Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 341–356

ON THE CONNECTEDNESS OF THE LOCUS OF REAL RIEMANN SURFACES

Antonio F. Costa and Milagros Izquierdo

Universidad Nacional de Educacion a Distancia Depto. Matemáticas Fundamentales, Facultad Ciencias C/ Senda del Rey 9, E-28040 Madrid, Spain; acosta@mat.uned.es

Linköpings Universitet, Matematiska Institutionen S-581 83 Linköping, Sweden; miizq@mai.liu.se, mizquier@mat.uned.es

Abstract. It is well known that the functorial equivalence between pairs (X, σ), where X is a Riemann surface which admits an antiholomorphic involution (symmetry) σ: X → X, and real algebraic curves. We shall refer to such Riemann surfaces as real Riemann surfaces, following Klein’s terminology. We consider the sets ℳ_{g}^{ℛ} and ℳ_{g}^{2ℛ} of real curves and real hyperelliptic curves, respectively in the moduli space ℳ_{g} of complex algebraic curves of genus g.

In this paper we prove that any real hyperelliptic Riemann surface can be quasiconformally deformed, preserving the real and hyperelliptic character, to a real hyperelliptic Riemann surface (X, σ), such that X admits a symmetry τ, where Fix(τ) is connected and non-separating. As a consequence, we obtain the connectedness of the sets ℳ_{g}^{2ℛ} (⊂ ℳ_{g}) of all real hyperelliptic Riemann surfaces of genus g and ℳ_{g}^{ℛ} (⊂ ℳ_{g}) of all real Riemann surfaces of given genus g using a procedure different from the one given by Seppälä for ℳ_{g}^{2ℛ} and Buser, Seppälä and Silhol for ℳ_{g}^{ℛ} .

A Riemann surface X is called a p-gonal Riemann surface, where p is a prime, if there exists a p-fold covering map from X onto the Riemann sphere. We prove in this paper that the subset of real p-gonal Riemann surfaces, p ≥ 3, is not a connected subset of ℳ_{g} in general. This generalizes a result of Gross and Harris for real trigonal algebraic curves.

1. Introduction

Let X_{g} be a compact Riemann surface of genus g ≥ 2. A symmetry σ of X_{g} is an anticonformal involution of X_{g}. The topological type of a symmetry is determined by properties of its fixed-point set Fix(σ). By Harnack’s theorem the fixed-point set of σ consists of k ≤ g + 1 Jordan curves, called ovals. The space X_{g} − Fix(σ) consists of one component if the quotient Klein surface X_{g}/⟨σ⟩ is non-orientable and of two components if it is orientable. Let σ be a symmetry

2000 Mathematics Subject Classification: Primary 14H15; Secondary 30F10.

The first author is partially supported by DGICYT PB 99-0017. The second author is partially supported by the Swedish Natural Science Research Council (NFR). The second author thanks UNED for its hospitality.

of X_{g} with k ovals, then the species of σ is +k or −k according to whether X_{g} − Fix(σ) has two or one component, respectively.

Whereas a compact Riemann surface corresponds to a complex algebraic curve, a compact Riemann surface X_{g} with a symmetry σ corresponds to a real algebraic curve. Each conjugacy class of symmetries in Aut(X_{g}) corresponds to an equivalence class, under real birational isomorphisms, of real algebraic curves, a real form of the complex algebraic curve. The ovals of the symmetry correspond to the graph components of the real form. So, if a Riemann surface X_{g} admits two non-conjugate symmetries σ_{1} , σ_{2} with k_{1} and k_{2} ovals, respectively, then the complex algebraic curve corresponding to X_{g} has two real forms with k_{1} and k_{2} components, respectively.

A Riemann surface X_{g} is called a cyclic p-gonal Riemann surface, where p is a prime, if X_{g} is a cyclic p-fold covering of the Riemann sphere. When p = 2 the surface X_{g} is called hyperelliptic.

We study in this paper the sets ℳ_{g}^{pℛ} of complex isomorphism classes of real cyclic p-gonal curves of genus g by means of their uniformization groups. The study of moduli spaces of real algebraic curves was initiated by Felix Klein [21]. Seppälä [23] proved that ℳ_{g}^{2ℛ} is connected and Buser, Seppälä and Silhol [8] proved that ℳ_{g}^{ℛ} is connected, using the fact that ℳ_{g}^{2ℛ} is a connected subset of ℳ_{g}. There is another proof of this fact in [12] with the techniques described in [13]. We also give, for the sake of completeness, a proof of the connectedness of ℳ_{g}^{ℛ} different from the ones given in [8] and in [12], and following the ideas in [10].

Let ℳ_{εk}^{g} be the subset of ℳ_{g}^{ℛ} formed by all Riemann surfaces admitting a symmetry with species εk, where ε = ± and k is the number of ovals. The spaces ℳ_{εk}^{g} and ℳ_{εk}^{g} ∩ ℳ_{g}^{2ℛ} are connected (see [11], [8] and Theorems 2.1 and 2.2). In this paper we prove not only that ℳ_{g}^{2ℛ} is connected, but that ℳ_{−1}^{g} ∩ ℳ_{g}^{2ℛ} , the subset formed by all real hyperelliptic Riemann surfaces admitting a nonseparating symmetry with one oval, cuts every ℳ_{εk}^{g}∩ℳ_{g}^{2ℛ} for any possible species εk for a given genus g. We shall say that ℳ_{−1}^{g} ∩ ℳ_{g}^{2ℛ} is a spine for ℳ_{g}^{2ℛ} . The above property not only implies the connectedness of ℳ_{g}^{2ℛ} , but also gives a way to connect any pair of points in ℳ_{g}^{2ℛ} . In the same way we show that ℳ_{−1}^{g} is a spine for the space ℳ_{g}^{ℛ} (see [10]).

