lφ̃ ; τ = L_{τ,φ} and L_{τ,φ̃} = L_{τ}^{φ}.

Let α_{i} ∈ {S_{+}, S_{−}, k_{+}, k_{−}}. We denote Γ_{(α_{1},...,α_{n})} = Γ ∩ (A_{α_{1}} ⊕ ⋯ ⊕ A_{α_{n}}) and by p_{(α_{1},...,α_{n})} the corresponding orthogonal projection into the group A_{α_{1}} ⊕ ⋯ ⊕ A_{α_{n}}.

By definition of the invariants (2.24) and (2.25), we have:

(2.38)  H(φ)_{+} = p_{S_{+}}(Γ_{(S_{+},S_{−},k_{−})})

with the natural embedding H(φ)_{+} ⊂ A_{q} (use (2.33)) since H(φ)_{+} ⊥ Γ_{(S_{+},k_{+})}. Similarly, we have

(2.39)  H(φ)_{−} = p_{S_{−}}(Γ_{(S_{+},S_{−},k_{+})})

with the natural embedding H(φ)_{−} ⊂ A_{q} defined by the graph Γ_{(S_{+},S_{−},k_{+})}/Γ_{(S_{+},k_{+})}. The form q on A_{q} defines then the pairing ρ(φ) between H(φ)_{+} and H(φ)_{−}.

For τφ we similarly get:

(2.40)  H(τφ)_{+} = p_{S_{+}}(Γ_{(S_{+},S_{−},k_{+})})

with the natural embedding into

(2.41)  (A_{q(τφ)} = (Γ_{(S_{+},k_{−})})_{q_{S_{+}}⊕k_{−}}^{⊥}/Γ_{(S_{+},k_{−})}, q(τφ) = q_{S_{+}} ⊕ k_{−}|A_{q(τφ)});
(2.42)  H(τφ)_{−} = p_{S_{−}}(Γ_{(S_{+},S_{−},k_{−})})

with the natural embedding into A_{q(τφ)} defined by the graph Γ_{(S_{+},S_{−},k_{−})}/Γ_{(S_{+},k_{−})}. The form q(τφ) defines then the pairing ρ(τφ) in between H(τφ)_{+}, H(τφ)_{−}.

This calculates the genus invariants (2.25). For these calculations, the trivial (from the construction above) and crucial observation is:

Theorem 12. Assume that q = q(φ) = q_{1} ⊕ q′ and ξ^{(+)} gives an embedding of q_{ρ} into q_{1} ⊕ {0}. Then the calculations above of the genus invariants H(τφ)_{+}, H(τφ)_{−} and ρ(τφ) give the same result if one replaces q = q(φ) by q_{1}.

The Theorem 12 reduces calculations of the invariants H(τφ)_{±} and ρ(τφ) to some finite and simple calculations.

There is a case when these calculations are very simple

Corollary 13. Assume that S_{+} = 0, S_{−} = S. Then H(φ)_{+} = H(τφ)_{+} = 0, ρ(φ) = ρ(τφ) = 0 and H(φ)_{+} ⊥ H(τφ)_{+} are orthogonal complements to each other with respect to q_{S} = q_{S_{−}}.

In many cases the genus invariants (2.21) have a unique isomorphism class of integral involutions. By Theorems 1, 6 and 7, then the genus invariants define the connected component of moduli of real K3 surfaces with a non-symplectic involution or the connected component of moduli of the corresponding real right DPN-pairs. We shall use the invariant κ(A) = m of a finite Abelian group A = (ℤ/2)^{m} ⊕ (ℤ/4ℤ)^{n}. Using results of [21] and [18], [19], we get the following statement.

Theorem 14. The genus invariants (2.21) have a unique isomorphism class from In(S, θ, G) of integral involutions (L, φ, S) of the type (S, θ) satisfying (RSK3) (in particular, by Theorems 1, 6 and 7, the moduli of real K3 surfaces with a non-symplectic involution, the moduli of the corresponding positive real right DPN-pairs, the moduli of the corresponding real right DPN-pairs having these invariants are connected), if both conditions (a_{±}) below are satisfied:

(a_{+}) an even lattices K_{+} with the discriminant quadratic form k_{+} (see (2.32)) and the signature (1, r − p − 1) is unique up to isomorphisms, and the canonical homomorphism O(K_{+}) → O(k_{+}) is epimorphic;

(a_{−}) an even lattice K_{−} with the discriminant quadratic form k_{−} (see (2.35)) and the signature (1, 21 − r − s + p) is unique up to isomorphisms, and the canonical homomorphism O(K_{−}) → O(k_{−}) is epimorphic.

The condition (a_{±}) is valid if the corresponding condition (b_{±}) below satisfies:

(b_{+}) either r − a > −2a_{H_{+}} + p + l(A_{S_{+}}), or (r − a = −2a_{H_{+}} + p + l(A_{S_{+}}), a ≥ 2a_{H_{+}} − κ(A_{S_{+}}) + 3), or (r − a = −2a_{H_{+}} + p + l(A_{S_{+}}), a = 2a_{H_{+}} − κ(A_{S_{+}}) + 2, δ_{φS_{+}} = 0);

(b_{−}) either r + a < 2a_{H_{−}} + p − s − l(A_{S_{−}}) + 22, or (r + a = 2a_{H_{−}} + p − s − l(A_{S_{−}}) + 22, a ≥ 2a_{H_{−}} − κ(A_{S_{−}}) + 3), or (r + a = 2a_{H_{−}} + p − s − l(A_{S_{−}}) + 22, a = 2a_{H_{−}} − κ(A_{S_{−}}) + 2, δ_{φS_{−}} = 0).

Proof.

The first statements follows from Theorem 1.3.1 in [21] (see also Remark 1.6.2 there). The lattice K_{+} is indefinite and even, its discriminant form k_{+} is a 2-form. Moreover, the

discriminant group A_{k_{+}} ≅ (ℤ/2ℤ)^{m} ⊕ (ℤ/4ℤ)^{n} since the lattice S is 2-elementary and the discriminant groups A_{S_{±}} are annulated by 4. Applying Theorem 1.14.2 from [19] to this case, we see that (a_{+}) is valid if either rk K_{+} > l(A_{k_{+}}) or rk K_{+} = l(A_{k_{+}}), but the discriminant form k_{+} ≅ u_{+}(2) ⊕ k_{1}, v_{+}(2) ⊕ k_{1}.

The relation (8.9) in [21] gives that

l(A_{k_{+}}) = l(A_{S_{+}}) + a − 2a_{H_{+}}.

Thus, the condition rk K_{+} > l(A_{k_{+}}) means r − p > l(A_{S_{+}}) + a − 2a_{H_{+}}. Equivalently, r − a > −2a_{H_{+}} + p + l(A_{S_{+}}).

We have A_{k_{+}} ≅ (ℤ/2ℤ)^{m} ⊕ (ℤ/4ℤ)^{n}. Let us calculate m. Assume A_{S_{+}} ≅ (ℤ/2ℤ)^{α} ⊕ (ℤ/4ℤ)^{β} where α = κ(A_{S_{+}}). By (2.32), #A_{k_{+}} = 2^{m+2n} = #A_{S_{+}} · #A_{q}/(#H_{+})^{2} = 2^{α+2β+a−2a_{H_{+}}}. It follows, m + 2n = α + 2β + a − 2a_{H_{+}}. Moreover, the relation l(A_{k_{+}}) = l(A_{S_{+}}) + a − 2a_{H_{+}} used above gives m + n = l(A_{k_{+}}) = l(A_{S_{+}}) + a − 2a_{H_{+}} = α + β + a − 2a_{H_{+}}. It follows, m = α + a − 2a_{H_{+}} = κ(A_{S_{+}}) + a − 2a_{H_{+}}.

From classification of finite quadratic forms, we get that k_{+} ≅ u_{+}(2) ⊕ k_{1} or v_{+}(2) ⊕ k_{2}, if m ≥ 3. Moreover, it is shown in ([21], page 116), that k_{+} ≅ u_{+}(2) ⊕ q_{1}, v_{+}(2) ⊕ q_{2}, if m = 2 and δ_{φS_{+}} = 0. Thus, cited above conditions of Theorem 1.14.2 from [19] are equivalent to (b_{+}). Thus, (b_{+}) implies (a_{+}).

The proof of (b_{−}) implies (a_{−}) is the same.

We mention the useful formulae which follow from the proof:

(2.43)  A_{k_{±}} ≅ (ℤ/2ℤ)^{m_{±}+a−2a_{H_{±}}} ⊕ (ℤ/4ℤ)^{n_{±}}, if A_{S_{±}} ≅ (ℤ/2ℤ)^{m_{±}} ⊕ (ℤ/4ℤ)^{n_{±}}.

We also mention the following useful statement which follows from (Theorem 3.6.3 in [19]):

Proposition 15. The condition (a_{±}) of Theorem 14 is valid, if A_{k_{±}} is 2-elementary. In particular, this is true (by (2.43)), if the lattice S_{±} is 2-elementary.

At last, we mention the main geometric interpretation of the invariants (r(φ), a(φ), δ(φ)) = (r, a, δ). We denote by S_{g} an oriented surface of the genus g.

We have (see Theorem 3.10.6 in [19]) for the real part of X_{φ}(ℝ) = X^{φ} of X with the real structure defined by φ the same result as for the holomorphic non-symplectic involution τ:

(2.44)  X_{φ}(ℝ) = \begin{cases}∅, & if (r, a, δ) = (10, 10, 0) \\ T_{1} ⨿ T_{1}, & if (r, a, δ) = (10, 8, 0) \\ T_{g} ⨿ (T_{0})^{k}, & otherwise, where \\ & g = (22 − r − a)/2, k = (r − a)/2 \end{cases} ;

and

(2.45)  X_{φ}(ℝ) ∼ s_{φ} mod 2 in H_{2}(X, ℤ).

We have the same formula for X_{τφ}(ℝ) using invariants (r(τφ), a(τφ), δ(τφ)) of τφ.

2.4. Possible types of (S, θ) with rk S ≤ 2. By classification in [19] and also in [20], [22] (see the beginning of Sect. 2.3 and Figure 1), there are the following and only the following possibilities for S with r(S) = rk S ≤ 2. We have

(r(S), a(S), δ(S)) = (1, 1, 1), (2, 2, 0), (2, 2, 1), (2, 0, 0).

We order these cases according to the natural ordering of the corresponding quotients Y = X/{1, τ}, see below.

We consider all possible θ for these cases.

The case: (r(S), a(S), δ(S)) = (1, 1, 1). Then S ≅ ⟨2⟩. Since S_{−} should be hyperbolic, S_{−} = S and θ = −1 on S. Then S_{+} = {0}. We consider this case in Sect. 3 below. For this case Y = X/{1, τ} = ℙ^{2}.

The case (r(S), a(S), δ(S)) = (2, 2, 0). Then S ≅ U(2). We consider this case in Sects. 4 and 5. By Proposition 4, we have Y = ℙ^{1} × ℙ^{1} in the non-degenerate case, which we are considering in this paper (in the degenerate case Y = 𝔽_{2}). Let us consider possible θ. Dividing form of S by 2, we get the unimodular lattice U. It follows that if rk S_{−} = 1, then S_{−} ≅ ⟨4⟩ and S_{+} ≅ ⟨−4⟩. If rk S_{−} = 2, then θ = −1 on S. Thus, we get two cases

The case S ≅ U(2), the involution θ is −1 on S. We consider this case in Sect. 4. Then Y = X/{1, τ} = ℙ^{1} × ℙ^{1} in the non-degenerate case which we consider. Moreover, Y = X/{1, τ} = ℙ^{1} × ℙ^{1} over ℝ is a hyperboloid, if Y (ℝ) ≠ ∅.

The case S ≅ U(2), S_{−} ≅ ⟨4⟩, S_{+} ≅ ⟨−4⟩. We consider this case in Sect. 5. For this case, Y = X/{1, τ} = ℙ^{1} × ℙ^{1} and Y over ℝ is an ellipsoid. Really, if Y = 𝔽_{2}, then any anti-holomorphic involution of Y acts as −1 in H^{2}(Y, ℝ) = ℝ^{2}, and then θ = −1 in S ⊗ ℝ.

The case (r(S), a(S), δ(S)) = (2, 2, 1). Then S ≅ ⟨2⟩ ⊕ ⟨−2⟩. Assume that rk S_{−} = 1. Since the lattice S(2^{−1}) ≅ ⟨1⟩ ⊕ ⟨−1⟩ is unimodular and odd, it follows that S_{−} = ⟨2⟩ and S_{+} ≅ ⟨−2⟩. Again the lattice S_{+} has elements with square −2 which is impossible for θ. Thus, θ = −1 on S and S_{+} = {0}. We consider this case in Sect. 6. For this case Y = X/{1, τ} = 𝔽_{1}.

The case (r(S), a(S), δ(S)) = (2, 0, 0). Then S ≅ U where U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Since S_{−} is

hyperbolic, rk S_{−} = 1 or 2. Let rk S_{−} = 1. Since S is unimodular and even, S_{−} ≅ ⟨2⟩ and S_{+} ≅ ⟨−2⟩. Then S_{+} has elements with square −2 which is impossible for θ. Thus, rk S_{−} = 2 and θ = −1 on S. Then S_{−} = S and S_{+} = {0}. We consider this case in Sect. 7 below. For this case, Y = X/{1, τ} = 𝔽_{4}.

3. Connected components of moduli of real non-singular curves of degree 6 in ℙ^{2}

Here we consider the case (r(S), a(S), δ(S)) = (1, 1, 1). Then S ≅ ⟨2⟩. The involution θ = −1 on S. Since S has no elements with square −4, the group G is trivial.

For this case, Y = X/{1, τ} is a rational surface with Picard number rk S = 1. It follows that Y ≅ ℙ^{2}. Then | − 2K_{Y} | consists of curves of degree 6. Thus, X is a double covering of ℙ^{2} ramified in a non-singular curve of degree 6 and τ is the involution of the double covering. The lattice S = ℤh where h is the preimage of a line in ℙ^{2}, evidently h^{2} = 2.

The ℙ^{2} has only one real structure which is defined by a homogeneous real coordinates (x_{0} : x_{1} : x_{2}) in ℙ^{2}. Then ℙ^{2}(ℝ) = ℝℙ^{2} is the real projective plane. A curve A is the zero set P = 0 of a real homogeneous polynomial P (x_{0}, x_{1}, x_{2}) of degree 6. A positive curve A^{+} is equivalent to a half P ≥ 0 of the ℝℙ^{2}. It defines the real double covering (X, τ, φ) of ℙ^{2} ramified along A by the condition that A^{+} is the image of the real part X_{φ}(ℝ). A curve A is defined by the polynomial P up to λP where λ ∈ ℝ^{∗}. A positive curve A^{+} is defined by the polynomial P up to λP where λ ∈ ℝ_{++}. Here ℝ_{++} is the set of positive real numbers.

Classification of non-singular A and A^{+} of degree 6 is well-known. The isotopy classification was obtained by Gudkov [6]. Classification of connected components of moduli was obtained in [19]. See also Arnol’d [3] and Rokhlin [25] for the geometric interpretation of invariants. The last classification (for positive curves) is equivalent to description of connected components of

((ℝ^{28} − Discr) /ℝ_{++}) /P GL(3, ℝ)

where ℝ^{28} is the space of real homogeneous polynomials P (x_{0}, x_{1}, x_{2}) of degree 6, and Discr is the subspace of singular polynomials, defining singular curves P = 0. Since both groups ℝ_{++} and P GL(3, ℝ) are connected, this classification is equivalent to the classification of connected components of ℝ^{28} − Discr which is called the rigid isotopy classification due to V.A. Rokhlin [25].

For completeness, we outline general arguments of Sect. 2, in this case.

We have S_{+} = 0, S_{−} = S, H_{+} = 0, ρ = 0. The invariants s = 1, p = 0, l(A_{S_{+}}) = 0, l(A_{S_{−}}) = 1. The group H = H_{−} ⊂ ℤh/2ℤh. We get two possibilities for the group H.

(1) H = 0 (equivalently, δ_{h} = 1). Then a_{H} = 0, δ_{H} = 0, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φS} = 0, then v = 0 (equivalently, δ_{φ} = 0), c_{v} ≡ 0 (mod 4), δ_{φS_{+}} = 0, δ_{φS_{−}} = 1, ε_{v_{+}} ≡ 0 (mod 2).

(2) H = [h] = {0, h} mod 2S (equivalently, δ_{h} = 0). Then a_{H} = 1, δ_{H} = 1, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ −1 (mod 8). If δ_{φS} = 0, then v = h mod 2S (equivalently, δ_{φh} = 0, see [19]), c_{v} ≡ −1 (mod 4), δ_{φS_{+}} = 1, δ_{φS_{−}} = 0, ε_{v_{−}} ≡ 0 (mod 2).

In particular, from these relations, a_{H_{+}} = 0, a_{H} = a_{H_{−}} = 1 − δ_{h}.

Since S_{±} are 2-elementary, by Proposition 15, Theorem 14 and results of Sect. 1, we get that connected components of moduli of (X, τ, φ) (or A^{+}) are in one-to-one correspondence with the genus invariants

(3.1)  (r, a, δ_{h}, δ_{φS}, v) = (r(φ), a(φ), δ(φ)_{h}, δ_{φS}, v(φ)).

The complete list of the invariants (where r, a(≥ 0) ∈ ℤ, δ_{h}, δ_{φS} ∈ {0, 1}, the v is defined only if δ_{φS} = 0) is given by the conditions:

Inequalities and general relations: 1 ≤ r ≤ 20; 1 − δ_{h} + δ_{φS} ≤ a; a ≤ r; r + a ≤ 22 − 2δ_{h}. r + a ≡ 0 (mod 2); if δ_{h} = 1 and δ_{φS} = 0, then v = 0 (mod 2S) and 2 − r ≡ 0 (mod 4); if δ_{h} = 0 and δ_{φS} = 0, then v = h (mod 2S) and 2 − r ≡ −1 (mod 4). Boundary conditions: If a = 1 − δ_{h} and v = 0 (mod 2S), then 2 − r ≡ 0 (mod 8); If a = 1 − δ_{h} and v = h (mod 2S), then 2 − r ≡ −1 (mod 8); If a = 2 − δ_{h} and δ_{φS} = 1, then 2 − r ≡ −1 + δ_{h} ± 1 (mod 8); If a = 3 − δ_{h} and δ_{φS} = 1, then 2 − r ≢ −1 + δ_{h} + 4 (mod 8). If a = r, δ_{φS} = 0 and v = 0, then 2 − r ≡ 0 mod 8. If r + a = 22 − 2δ_{h}, δ_{φS} = 0 and v = h (mod 2S), then 2 − r ≡ −1 (mod 8).

All possible invariants (3.1) satisfying these conditions are given in Figures 2 and 3.

The relations between related involutions are:

(3.2)  r(φ) + r(τφ) = 21; a(τφ) − a(φ) = 2δ(φ)_{h} − 1; δ(φ)_{h} + δ(τφ)_{h} = 1;
(3.3)  δ_{φS} = δ_{τφS}; v(φ) + v(τφ) ≡ h (mod 2S), if δ_{φS} = δ_{τφS} = 0.

Because of the relation δ(φ)_{h} +δ(τφ)_{h} = 1, it is sufficient to describe only involutions with δ(φ)_{h} = 0 or with δ(φ)_{h} = 1.

It follows that there are 49 connected components of moduli of positive curves with δ_{φS} = 1, and 15 with δ_{φS} = 0 (for the fixed δ_{h} = 1 or 0). Thus, there are 128 connected components of moduli of positive curves (or involutions (X, τ, φ)) and 64 connected components of moduli of real non-singular curves of degree 6 on the real projective plane.

According to Arnol’d [3], the invariant δ_{h} = 0, if A^{+} ≠ ∅ and the positive curve A^{+} is orientable. Otherwise, δ_{h} = 1. Using this, Bezout Theorem and (2.44), (2.45), (3.2), (3.3), one can draw the corresponding pictures. It is enough to draw them for δ_{h} = 0 when the positive curve is orientable. We have

If (δ_{h}; r, a, δ_{φS}) = (0; 11, 11, 0), the positive curve is the whole ℝℙ^{2}.

If (δ_{h}; r, a, δ_{φS}) = (0; 11, 9, 0) the positive curve is drawn in Figure 4.

If δ_{h} = 0 and (r, a, δ_{φS}) is different from (11, 11, 0) and (11, 9, 0) the positive curve is drawn in Figure 5 where g = (22 − r − a)/2 and k = (r − a)/2.

The invariant δ_{φS} = 0, if and only if the real curve A(ℝ) divides A(ℂ). See [3] and [25].

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]
Figure 2. ℙ^{2}: All possible (r, a, δ_{φS}, v) with H = 0 (i. e. δ_{h} = 1)

[figure: means δ_{φS} = 0 and v = h,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]
Figure 3. ℙ^{2}: All possible (r, a, δ_{φS}, v) with H = [h] (i. e. δ_{h} = 0)

Figure 4. ℙ^{2}: A^{+} for H = [h] and (r, a, δ_{φS}, v) = (11, 9, 0, h)

More exactly, by [3] the invariant δ_{φS} = 0, if the curve A(ℝ) divides A(ℂ). By the results above, connected component of moduli is defined by the isotopy type and by the invariant δ_{φS}. Thus, to prove the opposite statement, it is sufficient to construct a dividing curve on the plane for each isotopy type with δ_{φS} = 0. This construction is explained in [25].

The connected component of moduli of a positive non-singular real curve A of the degree 6 on the real projective plane is defined by its isotopy type and by dividiness of A(ℂ) by A(ℝ). All possibilities are described above.

Applying results of Sect. 2, below we get similar results for all lattices S of rk S = 2.

Figure 5. ℙ^{2}: A^{+} for H = [h] and (r, a, δ_{φS}, v) ≠ (11, 9, 0, h), (11, 11, 0, h).

4. Connected components of moduli of real non-singular curves of bidegree (4, 4) on hyperboloid

Here we consider the case S ≅ U(2) and θ = −1 on S. Then Y = X/{1, τ} = ℙ^{1} × ℙ^{1} in the non-degenerate case which we only consider, and A = X^{τ} is a non-singular curve of bidegree (4, 4) in Y.

Let S = ℤe_{1} + ℤe_{2} where e_{1}^{2} = e_{2}^{2} = 0 and (e_{1}, e_{2}) = 2. The generators e_{1} and e_{2} of S = H_{2}(X; ℤ)^{τ} are classes of preimages of pt × ℙ^{1} and ℙ^{1} × pt respectively.

Since θ = −1 on S, then the real structure on Y (defined by the anti-holomorphic involution φ of X) is one of the four structures: (usual, usual) = hyperboloid, (usual, spin), (spin, usual) and (spin, spin), where usual means the usual real structure on ℙ^{1}, which is given by the anti-holomoprhic involution (z_{0} : z_{1}) ↦ (\overline{z_{0}} : \overline{z_{1}}) in some homogeneous coordinates (z_{0} : z_{1}) on ℙ^{1}, while spin is given by (z_{0} : z_{1}) ↦ (\overline{z_{1}} : −\overline{z_{0}}). In hyperpoloid case Y (ℝ) = ℝℙ^{1} × ℝℙ^{1} is a 2-torus. In all three spin cases Y (ℝ) and X(ℝ) are empty, and A(ℝ) is also empty.

If Y has the usual real structure (i.e. it is a hyperboloid), then A is the zero set of a real bi-homogeneous polynomial P (x_{0} : x_{1}; y_{0} : y_{1}) of bidegree (4, 4). For a positive curve (A, φ), equivalently for A^{+} = π(X_{φ}(ℝ)), like for real non-singular curves of degree 6 in ℙ^{2}, we can choose P by the condition that A = {P = 0} in ℙ^{1} × ℙ^{1} and A^{+} = {P ≥ 0} on ℝℙ^{1} × ℝℙ^{1}. The polynomial P is defined up to ℝ_{++} × ((P GL(2, ℝ) × P GL(2, ℝ)) ⋊ ℤ/2ℤ). Here ℝ_{++} denote all positive real numbers. Thus, classification of connected components of moduli of positive curves A^{+} on a hyperboloid is equivalent to the description of connected components of

((ℝ^{25} − Discr)/ℝ_{++}) / ((P GL(2, ℝ) × P GL(2, ℝ)) ⋊ ℤ/2ℤ)

where the discriminant Discr is the set of all polynomials giving singular (over ℂ) curves. The group (P GL(2, ℝ)×P GL(2, ℝ))⋊ℤ/2ℤ has 8 connected components and has dimension 6 over ℝ.

If Y has one of spin structures, then Discr has codimension two. Hence the moduli of positive curves and the moduli of curves are connected. (We will also give another proof of that below.)

Now we outline general arguments of Subsect. 2.3 in the (S, θ) = (U(2), −1) case (for finding connected components of moduli).

Since Δ(S_{+}, S)^{(−4)} = ∅ and Δ(S_{−}, S)^{(−4)} = {±(e_{1}−e_{2})}, the group W^{(−4)}(S_{+}, S) is trivial, and the group W^{(−4)}(S_{−}, S) consists of identity and g where g(e_{1}) = e_{2} and g(e_{2}) = e_{1}. By Lemma 9, we have G = {identity, g} ≅ ℤ/2ℤ.

We have s = 2, p = 0. A_{S_{−}} = S_{−}^{∗}/S_{−} = S^{∗}/S is generated by e_{1}^{∗} = \frac{1}{2}e_{2} and e_{2}^{∗} = \frac{1}{2}e_{1}, and hence it is isomorphic to (ℤ/2ℤ)^{2} and l(A_{S_{−}}) = 2. We have A_{S_{+}} = 0, H_{+} = 0, ρ = 0. Since Γ_{−} = 0, H = H_{−} (⊂ S/2S) is one of the following 5 subgroups;

0, [e_{1}], [e_{2}], [h] = [e_{1} + e_{2}], S/2S = [e_{1}, e_{2}],

where we set h = e_{1} + e_{2} and consider e_{1}, e_{2} and h mod 2S. Since H_{+} = 0, we have q_{ρ} = (−q_{S_{−}})|H_{−}. Since δ(S) = 0, we always have δ_{H} = 0 and δ_{φS_{±}} = δ_{φ}. If δ_{φ} = 0, then ε_{v_{±}} ≡ 0 (mod 2).