The above result has been inspired by the following fact on elliptic curves: the set of real elliptic curves defined by rombic lattices, i.e., admitting a symmetry with one nonseparating oval, is a spine for the locus of real elliptic curves (see [20]).

On the contrary, we shall prove that the set ℳ_{g}^{pℛ} , p ≥ 3, of real cyclic pgonal Riemann surfaces is not connected in general. This generalizes a result of Gross and Harris for real, trigonal algebraic curves [16]. As a consequence the set of real p-gonal Riemann surfaces is not connected using Lemma 2.1 in [1].

The results presented in this work imply the following fact on equations of algebraic curves. Given two algebraic curves admitting polynomial equations y^{p} = P (x) and y^{p} = Q(x) with real coefficients, we shall consider two types of allowed deformations for such equations. The first type of deformation is to modify continuously the real coefficients of P (x) and Q(x). The second type is to change the real form of a fixed complex algebraic curve. Then if p = 2 it is always possible to go from a curve to the other one, but this is not the case in general if p > 2.

2. NEC groups and moduli spaces of Riemann surfaces

Let X_{g} be a compact Riemann surface of genus g ≥ 2. The surface X_{g} can be represented as a quotient X_{g} = ℋ /Γ of the upper half plane ℋ under the action of a surface Fuchsian group Γ, that is, a cocompact orientation-preserving subgroup of the group 𝒢 = Aut(ℋ ) of conformal and anticonformal automorphisms of ℋ without elliptic elements. A discrete, cocompact subgroup Γ of Aut(ℋ ) is called an NEC (non-euclidean crystallographic) group. The subgroup of Γ consisting of the orientation-preserving elements is called the canonical Fuchsian subgroup of Γ; it is denoted by Γ^{+} . The algebraic structure of an NEC group and the geometric structure of its quotient orbifold are given by the signature of Γ:

(2.1)  s(Γ) = (h; ±; [m_{1}, …, m_{r}]; {(n_{11}, …, n_{1s_{1}}), …, (n_{k1}, …, n_{ks_{k}})}).

The orbit space ℋ /Γ is an orbifold with underlying surface of genus h, having r cone points and k boundary components, each with s_{j} ≥ 0 corner points. The signs “+” and “−” correspond to orientable and non-orientable orbifolds, respectively. The integers m_{i} are called the proper periods of Γ and they are the orders of the cone points of ℋ /Γ. The brackets (n_{i1}, …, n_{is_{i}}) are the period cycles of Γ and the integers n_{ij} are the link periods of Γ and the orders of the corner points of ℋ /Γ. The group Γ is called the fundamental group of the orbifold ℋ /Γ.

A group Γ with signature (2.1) has a canonical presentation with generators:

x_{1}, …, x_{r}, e_{1}, …, e_{k}, c_{ij}, 1 ≤ i ≤ k, 1 ≤ j ≤ s_{i} + 1,

and a_{1}, b_{1}, …, a_{h}, b_{h} if ℋ /Γ is orientable or d_{1}, …, d_{h} otherwise, and relators:

x_{i}^{m_{i}}, i = 1, …, r, c_{ij}^{2}, (c_{ij−1}c_{ij})^{n_{ij}}, c_{i0}e_{i}^{−1}c_{is_{i}}e_{i}, i = 1, …, k, j = 2, …, s_{i} + 1,

and x_{1} ⋯ x_{r}e_{1} ⋯ e_{k}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{h}b_{h}a_{h}^{−1}b_{h}^{−1} or x_{1} ⋯ x_{r}e_{1} ⋯ e_{k}d_{1}^{2} ⋯ d_{h}^{2} , according to whether ℋ /Γ is orientable or not.

This last relation is called the long relation.

The hyperbolic area of the orbifold ℋ /Γ coincides with the hyperbolic area of an arbitrary fundamental region of Γ and equals:

(2.2)  μ(Γ) = 2π(εh − 2 + k + ∑_{i=1}^{r} (1 − \frac{1}{m_{i}} ) + \frac{1}{2} ∑_{i=1}^{k} ∑_{j=1}^{s_{i}} (1 − \frac{1}{n_{ij}} )),

where ε = 2 if there is a “+” sign and ε = 1 otherwise. If Γ′ is a subgroup of Γ of finite index then Γ′ is an NEC group and the following Riemann–Hurwitz formula holds:

(2.3)  [Γ : Γ′] = μ(Γ′)/μ(Γ).

An NEC group Γ without elliptic elements is called a surface group and it has signature (h; ±; [−], {(−), …^{k}, (−)}). In such a case ℋ /Γ is a Klein surface, i.e., a surface with a dianalytic structure of topological genus h, orientable or not according to the sign “+” or “−”, and having k boundary components. Conversely, a Klein surface whose complex double has genus greater than one can be expressed as ℋ /Γ for some NEC surface group Γ. Furthermore, given a Riemann (respectively, Klein) surface represented as the orbit space X = ℋ /Γ, with Γ a surface group, a finite group G is a group of automorphisms of X if and only if there exists an NEC group Δ and an epimorphism θ: Δ → G with ker(θ) = Γ (see [6]). The NEC group Δ is the lifting of G to the universal covering π: ℋ → ℋ /Γ and is called the the universal covering transformation group of (X, G).