Thus, we have the following cases:

(1) H = 0: Then a_{H} = 0, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φS} = 0, then v = 0, equivalently, δ_{φ} = 0. We have c_{v} ≡ 0 (mod 4), δ_{φS_{±}} = δ_{φ} = 0.

(2) H = [e_{1}]: Then a_{H} = 1, q_{ρ} ≅ z, k_{ρ} = 1, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8).

(3) H = [e_{2}]: Then a_{H} = 1, q_{ρ} ≅ z, k_{ρ} = 1, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8).

(4) H = [h] = [e_{1} + e_{2}]: Then a_{H} = 1, q_{ρ} ≅ w, k_{ρ} = 1, μ_{ρ} = 1.

(5) H = S/2S = [e_{1}, e_{2}]: Then a_{H} = 2, q_{ρ} ≅ u_{+}(2), k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φS} = 0, then v = 0, and δ_{φS} = 0 is equivalent to δ_{φ} = 0.

From above, μ_{ρ} = 1 if and only if H = [h]. If μ_{ρ} = 0, then σ_{ρ} ≡ 0 (mod 8).

We see that genus (hence, by Theorem 14 and Proposition 15, the isomorphism class) of an integral involutions (L, φ, S) of the type (U(2), −1) satisfying (RSK3) is determined by the data (r, a, H, δ_{φS}, v), where the v is defined only if δ_{φS} = 0.

By Theorem 10, the complete list of the above invariants is given by the conditions:

Type 0 (δ_{φ} = 0) a_{H} + k_{ρ} ≤ a ≤ r and a ≡ 0 (mod 2). 2 ≤ r ≤ 18 and 2 − r ≡ 0 (mod 4). r + a ≤ 2a_{H} + 18. If a = a_{H} + k_{ρ}, μ_{ρ} = 0, then 2 − r ≡ 0 (mod 8). If a = r, then 2 − r ≡ 0 (mod 8). If r + a = 2a_{H} + 18, then 2 − r ≡ 0 (mod 8).

Type Ia (δ_{φS} = 0, δ_{φ} = 1) a_{H} = 1, k_{ρ} = 1. c_{v} ≡ 0 (mod 4) if μ_{ρ} = 0, and c_{v} ≡ 2 (mod 4) if μ_{ρ} = 1. 2 ≤ a ≤ r and a ≡ 0 (mod 2). 2 ≤ r ≤ 18 and 2 − r ≡ c_{v} (mod 4). r + a ≤ 20. If a = 2, μ_{ρ} = 0, then 2 − r ≡ 0 (mod 8).

Type Ib (δ_{φS} = 1) a_{H} + k_{ρ} + 1 ≤ a ≤ r, 1 ≤ r ≤ 19, r + a ≡ 0 (mod 2), r + a ≤ 2a_{H} + 18. If a = a_{H} + k_{ρ} + 1, μ_{ρ} = 0, then 2 − r ≡ ±1 (mod 8). If a = a_{H} + k_{ρ} + 2, μ_{ρ} = 0, then 2 − r ≢ 4 (mod 8).

We see a_{H} = 1 in Type Ia case. Then H is generated by the unique nonzero element, hence it is nothing but v. Hence we can use the invariant δ_{φ} instead of v in the collection of invariants above. Thus the isomorphism class is determined by the data

(r, a, H, δ_{φS}, δ_{φ}).

Since G = {identity, g}, the triplets (r, a, [e_{1}]) and (r, a, [e_{2}]) represent the same isomorphism class for each Type (0, Ia and Ib). The other different triplets represent different isomorphism classes.

Figure 6. ℍ: All possible (r, a, δ_{φS}, v) with H = 0

All possible data

(4.1)  (r, a, H, δ_{φS}, v)

(which are equivalent to (r, a, H, δ_{φS}, δ_{φ})) are given in Figures 6 — 9. We have δ_{φ} = 0, if and only if δ_{φS} = v = 0.

The relations between related involutions are as follows. Since δ(S) = 0 and θ = −1 on S, then s_{θ} = 0 mod 2L. Hence, by Theorem 11, we have:

(4.2)  r(φ) + r(τφ) = 20; a_{H(φ)} + a_{H(τφ)} = 2; a(φ) − a_{H(φ)} = a(τφ) − a_{H(τφ)};
(4.3)  H(τφ) = H(φ)^{⊥} w. r. t. b_{S_{−}} on A_{S_{−}}; δ_{φS} = δ_{τφS}; s_{φ} ≡ s_{τφ} mod 2L.

Hence, involutions of Type 0 (resp. Type Ib) with H = 0 (11 (resp. 39) classes) are related to involutions of Type 0 (resp. Type Ib) with H = S/2S. Involutions of Type 0 (resp. Type Ia, Type Ib) with H = [e_{i}] (9 (resp. 11, 30) classes) are related to involutions of Type 0 (resp. Type Ia, Type Ib) with H = [e_{i}]. (More precisely, the class (r, a) is related to the class (20 − r, a).) Involutions of Type 0 (resp. Type Ia, Type Ib) with H = [h] (11 (resp. 12, 36) classes) are related to involutions of Type 0 (resp. Type Ia, Type Ib) with H = [h]. (More precisely, the class (r, a) is related to the class with (20 − r, a), too.)

Thus there are 42(= 11 × 2 + 9 + 11) classes (i.e., connected components of moduli of positives curves of bidegree (4, 4)) of Type 0, 23(= 11 + 12) classes of Type Ia, and 144(= 39 × 2 + 30 + 36) classes of Type Ib.

Moreover, if we identify related involutions, there are 26(= 11 + 7 + 8) classes (i.e., connected components of moduli of real non-singular curves of bidegree (4, 4)) of Type 0, 14(= 8 + 6) classes of Type Ia, and 76(= 39 + 17 + 20) classes of Type Ib.

If two positive curves are in one connected component of moduli or are related, the real structure on Y stays the same. By (2.44) and (2.45), the A^{+} is empty, if and only if (r, a, δ_{φ}) = (10, 10, 0). It follows that the component (r, a, H, δ_{φS}, δ_{φ}) = (10, 10, [h], 0, 0) corresponds to the real structure (spin, spin), the component (10, 10, [e_{i}], 0, 0) corresponds to the real structure (usual, spin) (or (spin, usual)). And all the remaining components correspond to the real structure (usual, usual) (namely, hyperboloid). The component (10, 10, S/2S, 0, 0) consists of empty A^{+} on hyperboloid.

The isotopy classification of real non-singular curves of bidegree (4, 4) on a hyperboloid was obtained by Gudkov [7]. Zvonilov [35] clarified all the complex schemes of curves of bidegree (4, 4) on hyperboloid and ellipsoid, where complex schemes mean real schemes (i.e. real isotopy types) with divideness and their complex orientations (if dividing). Curves on hyperboloid and ellipsoid were studied in [33], [34], [15], [16], [17]. In these articles the

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]
Figure 7. ℍ: All possible (r, a, δ_{φS}, v) with H = [e_{1}, e_{2}] = S/2S

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 0 and v = e_{i},; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]
Figure 8. ℍ: All possible (r, a, δ_{φS}, v) with H = [e_{i}] (i = 1 or 2)

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 0 and v = e_{1} + e_{2},; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]
Figure 9. ℍ: All possible (r, a, δ_{φS}, v) with H = [e_{1} + e_{2}] = [h]

notions: torsion (s, t) (∈ ℤ × ℤ) of a connected component of A(ℝ), oval, non-oval, odd branch and even branch, are well-known. We quote the isotopy classification of non-singular curves of bidegree (4, 4) on hyperboloid from [35] in Table 1 where we use notations due to Viro [31] and Zvonilov [35], and l_{1} and l_{2} denote non-ovals with torsions (1, 0) and (0, 1) respectively (e. g. see Figure 15).

If a curve A(ℝ) on hyperboloid has only ovals or A(ℝ) = ∅, then one of A^{+} and A^{−} contains the outermost component. Thus we divide isotopy types of positive curves A^{+} into the following 4 cases:

⟨1⟨m⟩ ⊔ n⟩,
where (m, n) = (9, 0), (5, 4), (1, 8), (8, 0), (5, 3), (4, 4), (1, 7);
or m ≥ 1, n ≥ 0 and m + n ≤ 7.
⟨m⟩,
where 0 ≤ m ≤ 9.
⟨1⟨1⟩ ⊔ 1⟨1⟩⟩
⟨l_{1} + l_{2}, m, l_{1} + l_{2}, n⟩,
where 0 ≤ m ≤ n and m + n ≤ 8.
⟨4(l_{1} + l_{2})⟩
⟨l_{1}, m, l_{1}, n⟩,
where (m, n) = (0, 8), (4, 4), (0, 7), (3, 4);
or 0 ≤ m ≤ n and m + n ≤ 6.
⟨4(l_{1})⟩
⟨2(l_{1} + 2l_{2})⟩

Table 1. The list of isotopy types of real non-singular curves of bidegree (4, 4) on a hyperboloid.

(i): A(ℝ) has only ovals or A(ℝ) = ∅, and A^{+} contains the outermost component. (In

this case we say A^{+} is outer.) (ii): A(ℝ) has only ovals or A(ℝ) = ∅, and A^{+} does not contain the outermost compo-

nent. (In this case we say A^{+} is inner.) E. g. see Figures 10, 11. (iii): A(ℝ) has even branches. E. g. see Figures 12, 13 (iv): A(ℝ) has odd branches. E. g. see Figures 14, 15.

The subgroup H = H_{−} is determined by the invariants δ_{e_{1}}, δ_{e_{2}} and δ_{h}. We have the following interpretation of the invariant δ_{x_{−}}.

Theorem 16. Let X be a compact Kähler surface with an anti-holomorphic involution φ, and X(ℝ) be the fixed point set of φ. We assume that H_{1}(X; ℤ) = 0 and X(ℝ) ≠ ∅. We set L = H_{2}(X; ℤ) and L_{φ} = {x ∈ L | φ_{∗}(x) = −x}. Let C be a 1-dimensional complex submanifold of X with φ(C) = C, and C(ℝ) be the fixed point set of φ on C. Let [C] and [C(ℝ)] denote the homology classes represented by C and C(ℝ) respectively. We set x_{−} := [C] (∈ L_{φ}). Then we have

δ_{x_{−}} = 0 ⟺ [C(ℝ)] = 0 in H_{1}(X(ℝ); ℤ/2ℤ),

where we define the symbol δ_{x_{−}} in the same way as δ_{x_{−}} in Subsect. 2.3.

Proof. Suppose that [C(ℝ)] ≠ 0 in H_{1}(X(ℝ); ℤ/2ℤ). Then there exists an embedded circle D on X(ℝ) such that [C(ℝ)] · [D] ≠ 0, where [C(ℝ)] · [D] means the ℤ/2ℤ-intersection number in X(ℝ). We set E_{−} = {x ∈ H^{2}(X; ℤ) | φ^{∗}(x) = −x}. E_{−} is the Poincaré dual to L_{φ}. By Théorème 2.4 in [14], there exists a surjective canonical morphism

φ_{X} : E_{−} → H^{1}(X(ℝ); ℤ/2ℤ)

such that

(φ_{X}(γ), φ_{X}(γ′)) ≡ Q(γ, γ′) (mod 2) ∀γ, γ′ ∈ E_{−},

where ( , ) and Q are the forms induced by the cup products on H^{1}(X(ℝ); ℤ/2ℤ) and H^{2}(X; ℤ). Moreover, by Théorème 2.5 in [14], the following diagram commutes:

Pic(X)^{G}  ⟶^{c_{1}}  E_{−}
α ↓  φ_{X} ↓
H_{alg}^{1}(X(ℝ); ℤ/2ℤ) ⟶^{i} H^{1}(X(ℝ); ℤ/2ℤ),

where α is defined as in [14],p.562 (see also below), c_{1} is the first Chern class map, and i is the inclusion map. In the sequel, A^{P} denotes the Poincaré dual element to a (co)homology class A. Since φ_{X} is surjective, there exists γ (∈ E_{−}) such that φ_{X}(γ) = [D]^{P} (∈ H^{1}(X(ℝ); ℤ/2ℤ)). We consider the divisor class [C] in Pic(X)^{G}. Then, as is well known, its first Chern class c_{1}([C]) is the Poincaré dual to x_{−}. By the definition (see [14]) of α, we see

α([C]) = η(C) = [C(ℝ)]^{P} (∈ H_{alg}^{1}(X(ℝ); ℤ/2ℤ)).

On the other hand, by the above commutative diagram, we see

α([C]) = φ_{X}(c_{1}([C])) = φ_{X}(x_{−}^{P}).

Hence, we have [C(ℝ)] · [D] = ([C(ℝ)]^{P} , [D]^{P}) = (φ_{X}(x_{−}^{P}), φ_{X}(γ)) ≡ Q(x_{−}^{P}, γ) (mod 2) = (x_{−}, γ^{P}). Thus we have δ_{x_{−}} = 1. This completes the proof of the implication ⟹. The converse assertion can be proved by the same argument as the proof of Lemma 2 in [17]. Suppose that [C(ℝ)] = 0 in H_{1}(X(ℝ); ℤ/2ℤ). Then X(ℝ) and C(ℝ) satisfy the conditions a) and b) of Remark 2.2 in [11]. By that remark, Lemma 2.3 is applicable to the involution φ : X → X and C. Hence, we see (x_{−})_{mod 2} (∈ H_{2}(X; ℤ/2ℤ)) is orthogonal to Imα_{2}. Since H_{1}(X; ℤ) = 0, as in the proof of Lemma 3.7 in [12], we have Imα_{2} = {x ∈ H_{2}(X; ℤ/2ℤ) | φ_{∗}(x) = x}. Thus we have δ_{x_{−}} = 0. □

By Theorem 16, we have H = 0 if and only if A^{+} is outer, H = S/2S if and only if A^{+} is inner, H = [h] if and only if A(ℝ) has even branches, H = [e_{1}] if and only if A(ℝ) has odd branches with odd s, and H = [e_{2}] if and only if A(ℝ) has odd branches with odd t.

When A is a dividing curve on hyperboloid with non-ovals, we define the number l̂ (∈ ℤ/2ℤ) as follows (see [15]): Fix a complex orientation of A(ℝ). When the number of nonovals of A(ℝ) is 2, we define l̂= 0 if the complex orientations of the 2 non-ovals are different in ℝℙ^{1} × ℝℙ^{1}, l̂ = 1 if otherwise. When the number of non-ovals of A(ℝ) is 4, we fix a non-oval E and define l̂ = 0 if the number of non-ovals whose complex orientations are the same as E in ℝℙ^{1} × ℝℙ^{1} is even, l̂= 1 if odd.

We quote the following lemma.

Lemma 17 ([15]). For a positive curve (A, φ) of bidegree (4, 4) on hyperboloid, we have the following:

(1) [X_{φ}(ℝ)] = [X_{τφ}(ℝ)] in H_{2}(X; ℤ/2ℤ)

(2) If A is dividing, then

[X_{φ}(ℝ)] (= [X_{τφ}(ℝ)]) = \{ \begin{matrix}0 & (if A(ℝ) has only ovals) \\ l̂h & (if A(ℝ) has even branches) \\ l̂e_{1} & (if A(ℝ) has odd branches with odd s) \\ l̂e_{2} & (if A(ℝ) has odd branches with odd t) \end{matrix}

in H_{2}(X; ℤ/2ℤ). □

Then we have the following:

Proposition 18. Let A be a non-singular real curve of bidegree (4, 4) on hyperboloid. If A is dividing, then the positive curves (A, φ) satisfies δ_{φS} = 0.

Figure 10. ℍ: A^{+} for H = S/2S = [e_{1}, e_{2}] with (r, a, δ_{φS}, v) ≠ (10, 10, 0, 0) nor (10, 8, 0, 0).

Figure 11. ℍ: A^{+} for H = S/2S = [e_{1}, e_{2}] with (r, a, δ_{φS}, v) = (10, 8, 0, 0).

Proof. If A is dividing, then by (2) of Lemma 17, we have [X_{φ}(ℝ)] ≡ s_{φ} (mod 2H_{2}(X; ℤ)) for some s_{φ} ∈ S. Hence we have δ_{φS} = 0. □

Moreover, we have the following interpretation of v when A is a dividing curve with nonovals on hyperboloid: v = 0 if and only if l̂= 0, v = h (mod 2S) if and only if A(ℝ) has even branches and l̂= 1, v = e_{1} (mod 2S) if and only if A(ℝ) has odd branches with odd s and l̂= 1, v = e_{2} (mod 2S) if and only if A(ℝ) has odd branches with odd t and l̂= 1.

By the above geometric interpretations and Zvonilov’s classification [35], from the invariants (4.1) of the connected component of moduli of a positive curve A^{+} on hyperboloid, we can uniquely determine the complex scheme of A. In particular, one can draw the picture of the positive curve A^{+}. By (4.2) and (4.3), it is enough to draw them for H = S/2S,

[h] and [e_{1}]. See Figures 10 — 15. Remind that A(ℝ) is an empty curve on hyperboloid if H = S/2S and (r, a, δ_{φS}, v) = (10, 10, 0, 0), Y has the (spin,spin) structure if H = [h] and (r, a, δ_{φS}, v) = (10, 10, 0, 0), and Y has the (usual,spin) (or (spin,usual)) structure if H = [e_{i}] and (r, a, δ_{φS}, v) = (10, 10, 0, 0).

Moreover, there is known (due to Zvonilov [35]) a dividing curve on hyperboloid for each connected component with δ_{φS} = 0. Hence, the opposite statement of Proposition 18 is true:

Theorem 19. Let A be a non-singular real curve of bidegree (4, 4) on hyperboloid. Then, A is dividing or A(ℝ) = ∅, if and only if the positive curves (A, φ) (or A^{+}) has δ_{φS} = 0. □

Figure 12. ℍ: A^{+} for H = [h] = [e_{1} + e_{2}] with (r, a, δ_{φS}, v) ≠ (10, 10, 0, 0) nor (10, 8, 0, 0).

Figure 13. ℍ: A^{+} for H = [h] = [e_{1} + e_{2}] with (r, a, δ_{φS}, v) = (10, 8, 0, 0).

Figure 14. ℍ: A^{+} for H = [e_{1}] with (r, a, δ_{φS}, v) ≠ (10, 10, 0, 0) nor (10, 8, 0, 0).

It turns out that if δ_{φS} = 0 and H = [h], then (r, a) determines v. But when δ_{φS} = 0 and H = [e_{i}], (r, a) does not always determine v. As stated above, when δ_{φS} = 0, H = [e_{i}] and (r, a, v) ≠ (10, 10, 0), v = 0 if and only if l̂= 0.

Thus, we finally get

Theorem 20. A connected component of moduli of a positive real non-singular curve A^{+} of bidegree (4, 4) on hyperboloid is defined by the isotopy type of A^{+} ⊂ ℝℙ^{1} × ℝℙ^{1} (up to the action of (P GL(2, ℝ) × P GL(2, ℝ)) ⋊ ℤ/2ℤ), divideness of A(ℂ) by A(ℝ), and by

Figure 15. ℍ: A^{+} for H = [e_{1}] with (r, a, δ_{φS}, v) = (10, 8, 0, 0).

the invariant l̂ mod 2 defined by the complex orientation (if A(ℝ) has odd branches and is dividing).

5. Connected components of moduli of real non-singular curves of bidegree (4, 4) on ellipsoid

Here we consider the case S ≅ U(2) and rk S_{−} = 1 (ellipsoid cases). Like for hyperboloid, Y = ℙ^{1} × ℙ^{1}, and we use the same notations for S. Obviously, θ(e_{1}) = −e_{2} and θ(e_{2}) = −e_{1}, we have S_{+} = ℤ(e_{1} − e_{2}) and S_{−} = ℤ(e_{1} + e_{2}). Thus S_{+} ≅ ⟨−4⟩ and S_{−} ≅ ⟨4⟩.

Since Δ(S_{+}, S)^{(−4)} = {±(e_{1} −e_{2})} and Δ(S_{−}, S)^{(−4)} = ∅, W^{(−4)}(S_{+}, S) consists of identity and g where g(e_{1}) = e_{2} and g(e_{2}) = e_{1}, and W^{(−4)}(S_{−}, S) is trivial. By Lemma 9, we have G = {identity, g} ≅ ℤ/2ℤ.

We call these data (S, θ, G) as the ellipsoid case.

For the ellipsoid case we have s = 2, p = 1. A_{S_{+}} ≅ ℤ/4ℤ is generated by \frac{1}{4}(e_{1} − e_{2}), and A_{S_{−}} ≅ ℤ/4ℤ is generated by \frac{1}{4}(e_{1} + e_{2}). We see Γ_{+} = [e_{1} − e_{2}] = 2(S_{+}^{∗} ∩ (\frac{1}{2}S_{+}))/2S_{+} and Γ_{−} = [e_{1}+e_{2}] = 2(S_{−}^{∗}∩(\frac{1}{2}S_{−}))/2S_{−}. Hence we have H_{±} = Γ_{±}. Since (e_{1}−e_{2}, 0) = (0, e_{1}+e_{2}) in H = H_{+} ⊕_{γ} H_{−}, we see

H = {0, (e_{1} − e_{2}, 0) = (0, e_{1} + e_{2})} ≅ ℤ/2ℤ,

a_{H} = 1 and q_{ρ} ≅ w. Hence δ_{H} = 0, k_{ρ} = μ_{ρ} = 1.

If δ_{φ} = 0 (Type 0), then c_{v} ≡ 0 (mod 4). If δ_{φS} = 0 and δ_{φ} = 1 (Type Ia), then v = (e_{1} − e_{2}, 0) = (0, e_{1} + e_{2}) and c_{v} ≡ 2 (mod 4).

We have δ_{φS} = δ_{φS_{±}}. Assume that δ_{φS} = 0. Then

γ_{±} = \{ \begin{matrix}0 & if δ_{φ} = 0, \\ q_{1}(2)^{2} & if δ_{φS} = 0 and δ_{φ} = 1. \end{matrix}

If δ_{φ} = 0, then (q_{S_{±}})_{v} = q_{S_{±}}. Hence we have ε_{v_{±}} = 0. If δ_{φS} = 0 and δ_{φ} = 1, then (q_{S_{±}})_{v} is defined on ℤ/4ℤ, and the square of a generator is ±\frac{5}{4} (mod 2ℤ). Hence (q_{S_{±}})_{v} is the discriminant quadratic form of the 2-adic lattice K_{±5}^{(2)}(2^{2}) = ⟨±5 · 2^{2}⟩ ([19]). Thus we have ε_{v_{±}} = 1.

We see that the genus of an integral involutions (L, φ, S) is determined by the data

(r, a, δ_{φS}, δ_{φ}).

By Theorem 10, the complete list of the above invariants for ellipsoid is given by the conditions:

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 0 and v = e_{1} + e_{2},; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]
Figure 16. 𝔼: All possible (r, a, δ_{φS}, v) (here H = [e_{1} + e_{2}])

Type 0 (δ_{φ} = 0) 2 ≤ a ≤ r and a is even. r = 2, 6, 10, 14, 18. r + a ≤ 22. If a = r, then (r, a) = (2, 2), (10, 10). If r + a = 22, then (r, a) = (18, 4).

Type Ia (δ_{φS} = 0, δ_{φ} = 1) 2 ≤ a ≤ r and a is even. r = 4, 8, 12, 16, 20. r + a ≤ 22. If a = r, then (r, a) = (4, 4). If r + a = 22, then (r, a) = (20, 2), (12, 10).

Type Ib (δ_{φS} = 1) 3 ≤ a ≤ r. r + a is even. 2 ≤ r ≤ 20. r + a ≤ 22.

All possible data

(5.1)  (r, a, δ_{φS}, v)

which are equivalent to (r, a, δ_{φS}, δ_{φ}) are given in Figure 16. We have δ_{φ} = 0, if and only if δ_{φS} = v = 0. Thus there are 13 cases of Type 0, and 13 cases of Type Ia, and 45 cases of Type Ib.

The relations between related involutions are:

(5.2)  r(φ) + r(τφ) = 22, a(φ) = a(τφ),
(5.3)  δ_{φS} = δ_{τφS}, s_{φ} + s_{τφ} ≡ e_{1} + e_{2} mod 2L.

Hence, if a positive curve A^{+} satisfies δ_{φ} = 0, then the related positive curve A^{−} satisfies (δ_{τφS}, δ_{τφ}) = (0, 1). If a positive curve A^{+} satisfies δ_{φS} = 1 and r(φ) ≤ 11, then the related positive curve A^{−} satisfies δ_{τφS} = 1 and r(τφ) ≥ 11.

For the ellipsoid case, conditions (b_{+}) and (b_{−}) of Theorem 14 are not valid in all cases. We have to check conditions (a_{+}) and (a_{−}) in the rest cases directly. As the result, we have

Theorem 21. For the ellipsoid case, genus always defines the class of an integral involution. In particular (by Theorems 1, 6 and 10), the genus invariants

(r, a, δ_{φS}, δ_{φ})

define the connected component of muduli of positive non-singular curves of bidegree (4, 4) on ellipsoid up to the action of the automorphism group of ellipsoid. There are exactly 13 connected components of moduli with δ_{φ} = 0, exactly 13 connected components of moduli with (δ_{φS}, δ_{φ}) = (0, 1), and exactly 45 connected components of moduli with δ_{φS} = 1.

Identifying related positive curves, we get that there are exactly 38 connected components of moduli of real non-singular curves of bidegree (4, 4) on ellipsoid up to the action of the automorphism group of ellipsoid. There are 13 of them with δ_{φ} = 0 (equivalently, (δ_{φS}, δ_{φ}) = (0, 1)), and 25 connected components with δ_{φS} = 1: they are distinguished from 45 by the inequality r(φ) ≤ 11 (equivalently, by r(φ) ≥ 11).

Proof. For the ellipsoid case, a_{H_{+}} = 1, l(A_{S_{+}}) = 1, p = 1. Moreover, κ(A_{S_{+}}) = 0. Thus, the condition (b_{+}) of Theorem 14 means that either r − a > 0, or r − a = 0 and a ≥ 5, or r − a = 0, a = 4 and δ_{φS_{+}} = 0 (for our case, δ_{φS_{+}} = δ_{φS}). From the list of possible invariants above, we see that this condition is not satisfied only in the following cases:

(a) δ_{φ} = 0 and r = a = 2;

(b) δ_{φS} = 1 and r = a = 3;

(c) δ_{φS} = 1 and r = a = 4.