Given an NEC group Γ, we denote by R(Γ) the set of monomorphisms r: Γ → Aut(ℋ ) such that r(Γ) is discrete and cocompact. Two elements r_{1}, r_{2} ∈ R(Γ) are said to be equivalent if there exists g ∈ Aut(ℋ ) such that for each γ ∈ Γ, r_{1}(γ) = gr_{2}(γ)g^{−1} . The orbit space T(Γ) is called the Teichmüller space of Γ and it is homeomorphic to a real ball. Notice that g ∈ Aut(ℋ ) , where Aut(ℋ ) are conformal and anticonformal automorphisms, then the groups uniformising the same orientable (but not oriented) Riemann surface appear in the same Teichmüller space. Let Γ ≤ Δ be NEC groups, the inclusion mapping i: Γ → Δ induces an embedding m_{i}: T(Δ) → T(Γ) defined by m_{i}[r] = [r ∘ i] (see [6]).

Let A(Γ) denote the automorphism group of Γ (A(Γ)^{+} the orientationpreserving automorphism group if Γ is a Fuchsian group), and I(Γ) the subgroup of inner automorphisms. The modular group M(Γ) = A(Γ)/I(Γ) (M(Γ)^{+} = A(Γ)^{+}/I(Γ) if Γ is a Fuchsian group), acts on T(Γ). The moduli space of Γ is the quotient space

ℳ_{g} = T(Γ)/M(Γ)

and ℳ_{g} = T(Γ)/M(Γ)^{+} if Γ is a Fuchsian group. For Γ being a Fuchsian group, ℳ_{g} is the space of conformal classes of Riemann surfaces of genus g, or equivalently, the space of isomorphism classes of smooth complex projective algebraic curves.

Let now ℳ_{g}^{ℛ} denote the subspace of ℳ_{g} consisting of real Riemann surfaces of genus g, that is, Riemann surfaces X_{g} admitting a symmetry (an anticonformal involution) σ: X_{g} → X_{g}. If Γ is a uniformizing Fuchsian group of X_{g}, then σ induces an element of M(Γ) acting on T(Γ). If we denote σ^{∗} the action of σ on T(Γ), then [X_{g}] ∈ T(Γ)_{σ^{∗}}, where T(Γ)_{σ^{∗}} is the set of fixed points under σ^{∗}, and the converse holds. Now, ℳ_{g}^{ℛ} is the projection of T_{R} = ⋃_{σ^{∗}} T(Γ)_{σ^{∗}} on ℳ_{g}, where σ runs over the different topological types of involutions of Riemann surfaces of genus g.

Let ℳ_{εk}^{g} denote the space of real Riemann surfaces of genus g admitting a symmetry with species εk, −g ≤ εk ≤ g +1. Then we have the following theorem:

Theorem 2.1. Let X_{g} be a Riemann surface of genus g. Given a symmetry σ: X_{g} → X_{g} with species εk, then ℳ_{εk}^{g} is connected.

Proof. Let X_{g} = ℋ /Γ and X_{g}/⟨σ⟩ = ℋ /Γ′, then Γ is an index two subgroup of Γ′ and let i: Γ → Γ′ be the inclusion map. Thus ℳ_{εk}^{g} is m_{i}(T(Γ′))/M(Γ) and since T(Γ′) is connected then ℳ_{εk}^{g} is connected.

Let (X_{g}, ϕ) be a hyperelliptic Riemann surface of genus g. Given a symmetry σ of X_{g} with species εk, consider the finite group G = ⟨ϕ, σ⟩. Then with the same proof as in Theorem 2.1 we have

Theorem 2.2. Let X_{g} be a hyperelliptic Riemann surface of genus g. Given a symmetry σ: X_{g} → X_{g} with species εk, then ℳ_{εk}^{g} ∩ ℳ_{g}^{2ℛ} is connected.

Theorems 2.1 and 2.2 are well-known facts going back to Klein and Earle [21] and [11] (cf. [8]).

We have a natural decomposition of ℳ_{g}^{ℛ} and ℳ_{g}^{2ℛ} into the connected subspaces ℳ_{εk}^{g} and ℳ_{εk}^{g} ∩ ℳ_{g}^{2ℛ} . We shall prove in the next two sections that the subspaces ℳ_{−1}^{g} and ℳ_{−1}^{g} ∩ ℳ_{g}^{2ℛ} of real and real hyperelliptic Riemann surfaces having a non-separating symmetry with one oval intersect any other subspace ℳ_{εk}^{g} and ℳ_{εk}^{g} ∩ ℳ_{g}^{2ℛ} , respectively. To prove this we have to find a (hyperelliptic) Riemann surface admitting two symmetries σ_{1} , σ_{2} with species −1 and εk, respectively for every possible species εk. Notice that the possible species εk for symmetries of hyperelliptic Riemann surfaces are: ε = −1, 1 ≤ k ≤ g, ε = 1, k = g + 1, and ε = 1, k = 1 for g even, and ε = 1, k = 2 for g odd.