Let us consider all these cases.

Case (a): In this case, the lattice L^{φ} ≅ U(2), and the lattice K_{+} is the orthogonal complement to the primitive sublattice S_{+} ⊂ U(2) where S_{+} ≅ ⟨−4⟩. It follows that K_{+} ≅ ⟨4⟩. Obviously, the condition (a_{+}) of Theorem 14 is valid in this case.

In cases (b) and (c), the invariant δ_{φ} = 1, and the lattice L^{φ} ≅ ⟨2⟩⊕(r−1)⟨−2⟩. The lattice K_{+} is the orthogonal complement to the primitive sublattice S_{+} ≅ ⟨−4⟩ ⊂ L^{φ}. Equivalently, K_{+} is the orthogonal complement δ^{⊥} to a primitive element δ ∈ L^{φ} with δ^{2} = −4. Since δ_{φS} = δ_{φS_{+}} = 1, the discriminant form q_{K_{+}} must be isomorphic to q_{±1}(2) ⊕ q′ in this case. Here q_{±1}(2) is a non-degenerate quadratic form on the group of order 2. Equivalently, there exists x ∈ A_{K_{+}} of order 2 such that q_{K_{+}}(x) ≢ 0 mod 1. See ([21], page 116).

Considering the odd unimodular lattice L = L^{φ}(1/2) ≅ ⟨1⟩ ⊕ (r − 1)⟨−1⟩ instead of L^{φ}, we should prove the following for r = 3, 4: For all elements δ ∈ L with δ^{2} = −2 and with odd K = δ^{⊥}, the lattices K = δ^{⊥} are isomorphic, and the canonical homomorphism

(5.4)  ξ : O(K) → O(q_{K(2)})

is epimorphic. Now let us consider cases (b) and (c) separately.

Case (b). In this case, by simple calculations, the discriminant form q_{K(2)} ≅ q_{1}(2)⊕q_{−1}(4). Its automorphism group O(q_{K(2)}) = ±1 is the image of the automorphism subgroup ±1 of O(K). Thus, (5.4) is valid. To prove that all lattices K = δ^{⊥} are isomorphic, one should argue similarly to the case (c) below which is more complicated.

Case (c). In this case, q_{K(2)} ≅ q_{1}(2) ⊕ q_{−1}(2) ⊕ q_{−1}(4) (this is clear, but we shall give the proof in considerations below). By simple calculations, O(q_{K(2)}) ≅ (ℤ/2)^{2}.

Let us consider the group W (L) generated by reflections in all elements of L with squares −1 and −2. The Dynkin diagram of the fundamental chamber of W (L) is well-known (e. g. see [30]). It is (with usual notations)

(5.5)

• — • — ∘ — ⋆

Here the two • correspond to elements δ ∈ L with δ^{2} = −2 and with odd orthogonal complement K = δ^{⊥}. They (i. e. their O(L)-orbits) give all elements of L we are looking for. In particular, there exist no more than two possibilities for K up to isomorphisms. The ∘ corresponds to elements e ∈ L with e^{2} = −1. The ⋆ corresponds to elements v ∈ L with v^{2} = −2 and with even v^{⊥} in L.

Let e_{0}, e_{1}, e_{2}, e_{3} be the standard basis of L with the diagonal matrix diag(1, −1, −1, −1). The diagram (5.5) is given (from the left) by δ_{2} = −e_{2} + e_{3}, δ_{1} = −e_{1} + e_{2}, e = e_{1} and v = e_{0} −e_{1} −e_{2} −e_{3}. The permutation (e_{0}, e_{1}, e_{2}, e_{3}) → (e_{0}, e_{2}, e_{3}, e_{1}) gives the automorphism of L such that δ_{1} ↦ δ_{2}. Thus the lattices K corresponding to K = δ_{1}^{⊥} and K = δ_{2}^{⊥} are isomorphic. Any element δ ∈ K with δ^{2} = −2 and with odd δ^{⊥} belongs to O(L)-orbit of δ_{1} or δ_{2}.

The lattice K = δ_{2}^{⊥} has the basis e_{0}, e_{1}, δ = e_{2}+e_{3} with the diagonal matrix diag(1, −1, −4). It follows that q_{K(2)} ≅ q_{1}(2) ⊕ q_{−1}(2) ⊕ q_{−1}(4) with the corresponding bases e_{0}/2, e_{1}/2, δ/4 mod K. As we have mentioned, it is easy to see that O(q_{K(2)}) ≅ (ℤ/2)^{2}. The automorphisms ±1 of the lattice K give the automorphisms ±1 of q_{K(2)}. To prove (5.4), it is sufficient to construct an automorphism of K which does not give the automorphism ±1 in O(q_{K(2)}).

This is the automorphism f: e_{0} ↦ 2e_{0} + e_{1} + δ, e_{1} ↦ e_{0} + δ, δ ↦ 2e_{0} + 2e_{1} + δ of the lattice K. One can easily check that this is an automorphism of K. We have that f : δ/4 ↦ e_{0}/2 + e_{1}/2 + δ/4. It follows that f does not give ±1 in O(q_{K(2)}). Thus, (5.4) is epimorphic.

In the case (b), the Dynkin diagram of W (L) is (5.6)

• — ∘ — ∘ .

Replacing ϕ by τϕ, we get the statement (a_{−}) or Theorem 13.

It follows the theorem.

Now we consider the geometric interpretation of the above calculations. Using (2.44), (2.45) for both positive curves A^{+}, A^{−} and (5.2), (5.3), we can easily draw the picture of A^{+} on ellipsoid for all invariants (5.1). See Figures 17 and 18. As usual, we set g = (22−r−a)/2 and k = (r − a)/2. Thus we get the isotopy classification of real non-singular curves of bidegree (4, 4) on ellipsoid. We can see that the isotopy type is defined by the topology type of A^{+}. The isotopy types of A are:

a nest of depth 4; ⟨0⟩ (or ∅); ⟨1⟨m⟩ ⊔ n⟩ where m + n ≤ 9, m ≤ n.

This isotopy classification was first obtained by Gudkov and Shustin [8]. As we have mentioned in Sect. 4, Zvonilov [35] classified complex schemes of these isotopy types. Below we partly repeat his results.

We have similar statement to Arnold [3]

Proposition 22. Let (A, φ) (or A^{+}) be a positive curve of bidegree (4, 4) on an ellipsoid. If A is dividing, then

[X_{φ}(ℝ)] = c(e_{1} + e_{2})

in H_{2}(X; ℤ/2ℤ) for some c (∈ ℤ/2ℤ). In particular, we have δ_{φS} = 0.

Proof. We use arguments like Wilson [32], Lemma 6.7. Let C^{+} and C^{−} be the connected components of A \ A(ℝ). C^{±} have the orientations induced from A. A^{+} and A^{−} are the halfs of Y (ℝ) \ A(ℝ). A^{+} = π(X_{φ}(ℝ)) and A^{−} = π(X_{τφ}(ℝ)). Y (ℝ) (ellipsoid) is homeomorphic

Figure 17. 𝔼: A^{+} for (r, a, δ_{φS}) ≠ (10, 10, 0), (12, 10, 0), (10, 8, 0), (12, 8, 0).

to the 2-sphere S^{2}, and orientable. We give A^{±} the orientations induced from an orientation of Y (ℝ). We regard C^{±} and A^{±} as ℤ-chain. Then we have ∂(C^{+} + A^{+}) = 2∂D where D is the sum of some 2-disks on Y (ℝ). Hence C^{+} + A^{+} − 2D is ℤ-cycle. We consider the anti-holomorphic involution θ = φ mod {1, τ} on Y. Then θ(pt × ℙ^{1}) = −(ℙ^{1} × pt), θ(ℙ^{1} × pt) = −(pt × ℙ^{1}), θ(C^{+}) = −C^{−} and θ fix ℤ-chains on Y (ℝ).

For any integer m, n, we have

(1 + θ)(m(pt × ℙ^{1}) + n(ℙ^{1} × pt)) = (m − n)(pt × ℙ^{1} − ℙ^{1} × pt).

Hence (1 + θ)(C^{+} + A^{+} − 2D) ∼ c(pt × ℙ^{1} − ℙ^{1} × pt) for some integer c, where ∼ means ℤ-homologous. Since A is of bidegree (4, 4), C^{+} + C^{−} ∼ 4(pt × ℙ^{1} + ℙ^{1} × pt). Thus we have

2[C^{+} + A^{+} − 2D − 2(pt × ℙ^{1} + ℙ^{1} × pt)] = c[pt × ℙ^{1} − ℙ^{1} × pt]

in H_{2}(ℙ^{1}×ℙ^{1}; ℤ). Since H_{2}(ℙ^{1}×ℙ^{1}; ℤ) is free, we have [C^{+}+A^{+}−2D−2(pt×ℙ^{1}+ℙ^{1}×pt)] = c[pt × ℙ^{1} − ℙ^{1} × pt] for some integer c. Hence we have

C^{+} + A^{+} ∼ c(pt × ℙ^{1} + ℙ^{1} × pt)

as ℤ/2ℤ-cycles for some c (∈ ℤ/2ℤ). Taking the transfer (like [32], Lemma 6.7) gives [X_{φ}(ℝ)] = c(e_{1} + e_{2}) in H_{2}(X; ℤ/2ℤ). □

Suppose that A is dividing. Then δ_{φS} = 0 by Proposition 22. Since s_{φ} ≡ [X_{φ}(ℝ)] mod 2H_{2}(X; ℤ) and s_{φ} + s_{τφ} ≡ (e_{1} − e_{2}) mod 2H_{2}(X; ℤ) (see above), we have

[X_{φ}(ℝ)] + [X_{τφ}(ℝ)] = (e_{1} + e_{2})

in H_{2}(X; ℤ/2ℤ). Hence, by Proposition 22, we see [X_{τφ}(ℝ)] = (e_{1} + e_{2}) if [X_{φ}(ℝ)] = 0, and [X_{τφ}(ℝ)] = 0 if [X_{φ}(ℝ)] = (e_{1} + e_{2}) in H_{2}(X; ℤ/2ℤ).

By Proposition 22, if δ_{φS} = 1, then the curves are not dividing. Hence, the following 6 isotopy types ⟨7⊔1⟨0⟩⟩, ⟨6⊔1⟨1⟩⟩, ⟨5⊔1⟨2⟩⟩, ⟨4⊔1⟨3⟩⟩, ⟨4⊔1⟨1⟩⟩ and ⟨3⊔1⟨2⟩⟩ with δ_{φS} = 1 are not dividing. By Zvonilov [35], these 6 isotopy types are of indeterminate type, i.e., there exist dividing curves of these isotopy types. Then these dividing curves must realize some connected components with δ_{φS} = 0. As a result, it turns out that if δ_{φS} = 0 and A(ℝ) ≠ ∅, then A is dividing. Thus we have the following characterization of the invariant δ_{φS}.

Theorem 23. Let A^{+} be a positive non-singular real curve of bidegree (4, 4) on ellipsoid. Then, A is dividing or A(ℝ) = ∅, if and only if δ_{φS} = 0. □

Finally we get

Figure 18. 𝔼: A^{+} for (r, a, δ_{φS}) = (10, 8, 0) (A^{+} = ∅ for (r, a, δ_{φS}) = (10, 10, 0)).

Theorem 24. A connected component of moduli of a positive real non-singular curve A^{+} of bidegree (4, 4) on ellipsoid is defined by the topology type (or isotopy type) of A^{+} and by divideness of A(ℂ) by A(ℝ) (or the invariant δ_{φS}). All these possibilities are presented in figures 𝔼 above.

6. Connected components of moduli of real non-singular curves in | − 2K_{𝔽_{1}}|

Here we consider the case S ≅ ⟨2⟩ ⊕ ⟨−2⟩ and θ = −1 on S.

Let us show that Y = X/{1, τ} = 𝔽_{1} in this case. We apply general results about right DPN-pairs which we have cited in Sect. 2.2. By (2.11), (2.12), A = X^{τ} is a non-singular irreducible curve of genus 9. Since A ≠ ∅, Y is a rational surface with the Picard number 2. Then Y ≅ 𝔽_{n} with n ≥ 0. Since A ⊂ Y is a right DPN-pair, exceptional curves of Y are rational with square −1, −2 or −4. Thus, Y ≅ 𝔽_{n} with n = 0, 1, 2 or 4. If n = 0, then Y = ℙ^{1} × ℙ^{1}, and we have seen that S = U(2) in this case. If n = 2, then preimage of the exceptional section gives an element of S with square −4 (it is also a root of S), but S does not have elements with square −4. If n = 4, then the exceptional rational curve with square −4 must be a component of A, but A is irreducible of genus 9. Thus, Y = 𝔽_{1}.

This case is (D)-non-degenerate because S has no elements with square −4.

For the exceptional section of 𝔽_{1} with square −1, we denote by e its preimage on X. Then e^{2} = −2. We consider the contraction 𝔽_{1} → ℙ^{2} of the exceptional section and denote by h the preimage on X of a line l ⊂ ℙ^{2}. Then h^{2} = 2, h · e = 0. We have S = ℤh + ℤe.

Since θ = −1, we have S_{+} = {0}. Obviously, s = 2, p = 0.

Since Δ(S_{+}, S)^{(−4)} = Δ(S_{−}, S)^{(−4)} = ∅, by Lemma 9, G is trivial.

The group A_{S_{−}} is generated by h^{∗} = \frac{1}{2}h and e^{∗} = −\frac{1}{2}e, and hence A_{S_{−}} ≅ (ℤ/2ℤ)^{2} and l(A_{S_{−}}) = 2. The characteristic element of q_{S_{−}} is h + e (mod 2S).

We have A_{S_{+}} = 0, H_{+} = 0, ρ = 0. Since Γ_{−} = 0, H = H_{−} (⊂ S/2S) is one of the following 5 subgroups:

0, [h], [e], [h + e], S/2S = [h, e],

where we consider h, e mod 2S. Since H_{+} = 0, we have q_{ρ} = (−q_{S_{−}})|H_{−}. For the groups H we have:

(1) H = 0. Then a_{H} = 0, q_{ρ} = 0, δ_{H} = 0, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φS} = 0, then v = 0, equivalently, δ_{φ} = 0, and c_{v} ≡ 0 (mod 4).

(2) H = [h]. Then a_{H} = 1, q_{ρ} ≅ q_{−1}(2), δ_{H} = 1, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ −1 (mod 8). If δ_{φS} = 0, then v = h, and hence c_{v} ≡ −1 (mod 4).

Figure 19. 𝔽_{1}: All possible (r, a, δ_{φS}, v) with H = 0

[figure: means δ_{φS} = 0 and v = h + e,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]
Figure 20. 𝔽_{1}: All possible (r, a, δ_{φS}, v) with H = S/2S = [h, e]

(3) H = [e]. Then a_{H} = 1, q_{ρ} ≅ q_{1}(2), δ_{H} = 1, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 1 (mod 8). If δ_{φS} = 0, then v = e, and hence c_{v} ≡ 1 (mod 4).

(4) H = [h+e]. Then a_{H} = 1, q_{ρ} ≅ z, δ_{H} = 0, k_{ρ} = 1, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φS} = 0, then v = 0 or h + e, and hence c_{v} ≡ 0 (mod 4).

(5) H = S/2S. Then a_{H} = 2, q_{ρ} ≅ q_{1}(2)⊕q_{−1}(2), δ_{H} = 1, k_{ρ} = 0, μ_{ρ} = 0, σ_{ρ} ≡ 0 (mod 8). If δ_{φ} = 0, then c_{v} ≡ 0 (mod 4). If δ_{φS} = 0 and δ_{φ} = 1 (Type Ia), then we also have c_{v} ≡ 0 (mod 4) as we mention below.

From above we always have μ_{ρ} = 0. We see δ_{φS_{+}} = δ_{φ}. We also see δ_{φS_{−}} = 0 if and only if δ_{φS} = 0, v = h + e and H = [h + e] or S/2S. If δ_{φS_{+}} = δ_{φ} = 0, then we have ε_{v_{+}} ≡ 0 (mod 2). If δ_{φS_{−}} = 0, then we have c_{v} ≡ 0 (mod 4) (see (4) and (5) above), and hence γ_{−} = q_{1}(2) ⊕ q_{−1}(2). We see (q_{S_{−}})_{v} = u_{+}(2), and hence ε_{v_{−}} ≡ 0 (mod 2).

We see that the genus (hence, by Theorem 14 and Proposition 15, isomorphism class) of an integral involutions (L, φ, S) of the type (⟨2⟩⊕⟨−2⟩, −1) satisfying (RSK3) is determined by the data

(6.1)  (r, a, H, δ_{φS}, v),

where the v (∈ H) is defined only if δ_{φS} = 0. We remind that δ_{φ} = 0 if and only if δ_{φS} = v = 0.

By Theorem 10, the complete list of the above invariants is given by the conditions: Type 0 (δ_{φ} = 0) H = 0 or [h + e].

[figure: means δ_{φS} = 0 and v = h,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]

Figure 21. 𝔽_{1}: All possible (r, a, δ_{φS}, v) with H = [h]

[figure: means δ_{φS} = 0 and v = e,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]

Figure 22. 𝔽_{1}: All possible (r, a, δ_{φS}, v) with H = [e]

[figure: means δ_{φS} = 0 and v = 0,; means δ_{φS} = 0 and v = h + e,; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]

Figure 23. 𝔽_{1}: All possible (r, a, δ_{φS}, v) with H = [h + e]

r = 2, 6, 10, 14, 18. a ≡ 0 (mod 2). a ≤ r. r + a ≤ 2a_{H} + 18. If H = 0, then a ≥ 0. If H = [h + e], then a ≥ 2. If H = 0 and a = 0, then 2 − r ≡ 0 (mod 8). If H = [h + e] and a = 2, then 2 − r ≡ 0 (mod 8). If a = r, then 2 − r ≡ 0 (mod 8).

Type Ia (δ_{φS} = 0, δ_{φ} = 1) a_{H} ≥ 1. c_{v} ≡ σ_{ρ} (mod 4). Hence, c_{v} ≡ 0 (mod 4) if H = S/2S. If H = [h], then a ≥ 1, r = 3, 7, 11, 15 and a ≡ 1 (mod 2). If H = [e], then a ≥ 1, r = 1, 5, 9, 13, 17 and a ≡ 1 (mod 2). If H = [h + e], then a ≥ 2, r = 2, 6, 10, 14, 18 and a ≡ 0 (mod 2). If H = S/2S, then a ≥ 2, r = 2, 6, 10, 14, 18 and a ≡ 0 (mod 2). If H = [h] or H = [e] and a = 1, then 2 − r ≡ σ_{ρ} (mod 8). If H = [h + e] or H = S/2S and a = 2, then 2 − r ≡ 0 (mod 8). a ≤ r, r + a ≤ 2a_{H} + 18. If δ_{φS} = 0, v = h + e, H = [h + e] or S/2S and r + a = 2a_{H} + 18, then 2 − r ≡ 0 (mod 8).

Type Ib (δ_{φS} = 1) a ≥ a_{H} + k_{ρ} + 1. r + a ≡ 0 (mod 2). If a = a_{H} + k_{ρ} + 1, then 2 − r ≡ σ_{ρ} ± 1 (mod 8). If a = a_{H} + k_{ρ} + 2, then 2 − r ≢ σ_{ρ} + 4 (mod 8). 1 ≤ r ≤ 19, a ≤ r, r + a ≤ 2a_{H} + 18. If δ_{φS} = 0, v = h + e, H = [h + e] or S/2S and r + a = 2a_{H} + 18, then 2 − r ≡ 0 (mod 8).

In Type Ia case, if H = S/2S, then c_{v} ≡ 0 (mod 4). Thus v = h + e. Hence, H determines v in Type Ia case. Thus, by the above, the isomorphism class of an integral involutions (L, φ, S) of the type (⟨2⟩ ⊕ ⟨−2⟩, −1) satisfying (RSK3) is determined by the data

(r, a, H, δ_{φS}, δ_{φ}).

We give all these possible data in Figures 19 — 23.

By Theorem 11 about related involutions, we have

\begin{matrix}r(φ) + r(τφ) = 20, a(φ) − a_{H(φ)} = a(τφ) − a_{H(τφ)}, \\ a_{H(φ)} + a_{H(τφ)} = 2, H(τφ) = H(φ)^{⊥} w.r.t. b_{S_{−}}, \\ δ_{φS} = δ_{(τφ)S}, s_{φ} + s_{τφ} = h + e mod 2L. \end{matrix} ; (6.2)

Thus, integral involutions of Type 0 with H = 0 (12 classes) are related to involutions of Type Ia with H = S/2S. Involutions of Type 0 with H = [h + e] (10 classes) are related to involutions of Type Ia with H = [h + e]. Involutions of Type Ia with H = [h] (13 classes) are related to involutions of Type Ia with H = [e]. Involutions of Type Ib with H = 0 (39 classes) are related to involutions of Type Ib with H = S/2S. Involutions of Type Ib with H = [h] (39 classes) are related to involutions of Type Ib with H = [e]. Finally, the class (r, a, H = [h+e], Type Ib) is related to (20−r, a, H = [h+e], Type Ib). (There are 30 classes of Type Ib with H = [h + e].)

Moreover, if we identify related involutions, there are 35(= 12 + 10 + 13) classes with δ_{φS} = 0 and 95(= 39 + 39 + 17) classes with δ_{φS} = 1.

Let us consider the geometric interpretation of the above calculations. We denote by s the exceptional section of 𝔽_{1} with s^{2} = −1 and by c the fiber of the natural fibration π : 𝔽_{1} → ℙ^{1}. The contraction of s (as an exceptional curve of the first kind) gives the natural morphism p : 𝔽_{1} → ℙ^{2}, and we denote P = p(s). A non-singular curve A ∈ | − 2K_{𝔽_{1}}| gives then a curve A_{1} = p(A) of degree 6 in ℙ^{2} with only one singular point P which is a quadratic singular

Figure 24. 𝔽_{1}: A^{+} for H = [h, e] with (r, a, δ_{φS}, v) ≠ (10, 10, 0, h + e); here
α = g, β = k.

Figure 25. 𝔽_{1}: A^{+} for H = [h, e] with (r, a, δ_{φS}, v) = (10, 10, 0, h + e).

Figure 26. 𝔽_{1}: A^{+} for H = [h]; here α = g − 1, β = k.

point resolving by one blow-up. A small deformation of A or A_{1} (in the same connected component of moduli) makes P non-degenerate. Using Bezoute theorem, one can easily draw all in principle possible pictures of A_{1}. For example, one can find these pictures in Figure 1 of [10]. Lifting these pictures to 𝔽_{1} and using (2.44), (2.45) and Theorem 16 applied to both positive curves A^{+} and A^{−}, we get from Figures 19 — 23 all pictures of A^{+} on 𝔽_{1} up to isotopy. They are all presented in Figures 24—29 where we denote g = (22 − r − a)/2 and k = (r − a)/2, as usual. The interval AA denotes s(ℝ), the real part of the exceptional section s, and the interval BB denotes the p^{−1}(l(ℝ)) of a real projective line l ⊂ ℙ^{2} which does not contain P.

We have the following geometric interpretation of the invariants δ_{h} and δ_{e} (see (2.18)) of A^{+} if A^{+} ≠ ∅. We have δ_{h} = 0 (equivalently, h mod 2 ∈ H), if and only if homotopically

Figure 27. 𝔽_{1}: A^{+} for H = [h + e] with (r, a, δ_{φS}) ≠ (10, 8, 0), (10, 10, 0);
here α = g − 1, β = k.

Figure 28. 𝔽_{1}: A^{+} for H = [h + e] with (r, a, δ_{φS}, v) = (10, 8, 0, 0).

Figure 29. 𝔽_{1}: A^{+} = ∅ for H = [h + e] with (r, a, δ_{φS}, v) = (10, 10, 0, 0).

l(ℝ) ⊂ A^{−} (i. e. some deformation of l(ℝ) is contained in A^{−}). Similarly, δ_{e} = 0, if and only if homotopically s(ℝ) ⊂ A^{−}. The invariants δ_{h} and δ_{e} for both positive curves A^{+} and A^{−} are sufficient to find the group H.

Thus, we get the isotopy classification of real non-singular curves A ∈ | − 2K_{𝔽_{1}}|. In this classification we don’t care about position of A(ℝ) with respect to the real part s(ℝ) of the exceptional section (one can see that more delicate classification in [10]).

We have the following interpretation of the invariant δ_{φS}: one has δ_{φS} = 0, if and only if the curve A is dividing: A(ℝ) divides A(ℂ) in two connected parts or A(ℝ) = ∅. Equivalently, A(ℝ) = 0 in H_{1}(A(ℂ), ℤ/2). Like in previous cases (because we know that invariants (6.1) define the connected component of moduli), it would be enough to construct a dividing curve A on 𝔽_{1} in each case when δ_{φS} = 0, and it should follow from known results about real curves of degree 6 (e. g. see [9] and [10]). Another general argument is to apply Donaldson’s trick [5], like in [4]. We can consider φ as holomorphic involution and τ as anti-holomorphic. Then the statement follows from Theorem 16. The same argument works in all previous cases, but we did not need it. Thus, we finally get

Theorem 25. A connected component of moduli of a positive real non-singular curve A ∈ |−2K_{𝔽_{1}}| is defined by the isotopy type of A^{+} ⊂ 𝔽_{1}(ℝ) and by the divideness by A(ℝ) of A(ℂ) (equivalently, by the invariant δ_{φS}). All these possibilities are presented in in the Figures 𝔽_{1} above and in (6.2).

7. Connected components of moduli of real non-singular curves in | − 2K_{𝔽_{4}}|

Here we consider the case S ≅ U and θ = −1.

Like in previous cases, we can prove that Y = X/{1, τ} = 𝔽_{4} in this case. This case is (D)-non-degenerate because S has no roots with square −4.

We have S_{+} = {0} and S_{−} = S, s = 2, p = 0.

Since Δ(S_{+}, S)^{(−4)} = Δ(S_{−}, S)^{(−4)} = ∅, by Lemma 9 the group G is trivial.