In the following we shall consider hyperelliptic Riemann surfaces (X_{g}, ϕ) with uniformizing surface Fuchsian group Γ, and where ϕ is the hyperelliptic involution.

Let σ_{1} , σ_{2} be symmetries of a hyperelliptic Riemann surface X_{g} = ℋ /Γ with species ε_{1}k_{1} and ε_{2}k_{2} , respectively. The involutions σ_{1} , σ_{2} and ϕ generate a finite group G. The group G is isomorphic either to D_{n} or to D_{n} × C_{2} , with n the order of σ_{1}σ_{2} . Notice that G = D_{n} if and only if ϕ = (σ_{1}σ_{2})^{n/2} . Then there exist an NEC group Δ (the universal covering transformation group) and an epimorphism θ: Δ → G with ker(θ) = Γ. If θ^{−1}(⟨σ_{1}⟩) = Λ_{1} , θ^{−1}(⟨σ_{2}⟩) = Λ_{2} , and θ^{−1}(⟨ϕ⟩) = Λ_{h} then s(Λ_{1}) = (h_{1}, ε_{1}, [−], {(−)^{k_{1}}}), s(Λ_{2}) = (h_{2}, ε_{2}, [−], {(−)^{k_{2}}}) and s(Λ_{h}) = (0, +, [2, …^{2g+2} , 2], {−}), where |Δ : Λ_{1}| = |Δ : Λ_{2}| = |Δ : Λ_{h}| = n (or 2n) and |Λ_{1} : Γ| = |Λ_{2} : Γ| = |Λ_{h} : Γ| = 2.

The aim is to give signatures Δ and epimorphisms θ: Δ → G such that s(Λ_{1}) = (h_{1}, −, [−], {(−)}), and s(Λ_{2}) = (h_{2}, ε, [−], {(−)^{k}}) where εk ranges over the possible species for a hyperelliptic Riemann surface.

In order to know the signature of Λ_{i} from the epimorphism θ we use the following procedure.

Let G′ be the set of generators of a canonical presentation of Δ, the Schreier ⟨σ_{i}⟩-coset graph S_{i} is the graph with vertices the ⟨σ_{i}⟩-cosets in G and labelled directed edges joining ⟨σ_{i}⟩α_{i} with ⟨σ_{i}⟩α_{j} with label g ∈ G′ if ⟨σ_{i}⟩α_{i}θ(g) = ⟨σ_{i}⟩α_{j}. Let c ∈ G′ be a reflection such that θ maps c to a conjugate of σ_{1} or σ_{2} in G. The action of c on the Λ_{i} -cosets is the same as the action of θ(c) on the ⟨σ_{i}⟩- cosets. Each coset Λ_{i}α fixed by c gives a reflection c_{α} = αcα^{−1} in Λ_{i}, called a reflection induced by c. In this way we obtain representatives of all conjugacy classes of reflections in Λ_{i}. Now each period cycle in (sΛ_{i}) gives one conjugacy class of reflections in Λ_{i}. Suppose that d is another reflection in Δ and that cd has finite order. Two induced reflections (c_{α} and d_{β}, c_{α} and c_{β} or d_{α} and d_{β}) are conjugate in Λ_{i} if ⟨σ_{i}⟩α and ⟨σ_{i}⟩β are in the same orbit under the action of θ(cd) on the ⟨σ_{i}⟩-cosets ([17]). In terms of the Schreier graph: two reflection loops c_{α} and d_{β} define conjugate reflections in Λ_{i} if the vertices of these reflection loops are joined by a path with the sides alternatively labelled by c and d. Finally, assume c is the reflection generator and e is the hyperbolic generator corresponding to an empty period cycle in s(Δ) (i.e., a period cycle without link periods). If σ_{1}σ_{2} has even order, then c can induce several reflections in Λ_{i}. In this case all the reflections c_{α} and c_{β} are conjugate in Λ_{i} if and only if G = D_{n} and θ(e) ≠ 1 (see [15] and [17]).

The sign of s(Λ_{i}), i = 1, 2, is determined by the following fact: the sign of s(Λ_{i}) is + if and only if in the Schreier graph S_{i} the product of the labels of each cycle (not containing reflection loops) is an orientation preserving element of Δ (see [18]).

3. The connection of the locus of real hyperelliptic Riemann surfaces

Notice that the possible species εk for symmetries of hyperelliptic Riemann surfaces are: ε = −1, 1 ≤ k ≤ g, ε = 1, k = g + 1, and ε = 1, k = 1 for g even, and ε = 1, k = 2 for g odd. With the notation above:

Theorem 3.1. We have ℳ_{−1}^{g} ⋂ ℳ_{−k}^{g} ⋂ ℳ_{g}^{2ℛ} ≠ ∅ for 1 ≤ k ≤ g for every

genus g ≥ 2.

Proof. We divide the proof in two cases according to the parity of g − k.

(1) g − k odd. Let Δ be an NEC group with signature

(3.1)  s(Δ) = (0; +; [2^{(g−k+1)/2}]; \{(4, 4, \overbrace{2, …, 2}^{k−1} )\}).

and canonical presentation as given in Section 2. Let θ: Δ → D_{4} be the epimorphism defined by θ(x_{j}) = ϕ = (σ_{1}σ_{2})^{2} , for all j, θ(c_{11}) = θ(c_{1k+2}) = σ_{1} , θ(c_{1i}) = σ_{2} for i even, and θ(c_{1i}) = σ_{2}ϕ for i odd, 2 ≤ i ≤ k − 1, θ(e_{1}) = ϕ^{i}, where i ∈ {0, 1} in order to fulfill the long relation.