Since U is unimodular, A_{S_{−}} = {0}. Obviously, A_{S_{+}} = {0}. Hence, H = H_{+} = H_{−} = 0, ρ = 0 and δ_{φS} = δ_{φ}.

By Theorem 14 and Proposition 15, the genus defines the isomorphism class of an integral involutions (L, φ, S) of the type (U, −1) satisfying (RSK3), and it defines the connected component of moduli. It is determined by the data

(7.1)  (r, a, δ_{φS} = δ_{φ}, v)

where v = 0 if δ_{φS} = 0 (otherwise, v is not defined). The complete list of the invariants (7.1) is given by the conditions:

δ_{φ} = 0 case r = 2, 6, 10, 14, 18, a ≡ 0 (mod 2), a ≤ r, r + a ≤ 20. If a = 0, a = r or r + a = 20, then 2 − r ≡ 0 (mod 8).

δ_{φ} = 1 case a ≥ 1, r + a ≡ 0 (mod 2), 1 ≤ r ≤ 19, a ≤ r, r + a ≤ 20. If a = 1, then 2 − r ≡ ±1 (mod 8). If a = 2, then 2 − r ≢ 4 (mod 8).

All possible data (7.1) are given on Figure 30. There are 14 isomorphism classes with δ_{φ} = 0 and 49 isomorphism classes with δ_{φ} = 1.

By Theorem 11, we have for related involutions

(7.2)  r(φ) + r(τφ) = 20, a(φ) = a(τφ) and δ_{φS} = δ_{τφS}, s_{φ} = s_{τφ}.

It corresponds to the symmetry with respect to the line r = 10 of Figure 30. Thus, if we identify related involutions, there are 10 isomorphism classes with δ_{φ} = 0 and 27 isomorphism classes with δ_{φ} = 1.

Now let us consider the geometric interpretation of the above results. Denote by s the exceptional rational section with s^{2} = −4 of 𝔽 = 𝔽_{4}, and by c the fiber of the natural

[figure: means δ_{φS} = 0 and v = 0; means δ_{φS} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; a; r]
Figure 30. 𝔽_{4}: All possible (r, a, δ_{φS}, v) (here H = 0)

Figure 31. 𝔽_{4}: A^{+} with (r, a, δ_{φS}, v) ≠ (10, 8, 0, 0), (10, 10, 0, 0).

fibration f : 𝔽_{4} → s. One has c^{2} = 0 and p_{g}(c) = 0. We have −2K_{𝔽} = 12c + 4s. Thus, for A ∈ | − 2K_{𝔽}|, one has A · s = −4. It follows A = s + A_{1} where A_{1} ∈ |12c + 3c|. We have c · A_{1} = 3 and s · A_{1} = 0. It follows that a non-singular A ∈ | − 2K_{𝔽}| has two irreducible components: A = s + A_{1}. Thus, we are describing connected components of moduli of non-singular curves A_{1} in the linear system |12c + 3s|. Any such A_{1} gives a non-singular A = s + A_{1} from | − 2K_{𝔽}|.

We mention that the lattice S ≅ U is ℤC + ℤE where C = π^{∗}(c) and E = π^{∗}(s)/2. We have C^{2} = 0, E^{2} = −2 and C · E = 1. As usual, we denote π : X → Y = 𝔽_{4} the quotient morphism.

If 𝔽(ℝ) is not empty, then 𝔽(ℝ) is a torus and s(ℝ) is a circle giving a generator of the torus. The curves c(ℝ) where f(c) ∈ s(ℝ) give another generator of the torus. It follows that a real curve A_{1}(ℝ) belongs to the open cylinder 𝔽(ℝ) − s(ℝ) with its infinity identified with two copies of s(ℝ).

Using c · A_{1} = 3 and (2.44), (2.45) for both involutions φ and τφ, we get at once that a positive curve A^{+} with the invariants (7.1) has the isotopy type given in Figures 31 and 32. In Figures 31 and 32 we assume that (r, a, δ_{φ}) ≠ (10, 10, 0). Then 𝔽(ℝ) is a torus. If (r, a, δ_{φ}) = (10, 10, 0), then 𝔽(ℝ) is empty. As usual, g = (22 − r − a)/2 and k = (r − a)/2.

We have the following interpretation of the invariant δ_{φ}: one has δ_{φ} = 0, if and only if the curve A is dividing: A(ℝ) divides A(ℂ) in two connected parts or A(ℝ) = ∅. Equivalently, A(ℝ) = 0 in H_{1}(A(ℂ), ℤ/2). Like in previous cases (because we know that invariants (7.1)

Figure 32. 𝔽_{4}: A^{+} with (r, a, δ_{φS}, v) = (10, 8, 0, 0) (𝔽_{4}(ℝ) = ∅ if
(r, a, δ_{φS}, v) = (10, 10, 0, 0)).

define the connected component of moduli), it would be enough to construct a dividing curve A on 𝔽_{4} in each case when δ_{φ} = 0. Unfortunately, this construction is not known (surely, it is not a hard problem). Thus, we have to find a general argument for divideness. Applying Donaldson’s trick [5], like in [4], we can consider φ as holomorphic involution and τ as anti-holomorphic. Then the statement follows from Theorem 16.

Thus, finally, we get

Theorem 26. The connected component of moduli of positive real non-singular curves A^{+}, A ∈ | − 2K_{𝔽_{4}}| up to the action of the automorphism group of 𝔽_{4} is defined by the isotopy type and by divideness of A(ℂ) by A(ℝ). All these possibilities are presented in Figures 𝔽_{4} above.

We remark that the corresponding real K3 surfaces (X, τ, φ) give an important class of K3 surfaces. They are elliptic K3 surfaces (preimages of c) with section (the preimage of s). The holomorphic involution τ gives the inverse map of the elliptic pencil.

8. Application: Real polarized K3 surfaces as deformations of general K3 double rational scrolls

Consider a non-degenerate real K3 surfaces (X, τ, φ) with a non-symplectic involution of the type (S, θ) where (S, θ) is of one of 5 types with rk S ≤ 2 described in Sect. 2.4. Assume that P ∈ S is a primitive nef element with P^{2} = n, n ∈ ℕ is even, and such that θ(P) = −P. Moreover, assume that the complete linear system |P | or |2P | gives the quotient map π : X → Y = X/{1, τ}. By standard results about linear systems on K3 surfaces, e. g. see [27], it is valid in the following and only the following cases:

The Case of ℙ^{2}, i. e. S = ⟨2⟩ and θ = −1: Then P = h and P^{2} = n = 2, both |P | and |2P | give the quotient map π : X → Y = ℙ^{2}. Thus, n = 2.

The Case of ellipsoid 𝔼, i. e. S = U(2) and θ(e_{1}) = −e_{2}: Then P = e_{1} + e_{2}, P^{2} = n = 4 and |P | gives the quotient map π to the ellipsoid. Thus, n = 4.

The Case of hyperboloid ℍ, i. e. S = U(2) and θ = −1: Then either P = n_{1}e_{1} + e_{2} (we denote this case as ℍ_{(1)}) or P = e_{1} + n_{1}e_{2} (we denote this case as ℍ_{(2)}), and P^{2} = n = 4n_{1}, n_{1} ≥ 1. Then |P | gives the quotient map π to the hyperboloid. Thus, n = 4n_{1} ≡ 0 mod 4 and n ≥ 4.

The Case of 𝔽_{1}, i. e. S = ⟨2⟩ ⊕ ⟨−2⟩ and θ = −1. We have P = n_{1}c + e = n_{1}h + e(1 − n_{1}) where c = h − e and P^{2} = n = 4n_{1} − 2, n_{1} ≥ 2. Then |P | gives the quotient map to 𝔽_{1}. We have n ≡ 2 mod 4 and n ≥ 6.

The case of 𝔽_{4}, i. e. S = U and θ = −1. We denote C = π^{∗}(c) and E = π^{∗}(s)/2 where C^{2} = 0, E^{2} = −2 and C · E = 1. We have two possibilities:

The case (𝔽_{4})^{(1)}: P = n_{1}C + E where n_{1} ≥ 3 and P^{2} = n = 2n_{1} − 2, then |2P | gives the quotient map and |P | gives the composition of π with the natural fibration 𝔽_{4} → ℙ^{1}. We have n ≡ 0 mod 2 and n ≥ 4 for this case

The case (𝔽_{4})^{(2)}: P = n_{1}C + 2E where n_{1} ≥ 5 is odd and P^{2} = n = 4n_{1} − 8, then |P | gives the quotient map π. We have n ≡ 4 mod 8 and n ≥ 12 for this case.

For each of these types:

(8.1)  T = {ℙ^{2}, 𝔼, ℍ_{(1)}, ℍ_{(2)}, 𝔽_{1}, (𝔽_{4})^{(1)}, (𝔽_{4})^{(2)}}

the element P is uniquely defined by n = P^{2}, if it exists. If for t ∈ T such P does exist we write n ∈ t. For t ∈ T we denote by π_{0}(M_{t}) the set of connected components of moduli of non-degenerate (X, τ, ϕ) of the type t ∈ T which we had described in Sects 3 — 7.

Since the branch curve is non-singular in all these cases, the P is ample. In all these cases |3P | gives an embedding of X to a projective space. If n ≥ 4, a small deformation of the pair (X, P) gives a K3 surface with the very ample P, i. e. |P | : X ⊂ ℙ^{n/2+1}.

We denote by M_{n,k} where n, k ∈ ℕ and n is even, the moduli space of real polarized K3 surfaces (X, kP) with the polarization kP where P is primitive and P^{2} = n, and we require that the linear system |kP | gives an embedding |kP | : X ⊂ ℙ^{k^{2}n/2+1}, i. e. kP is very ample. Thus, in all cases above, for k ≥ 3 the correspondence (X, τ, ϕ) ↦ (X, kP) gives an embedding of the moduli of (X, τ, φ) to the moduli M_{n,k}, k ≥ 3, of real polarized K3 surfaces (X, kP) with the polarization kP of the degree nk^{2} where P is primitive. Thus, for k ≥ 3 we get the natural deformation map

(8.2)  D_{n} : ⋃_{t∈T : n∈t} π_{0}(M_{t}) → π_{0}(M_{n,k}).

A polarized K3 surface (X, kP) which belongs to a connected component of moduli from the image of D can be obtained by a deformation inside M_{n,k} from a real K3 surface which is a general double covering (i. e. with a non-singular branch curve) of a rational scroll. All these scrolls and the types of their double coverings are given by the types t ∈ T such that n ∈ t. We call such K3 surfaces as general K3 double rational scrolls.

One of consequences of Theorem 3.10.1 in [19] (we shall discuss it in details later) is that the natural embedding

(8.3)  M_{n} = M_{n,1} → M_{n,k}, (X, P) ↦ (X, kP),

gives the isomorphism of the set of connected components of M_{n} = M_{n,1} and the set of connected components of M_{n,k} for k ≥ 2. A difference between M_{n} and M_{n,k} for k ≥ 2 is only on the boundary of M_{n} in M_{n,k}, and this boundary (in spite of it has the real codimension one in M_{n,k}) does not divide connected components of M_{n,k}, k ≥ 2.

Thus, the deformation map D_{n} in (8.2) is defined for all k ≥ 1. The image of D gives connected components of moduli of real polarized K3 surfaces (X, kP) for any k ≥ 1, which can be obtained as a deformation of a general real K3 double rational scroll. Using results of previous sections and [19], the deformation map D_{n} can be computed explicitly and gives all real polarized K3 surfaces (X, kP), k ≥ 1, which are deformations of general real K3 double rational scrolls. Moreover, we can even enumerate (classify) all types of these deformations.

A polarized real K3 surface (X, cong, kP), where P is primitive and k ≥ 1, defines an integral involution with a condition (L = H_{2}(X, ℤ), φ = cong_{∗}, S_{n} = [P]) where S_{n} = [P] = ℤP ≅ ⟨n⟩ and θ = ϕ|S_{n} = −1. Theorem 3.10.1 in [19] gives that the connected component of moduli of (X, cong, kP) is defined by the isomorphism class of its integral involution (L, φ, S_{n}) with the condition (S_{n}, −1). All invariants of the genus of (L, φ, S_{n}) are given by the invariants

(8.4)  (n; r, a, δ_{P} , δ_{φ}, δ_{φP})

which are the same as in (2.20). All possibilities for the genus invariants (8.4) are given in Theorem 3.4.3 from [19] where one should put l_{(+)} = 3, l_{(−)} = 19, t_{(+)} = 1 and t_{(−)} = r − 1. The genus (8.4) defines the class (Theorem 3.3.1 in [19]), if r ≤ 18 − δ_{P φ} (actually, similarly to Theorem 14 one can give much better sufficient conditions).

Assume that the real K3 surface (X, τ, φ) has the integral involution (L, φ, S) of the type (S, θ), and (S, θ) and P ∈ S is of one of types t ∈ T from (8.1) such that n ∈ t. Assume that (L, φ, S) has the invariants (2.21) which are

(8.5)  (r, a, H_{+}, H_{−}, δ_{φS}, v) where δ_{φ} = \{ \begin{matrix}0 & if (δ_{φS}, v) = (0, 0) \\ 1 & otherwise \end{matrix}

(we don’t need q_{ρ}, if rk S ≤ 2). Then the integral involution of the polarized K3 (X, P) is equal to (L, φ, S_{n} = ℤP) where S_{n} = ℤP ⊂ S is the sublattice generated by P. It follows that the invariants (r, a, δ_{φ}) of (X, P) in (8.4) are same as in (8.5). We have

(8.6)  δ_{P} = \{ \begin{matrix}0 & if P mod 2S_{−} ∈ H_{−} \\ 1 & otherwise \end{matrix} ,

and

(8.7)  δ_{φP} = \{ \begin{matrix}0 & if P ≡ v mod 2S_{−} \\ 1 & otherwise \end{matrix} .

Symbolically we write in this case

(8.8)  D_{n} : (t; r, a, H_{+}, H_{−}, δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ}, δ_{φP}).

Applying results of previous Sections (especially Theorem 1), we get

Theorem 27. Let us fix a type t ∈ T of a rational scroll. There exists a real polarized K3 surface (X, kP) with the invariants

(n; r, a, δ_{P} , δ_{φ}, δ_{φP})

which is a deformation of a general real K3 double rational scroll t, if and only if n ∈ t and

D_{n} : (t; r, a, H_{+}, H_{−}, δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ}, δ_{φP}).

All these polarized K3 surfaces (X, kP) belong to one connected component of moduli of real polarized K3 surfaces.

Thus, to find all real polarized K3 surfaces which are deformations of general real K3 double rational scrolls, we need to find out when one has (8.8). These calculations are routine, and we only formulate the result. Using notations of Sects. 3 — 7, we get:

(8.9)  Types of deformations of general real K3 double rational scrolls:

The Case of ℙ^{2}: n ∈ ℙ^{2}, if and only if n = 2; P = h;

D_{2} : (ℙ^{2}; r, a, H = 0, δ_{φS}, v) ⟹ (2; r, a, δ_{P} = 1, δ_{φ} = δ_{φS}, δ_{φP} = 1);
D_{2} : (ℙ^{2}; r, a, H = [h], δ_{φS}, v) ⟹ (2; r, a, δ_{P} = 0, δ_{φ} = 1, δ_{φP} = δ_{φS}).

See Figures 2 and 3 for the list of all possible invariants (ℙ^{2}; r, a, H, δ_{φS}, v).

The Case of ellipsoid 𝔼: n ∈ 𝔼, if and inly if n = 4; P = e_{1} + e_{2};

D_{4} : (𝔼; r, a, H = [e_{1} + e_{2}], δ_{φS}, v) ⟹ (4; r, a, δ_{P} = 0, δ_{φ}, δ_{φP}) where

δ_{φP} = \{ \begin{matrix}0 & if (δ_{φS}, v) = (0, e_{1} + e_{2}) \\ 1 & otherwise \end{matrix} .

See Figure 16 for the list of all possible invariants (𝔼; r, a, H = [e_{1} + e_{2}], δ_{φS}, v).

The Case of hyperboloid ℍ_{(1)}. n ∈ ℍ_{(1)}, if and only if n ≡ 0 mod 4 and n ≥ 4; P = (n/4)e_{1} + e_{2};

D_{n} : (ℍ_{(1)}; r, a, H = 0, δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ} = δ_{φS}, δ_{φP} = 1);
D_{n} : (ℍ_{(1)}; r, a, H = [e_{1}, e_{2}], δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 0, δ_{φ} = δ_{φS}, δ_{φP} = 1);
D_{n} : (ℍ_{(1)}; r, a, H = [e_{1}], δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ}, δ_{φP} = 1);
D_{n} : (ℍ_{(1)}; r, a, H = [e_{2}], δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ}, δ_{φP}) where

δ_{P} = \{ \begin{matrix}0 & if n ≡ 0 mod 8 \\ 1 & otherwise \end{matrix} , δ_{φP} = \{ \begin{matrix}0 & if (n mod 8, δ_{φS}, v) = (0 mod 8, 0, e_{2}) \\ 1 & otherwise \end{matrix} ;

D_{n} : (ℍ_{(1)}; r, a, H = [e_{1} + e_{2}] mod 2, δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ}, δ_{φP}) where

δ_{P} = \{ \begin{matrix}0 & if n ≡ 4 mod 8 \\ 1 & otherwise \end{matrix} , δ_{φP} = \{ \begin{matrix}0 & if (n mod 8, δ_{φS}, v) = (4 mod 8, 0, e_{1} + e_{2}) \\ 1 & otherwise \end{matrix} .

See figures 6 — 9 for the list of all possible invariants (ℍ; r, a, H, δ_{φS}, v). The Case of hyperboloid ℍ_{(2)}: Change e_{1} and e_{2} places in the previous case. The Case of 𝔽_{1}: n ∈ 𝔽_{1}, if and only if n ≡ 2 mod 4 and n ≥ 6; P = ((n + 2)/4) h +

((2 − n)/4) e;

D_{n} : (𝔽_{1}; r, a, H = 0, δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ} = δ_{φS}, δ_{φP} = 1);
D_{n} : (𝔽_{1}; r, a, H = [h, e], δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 0, δ_{φ} = 1, δ_{φP} = 1);
D_{n} : (𝔽_{1}; r, a, H = [h], δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ} = 1, δ_{φP}) where

δ_{P} = \{ \begin{matrix}0 & if n ≡ 2 mod 8 \\ 1 & otherwise \end{matrix} , δ_{φP} = \{ \begin{matrix}0 & if (n mod 8, δ_{φS}) = (2 mod 8, 0) \\ 1 & otherwise \end{matrix} ;

D_{n} : (𝔽_{1}; r, a, H = [e], δ_{φS}, v) ⟹ (n; r, a, δ_{P} , δ_{φ} = 1, δ_{φP}) where

δ_{P} = \{ \begin{matrix}0 & if n ≡ −2 mod 8 \\ 1 & otherwise \end{matrix} , δ_{φP} = \{ \begin{matrix}0 & if (n mod 8, δ_{φS}) = (−2 mod 8, 0) \\ 1 & otherwise \end{matrix} ;

D_{n} : (𝔽_{1}; r, a, H = [h + e], δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ}, δ_{φP} = 1).

See the list of all possible invariants (𝔽_{1}; r, a, H, δ_{φS}, v) in figures 19 — 23.

The Case of (𝔽_{4})^{(1)}: n ∈ (𝔽_{4})^{(1)}, if and only if n ≡ 0 mod 2 and n ≥ 4; P = (n/2+1)C+E;

D_{n} : ((𝔽_{4})^{(1)}; r, a, H = 0, δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ} = δ_{φS}, δ_{φP} = 1). The Case of (𝔽_{4})^{(2)}: n ∈ (𝔽_{4})^{(2)}, if and only if n ≡ 4 mod 8 and n ≥ 12;

P = (n/4 + 2)C + 2E;

D_{n} : ((𝔽_{4})^{(2)}; r, a, H = 0, δ_{φS}, v) ⟹ (n; r, a, δ_{P} = 1, δ_{φ} = δ_{φS}, δ_{φP} = 1). See the figure 30 for the list of all possible invariants (𝔽_{4}; r, a, H, δ_{φS}, v). We can apply this calculations to answer the following interesting question: What are real

K3 surfaces which are deformations of general real K3 double rational scrolls?

We have

Proposition 28. A real polarized K3 surface (X, kP) where P is primitive and n = P^{2} ≥ 6 cannot be a deformation of a general real K3 double rational scroll, if either (r, a) = (20, 2) (equivalently X(ℝ) = (T_{0})^{10}) or r + a = 22 and δ_{φP} = 0 (equivalently, X(ℝ) = (T_{0})^{m} and X(ℝ) ∼ P mod 2 in H_{2}(X, ℤ)).

Proof. If r = 20, then L_{φ} = L_{−} is positive definite or the rank 2. It cannot contain S_{−} of the rank two because S_{−} is hyperbolic. Thus, r = 20 may happen only for ℙ^{2} when n = 2, and for ellipsoid when n = 4. From the list of possible invariants (r, a, δ_{φ}) of real K3 surfaces (see Figure 1) we get that a = 2 and δ_{φ} = 1, if r = 20. By (2.44), this is equivalent to X(ℝ) = T_{0}^{10}.

Assume that r + a = 22 and n ≥ 6. This is possible only for (ℍ; H = [e_{1}, e_{2}]) and (𝔽_{1}; H = [h, e]), but all these cases give δ_{φP} = 1 by deformations (8.9). By (2.44), r +a = 22 is equivalent to X(ℝ) = (T_{0})^{m}, and δ_{φP} = 0 is equivalent to X(ℝ) ∼ P mod 2 in H_{2}(X, ℤ), by (2.45).

This proves the statement

We have

Theorem 29. A genus invariant (n; r, a, δ_{P} , δ_{φ}, δ_{φP}) of a real polarized K3 surface (X, kP) can be obtained by the deformation D_{n} of some general real K3 double rational scroll t ∈ T from (8.1), if and only if the following condition is valid:

(8.10)  n ≤ 4, if either (r, a) = (20, 2) or r + a = 22 and δ_{φP} = 0

(we just exclude cases of Proposition 28). All possible types of these deformations are given in (8.9).

Proof. Using Theorem 3.4.3 from [19], we can get the list of all possible invariants (n; r, a, δ_{φ}, δ_{φP}) of real polarized K3 when (r, a) ≠ (20, 2) and r + a < 22, if δ_{φP} = 0. They depend on n mod 8 and are presented in figures 33 — 39 (when r + a = 22 and δ_{φP} = 0, the result of [19] is more complicated: it depends on n mod 16 when (r, a) ≠ (20, 2); if (r, a) = (20, 2), it depends even on prime decomposition of n). By inspection (which requires some time and patience), one can see that all these invariants can be obtained applying all possible deformations (8.9) from ℙ^{2}, ℍ, 𝔽_{1} and 𝔽_{4}. Again applying Theorem 3.4.3 from [19] when n ≤ 4 and either (r, a) = (20, 2) or r+a = 22 and δ_{φP} = 0, we get all of them as deformations of ℙ^{2} and 𝔼. The list of all possible invariants for n = 4 is given in figures 36 and 41.

It proves the statement.

As a particular result, we get that any non-singular real quartic in ℙ^{3} can be obtained as a deformation of a general real double K3 cover of ellipsoid, hyperboloid or 𝔽_{4}. These deformations can be given explicitly [26] and give an effective method of construction of all (i. e. representatives of all connected components of moduli) examples of real quartics in ℙ^{3}.

References

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[figure: means δ_{φ} = 0 and δ_{φP} = 1,; means δ_{φP} = 0 and δ_{φ} = 1,; means δ_{φ} = δ_{φP} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 33. Polarized K3 when n ≡ 0 mod 8 and δ_{P} = 0 without (r, a) = (20, 2) and (r + a = 22, δ_{φP} = 0)

[figure: means δ_{φ} = 0,; means δ_{φ} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 34. Polarized K3 when n ≡ 0 mod 8 and δ_{P} = 1 (then δ_{φP} = 1)

[figure: means δ_{φ} = 0 and δ_{φP} = 1,; means δ_{φP} = 0 and δ_{φ} = 1,; means δ_{φ} = δ_{φP} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 35. Polarized K3 when n ≡ 4 mod 8 and δ_{P} = 0 without (r, a) = (20, 2) and (r + a = 22, δ_{φP} = 0)

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[figure: means δ_{φ} = 0,; means δ_{φ} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 36. Polarized K3 when n ≡ 4 mod 8 and δ_{P} = 1 (then δ_{φP} = 1)

[figure: means δ_{φP} = 0 and δ_{φ} = 1,; means δ_{φ} = δ_{φP} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 37. Polarized K3 when n ≡ 2 mod 8 and δ_{P} = 0 (then δ_{φ} = 1)

without (r, a) = (20, 2) and (r + a = 22, δ_{φP} = 0)

[figure: means δ_{φ} = 0,; means δ_{φ} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 38. Polarized K3 when n ≡ 2 mod 8 and δ_{P} = 1 (then δ_{φP} = 1)

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[figure: means δ_{φP} = 0 and δ_{φ} = 1,; means δ_{φ} = δ_{φP} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 39. Polarized K3 when n ≡ −2 mod 8 and δ_{P} = 0 (then δ_{φ} = 1)

without (r, a) = (20, 2) and (r + a = 22, δ_{φP} = 0)

[figure: means δ_{φ} = 0,; means δ_{φ} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 40. Polarized K3 when n ≡ −2 mod 8 and δ_{P} = 1 (then δ_{φP} = 1)

[figure: means δ_{φ} = 0 and δ_{φP} = 1,; means δ_{φP} = 0 and δ_{φ} = 1,; means δ_{φ} = δ_{φP} = 1; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; a; r]

Figure 41. Polarized K3 when n = 4 and δ_{P} = 0

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Department of Pure Mathematics, The University of Liverpool, Liverpool, L69 3BX, United Kingdom, Steklov Mathematical Institute, ul. Gubkina 8, Moscow, GSP-1, Russia

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

Department of Mathematics Education, Hakodate Campus, Hokkaido University of Education, 1-2 Hachiman-cho, Hakodate 040-8567, Japan

  E-mail address: sachi63@cc.hokkyodai.ac.jp