The Riemann formula applied to ker θ ≤ Δ yields:

(3.2)  8(−1 + \frac{g − k + 1}{4} + \frac{k − 1}{4} + \frac{3}{4}) = 2(g − 4 + 3) = 2(g − 1),

Then ker θ is a Fuchsian surface group of genus g. Let X_{g} = ℋ / ker θ, Λ_{1} = θ^{−1}(⟨σ_{1}⟩) and Λ_{2} = θ^{−1}(⟨σ_{2}⟩). The reflection c_{11} induces two reflections in Λ_{1} that are conjugate because θ(c_{11}c_{12}) = σ_{1}σ_{2} ; then there is only one conjugacy class of reflections and one empty period-cycle in s(Λ_{1}). In the Schreier graph of ⟨σ_{1}⟩-cosets there is an orientation-reversing cycle with labelled edges c_{12} , c_{13} and c_{11} . Hence there is a − sign and one empty period cycle in s(Λ_{1}), then the species of σ_{1} is −1.

Each generating reflection c_{1i}, k + 1 ≥ i ≥ 2, induces one conjugacy class of reflections in Λ_{2} . Hence there are k + 1 − 1 = k empty period-cycles in the signature of θ^{−1}(⟨σ_{2}⟩) = Λ_{2} . In the Schreier graph of ⟨σ_{2}⟩-cosets there is a cycle with labelled edges x_{1} and c_{12} and there is a − sign in s(Λ_{2}). The symmetry σ_{2} has species −k.

Notice that X_{g}/⟨ϕ⟩ is an orbifold with 4(\frac{1}{2}(g − k + 1)) + 2(k) = 2g + 2 conic points of order 2 and genus \frac{1}{2}(g − 4 + 3 − g − 1 + 2) = 0.

(2) g − k even. Let Δ be a group with signature

(3.3)  s(Δ) = (0; +; [2^{(g−k)/2}]; \{( \overbrace{2, …, 2}^{k+3} )\}).

Let θ: Δ → D_{2} ×C_{2} be the epimorphism defined by θ(x_{j}) = ϕ, for all j, θ(c_{11}) = θ(c_{1k+4}) = σ_{1} , θ(c_{1i}) = σ_{2} for i even, and θ(c_{1i}) = σ_{2}ϕ for i odd, 2 ≤ i ≤ k + 2, θ(c_{1k+3}) = σ_{1}ϕ, θ(e_{1}) = ϕ^{i}, where i ∈ {0, 1} in order to fulfil the long relation.

The Riemann formula applied to ker θ ≤ Δ gives

(3.4)  8(−1 + \frac{k + 3}{4} + \frac{g − k}{4} ) = 2(g − 4 + 3) = 2(g − 1).

Then ker θ is a Fuchsian surface group of genus g. We define X_{g} = ℋ / ker θ, Λ_{1} = θ^{−1}(⟨σ_{1}⟩) and Λ_{2} = θ^{−1}(⟨σ_{2}⟩). The same argument as in the case (1) shows that there is only one empty period-cycle in s(Λ_{1}). In the Schreier graph of ⟨σ_{1}⟩-cosets there is an orientation-reversing 3-cycle with edges labelled x_{1} , c_{1k+3} and c_{11} . Then the signature of Λ_{1} is (w, −, [−], {(−)}) and the species of σ_{1} is −1.

The generating reflection c_{12} induces one conjugacy class of reflections in Λ_{2} . Each generating reflection c_{1,2i}, 2 ≥ i ≥ \frac{1}{2}(k − 1), induces two conjugacy classes of reflections in Λ_{2} . Hence there are \frac{1}{2}(2(k − 1)) + 1 = k empty period-cycles in the signature of θ^{−1}(⟨σ_{2}⟩) = Λ_{2} . In the Schreier graph of ⟨σ_{2}⟩-cosets there is an orientation-reversing 3-cycle with edges labelled x_{1} , c_{12} and c_{13} . The symmetry σ_{2} has species −k.

Again, X_{g}/⟨ϕ⟩ is an orbifold with 4(\frac{1}{2}(g − k)) + 2(k + 1) = 2(g + 1) conic points of order 2 and genus \frac{1}{2}(g − 4 + 3 − g − 1 + 2) = 0, that is, X_{g} is a hyperelliptic surface.

The surface X_{g} is a point in ℳ_{−1}^{g} ⋂ ℳ_{−k}^{g} ∩ ℳ_{g}^{2ℛ} , so then ℳ_{−1}^{g} ⋂ ℳ_{−k}^{g} ∩ ℳ_{g}^{2ℛ} ≠ ∅.

Theorem 3.2 ([3]). We have ℳ_{−1}^{g} ⋂ ℳ_{g+1}^{g} ⋂ ℳ_{g}^{2ℛ} ≠ ∅ for every genus

g ≥ 2.

Proof. The proof is as for Theorem 3.1, Case 1, but now taking as Δ an NEC group with signature

(3.5)  s(Δ) = (0; +; [−]; \{(4, 4, \overbrace{2, …, 2}^{g} )\}),

and an epimorphism θ: Δ → D_{4} defined by θ(c_{1,1}) = σ_{1} , θ(c_{1,2j}) = σ_{2} , θ(c_{1,2j+1}) = σ_{2}ϕ, 1 ≤ j ≤ ⌊\frac{1}{2}(g + 1)⌋, θ(c_{1,0}) = σ_{2}ϕ^{i}, where i = 0 for even g and i = 1 for odd g and θ(e) = 1.

Theorem 3.3. We have ℳ_{−1}^{g} ⋂ ℳ_{0}^{g} ⋂ ℳ_{+k}^{g} ⋂ ℳ_{g}^{2ℛ} ≠ ∅, for k = 1 for even

genera g, and k = 2 for odd genera g ≥ 2.

Proof. First of all, notice that a symmetry of a hyperelliptic Riemann surface of genus g that separates, satisfies the fact that the number of its ovals is either g + 1 or 1 if the genus g is even, and either g + 1 or 2 if the genus g is odd.

Consider an NEC group with signature s(Δ) = (0; +; [−]; {(2, 2g+2, 2g+2)}), and an epimorphism θ: Δ → D_{2(g+1)} × C_{2} defined by θ(c_{11}) = σ_{1} , θ(c_{12}) = σ_{1}ϕ, θ(c_{13}) = σ_{2} .

The Riemann formula applied to ker θ ≤ Δ gives:

(3.6) 8(g + 1)(−1 + \frac{1}{4} + \frac{2g + 1}{2(g + 1)}) = 8(g + 1)(\frac{−3(g + 1) + 4g + 2}{4(g + 1)}) = 2(g − 1).

We define X_{g} = ℋ / ker θ, Λ_{1} = θ^{−1}(⟨σ_{1}⟩), Λ_{2} = θ^{−1}(⟨σ_{2}⟩) and Λ_{3} = θ^{−1}(⟨σ_{2}ϕ⟩). The same argument as in Theorem 3.1 shows that there is only one empty period-cycle in s(Λ_{1}). In the Schreier graph of ⟨σ_{1}⟩-cosets there is an orientation-reversing 5-cycle with edges labelled c_{12} , c_{13} , c_{11} , c_{12} and c_{13} . Then the signature of Λ_{1} is (w, −, [−], {(−)}) and the species of σ_{1} is −1.

The generating reflection c_{12} induces four reflections in Λ_{2} . These reflections are conjugated by θ((c_{12}c_{13})^{g+1}) = (σ_{1}σ_{2})^{g+1}ϕ^{g+1} and θ((c_{13}c_{11})^{g+1}) = (σ_{1}σ_{2})^{g+1} . Then Λ_{2} contains two conjugacy classes of reflections induced by c_{12} if g is odd, and one if g is even. The Schreier graph of ⟨σ_{2}⟩ is bipartite: the set of vertices {(σ_{2})(σ_{1}σ_{2})^{i}ϕ^{ε} | 1 ≤ i ≤ 2g + 2, ε ∈ {0, 1}} can be separated in the sets A = \{\overline{(i, 0)}, \overline{(i, 1)}, 1 ≤ i ≤ g + 1\} and B = \{\overline{(i, 0)}, \overline{(i, 1)}, g + 2 ≤ i ≤ 2g + 2\}, where the ⟨σ_{2}⟩-cosets are represented by the corresponding exponent of σ_{1}σ_{2} . The symmetry σ_{2} has species +2 if the genus g is odd, and σ_{2} has species +1 if the genus g is even.

The Schreier graph of ⟨σ_{2}ϕ⟩ has no reflection loops, then σ_{2}ϕ has species 0. Again, ℋ /θ^{−1}(⟨σ_{1}, σ_{2}⟩) is an orbifold with (2g+2)(1) = 2(g + 1) conic points

of order 2 and genus 0.

The surface X_{g} defined by (Δ, θ) admits the required three symmetries σ_{1} , σ_{2} , and σ_{2}ϕ.

We can now establish some direct consequences of the results 3.1, 3.2 and 3.3. Two real hyperelliptic Riemann surfaces (X_{i}, σ_{i}) i = 1, 2, are quasiconfor-

mally equivalent in the hyperelliptic locus if there is a quasiconformal deformation F_{t} from X_{1} to X_{2} such that X_{t} = F_{t}(X_{1}) is a hyperelliptic surface and F_{t} ∘ σ_{1} ∘ F_{t}^{−1} is a symmetry of X_{t}. Now we have the result quoted in the abstract:

Theorem 3.4. Every real hyperelliptic Riemann surface is quasiconformally equivalent in the hyperelliptic locus to a real hyperelliptic Riemann surface (X, σ), such that X admits an antiholomorphic involution τ, where Fix(τ) has one nonseparating connected component.

Proof. Each set ℳ_{εk}^{g} ⋂ ℳ_{g}^{2ℛ} corresponds to the real hyperelliptic Riemann surfaces with the same topological type. By Lemma 1.2 in [22] the set ℳ_{εk}^{g} ⋂ ℳ_{g}^{2ℛ} is a quasiconformal class of real hyperelliptic Riemann surfaces (this is a consequence of deep results in Teichmüller theory); then the theorem follows from Theorems 3.1, 3.2 and 3.3.

A consequence of the above theorem is the following:

Corollary 3.5. The space ℳ_{g}^{2ℛ} of real hyperelliptic algebraic curves is a connected subspace of the moduli space ℳ_{g} of complex algebraic curves. Furthermore the subset formed by all real hyperelliptic Riemann surfaces admitting a nonseparating symmetry with one oval ℳ_{−1}^{g} ⋂ ℳ_{g}^{2ℛ} cuts every subset ℳ_{εk}^{g} ⋂ ℳ_{g}^{2ℛ} for any possible species εk for a given g, i.e., ℳ_{−1}^{g} is a spine for ℳ_{g}^{2ℛ} .

4. On the existence of spines in ℳ_{g}^{ℛ}

We have proved in the previus section that ℳ_{−1}^{g} ⋂ ℳ_{εk}^{g} ≠ ∅ for εk being negative, 0, g + 1, and +1 if g is even or +2 if g is odd. To prove that ℳ_{−1}^{g} is a spine of ℳ_{g}^{ℛ} we must find surfaces X_{g} admitting two symmetries σ_{1} , σ_{2} such that σ_{1} has species −1 and σ_{2} has species +k, with k = g + 1 − 2t and 1 ≤ t ≤ ⌊\frac{1}{2}(g − 2)⌋.

Let σ_{1} , σ_{2} be symmetries of a Riemann surface X_{g} with species +1 and +k, respectively. The involutions σ_{1} , σ_{2} generate a finite group G. G = D_{n} = ⟨σ_{1}, σ_{2}⟩, with n the order of σ_{1}σ_{2} .

Theorem 4.1 ([10]). We have ℳ_{−1}^{g} ⋂ ℳ_{+k}^{g} ≠ ∅, for k = g + 1 − 2t, 1 ≤ t ≤ \frac{1}{2}(g − 2) for even genera g, and 1 ≤ t ≤ \frac{1}{2}(g − 3) for odd genera g.

Proof. The method of the proof is similar to the one used in Theorems 3.1, 3.2 and 3.3. We consider suitable NEC groups and epimorphisms θ from them to dihedral groups D_{n} = ⟨σ_{1}, σ_{2}⟩. The surface X = ℋ / ker θ admits two symmetries given by X → X/⟨σ_{i}⟩, i = 1, 2, with species −1 and +k.

We divide the proof in three cases according to the parity of g and k. We shall only give the signature of the groups Δ and the epimorphisms θ: Δ → D_{n} in each case.

(1) g even, k ≡ g + 1(mod 2) and k ≤ g − 1 ([19]). Let Δ be a group with signature

(4.1)  s(Δ) = (\frac{1}{2}(g + 1 − k); −; [−]; \{(2, 2)(−)^{(k−1)/2}\}),

and θ: Δ → D_{2} the epimorphism defined by θ(d_{j}) = σ_{1} , for all j, θ(c_{11}) = θ(c_{13}) = σ_{2} , θ(c_{12}) = σ_{1} , θ(e_{1}) = 1, θ(c_{i1}) = σ_{2} , θ(e_{i}) = 1, for all i ≥ 2.

(2) g odd, k ≡ g + 1(mod 4). Let Δ be a group with signature

(4.2)  s(Δ) = (\frac{1}{4}(g + 1 − k); −; [−]; \{( 4, \overbrace{2, …, 2}^{k−1} , 4)\}).

Let θ: Δ → D_{4} be the epimorphism defined by θ(d_{j}) = σ_{1} , for all j, θ(c_{1,1}) = θ(c_{1,k+2}) = σ_{1} , θ(c_{1,2i}) = σ_{2} , θ(c_{1,2i+1}) = σ_{2}(σ_{1}σ_{2})^{2} , 1 ≤ i ≤ \frac{1}{2}k, θ(e_{1}) = 1.

(3) g odd, k ≡ g − 1(mod 4), 3 < k. Let Δ be a group with signature either

(4.3) s(Δ) = (\frac{1}{4}(g − 1 − k); −; [−]; \{(4, \overbrace{2, …, 2}^{k−3} , 4), (−)\}),  if k < g − 1

or

(4.4)  s(Δ) = (0; +; [−]; \{(4, \overbrace{2, …, 2}^{k−3} , 4), (−)\}),  if k = g − 1.

Let θ: Δ → D_{4} be the epimorphism defined by θ(d_{j}) = σ_{1} , for all j, θ(c_{1,1}) = θ(c_{1,k}) = σ_{1} , θ(c_{1,2i}) = σ_{2} , θ(c_{1,2i+1}) = σ_{2}(σ_{1}σ_{2})^{2} , 1 ≤ i ≤ \frac{1}{2}k − 1, θ(e_{1}) = 1, θ(c_{2,1}) = σ_{2} and θ(e_{2}) = 1.

We can now establish

Theorem 4.2. Every real Riemann surface is quasiconformally equivalent in the real locus to a real Riemann surface (X, σ), such that X admits an antiholomorphic involution τ, where Fix(τ) has one non-separating connected component.

Proof. Each set ℳ_{εk}^{g} corresponds to the real Riemann surfaces with the same topological type. By Lemma 1.2 in [22] there is a quasiconformal class of real Riemann surfaces. Thus the theorem follows from Theorems 3.1, 3.2, 3.3 and 4.1.

A consequence of the above theorem is the following:

Corollary 4.3. The space ℳ_{g}^{ℛ} of real algebraic curves is a connected subspace of the moduli space ℳ_{g} of complex algebraic curves. Furthermore the subset formed by all real Riemann surfaces admitting a non-separating symmetry with one oval, ℳ_{−1}^{g}, cuts every subset ℳ_{εk}^{g} for any possible species εk for a given g, i.e., ℳ_{−1}^{g} is a spine for ℳ_{g}^{ℛ} .

Given the decomposition ℳ_{g}^{ℛ} = ⋃ ℳ_{k}^{g} , we try to find another spine different from ℳ_{−1}^{g}, i.e., we are looking for ε′k′ ≠ −1 such that ℳ_{εk}^{g} ⋂ ℳ_{ε′k′}^{g} ≠ ∅ for all possible species εk.

Theorem 4.4 ([10]). Let g > 2 be an even integer. Then the only spine in the decomposition of the subset ℳ_{g}^{ℛ} = ⋃ ℳ_{k}^{g} of real Riemann surfaces of ℳ_{g} is the subspace ℳ_{−1}^{g}.

Proof. Assume that g is an integer and ε′k′ ≠ −1 is such that ℳ_{εk}^{g} ⋂ ℳ_{ε′k′}^{g} ≠ ∅ for all possible species εk for g. First of all, by Theorem 3.3 in [3], if ℳ_{g+1}^{g} ⋂ ℳ_{ε′k′}^{g} ≠ ∅, then the species ε′k′ ≥ −1. By Theorem 3.4 in [3], if ℳ_{−g}^{g} ⋂ ℳ_{ε′k′}^{g} ≠ ∅, then the species ε′k′ ≤ 0. On the other hand, by Theorem 3.2 in [19] ℳ_{0}^{g} ⋂ ℳ_{εk}^{g} = ∅ for even genera g and even k ≠ 0. Then if g is even there is no such a ε′k′.

With the same proof we can show the following

Remark 4.5. Let g > 2 be an even integer. Then the only spine in the decomposition of the subset ℳ_{g}^{2ℛ} of real hyperelliptic Riemann surfaces of ℳ_{g} is the subspace ℳ_{−1}^{g} ⋂ ℳ_{g}^{2ℛ} .

Notice that in the proof of the above theorems the hypothesis on the parity of g is only used to avoid the possibility k′ = 0, then the only possible spine for g odd is ℳ_{0}^{g}. The results of [19] assert that ℳ_{0}^{g} is in fact a spine.

Theorem 4.6 ([10]). Let g be an odd integer. Then a real algebraic curve C_{g} of genus g is quasiconformally equivalent to a real curve C′_{g} such that the complexification of C′_{g} admits a real form which is purely imaginary.

Proof. We have to prove that there exist real Riemann surfaces admitting two symmetries with species 0 and εk, where εk runs over all possibilities. The signatures and epimorphisms listed below give us the required real Riemann surfaces ([19]).

(1.1) ε = −, k odd. Let Δ be a group with signature

(4.5)  s(Δ) = (0; +; [4, \overbrace{2, …, 2}^{(g−k)/2} ]; \{(2^{k})\}).

We construct an epimorphism θ: Δ → D_{4} = ⟨σ_{1}, σ_{2}⟩ by sending the elliptic generator of order 4 to σ_{1}σ_{2} , the elliptic generators of order 2 to (σ_{1}σ_{2})^{2} , the generating reflection generators alternately to σ_{2} and σ_{2}(σ_{1}σ_{2})^{2} , and e to σ_{1}σ_{2} or (σ_{1}σ_{2})^{3} (depending on the parity of \frac{1}{2}(g − k)).

(1.2) ε = −, k even. Let Δ be a group with signature

(4.6)  s(Δ) = (0; +; [2^{(g+3−k)/2}]; \{( \overbrace{2, …, 2}^{k} )\}).

We define an epimorphism θ from Δ to D_{2} × C_{2} = ⟨σ_{1}, σ_{2}, ϕ⟩ by mapping all but one elliptic generators to ϕ, one elliptic generator to σ_{1}σ_{2}ϕ, the generating reflections and glide reflection to σ_{2} alternatively σ_{2}ϕ and e to σ_{1}σ_{2}ϕ^{h}, where h = 0 if \frac{1}{2}(g + 3 − k) is even and h = 1 if \frac{1}{2}(g + 3 − k) is odd.

(2) ε = +. In this case k is even, since k ≡ g + 1(mod 2). Let Δ be a group with signature

(4.7)  s(Δ) = (0; +; [2^{g+3−k}]; {(−)^{k/2}}).

We define an epimorphism θ from Δ to D_{2} = ⟨σ_{1}, σ_{2}⟩ by mapping the elliptic generators to σ_{1}σ_{2} , the generating reflections to σ_{2} , and the connecting generators e_{i} to 1.

Notice that the Schreier graph of ⟨σ_{1}⟩ in D_{2} contain no reflection loops, so the species of σ_{1} is 0.

The above proof is different from the one given in [10] in the cases 1.1 and 1.2. The surfaces in the cases 1.1 and 1.2 of Theorem 4.6 are hyperelliptic, being the hyperelliptic involution ϕ = (σ_{1}σ_{2})^{2} in the case 1.1 and ϕ in 1.2. Then we have the following result:

Remark 4.7. Let g be an odd integer. Then a real hyperelliptic algebraic curve C_{g} of genus g is quasiconformally equivalent in the hyperelliptic locus to a real hyperelliptic curve C′_{g} such that the complexification of C′_{g} admits a real form which is purely imaginary.

5. The locus of real cyclic p-gonal Riemann surfaces

Let p be a prime integer. A cyclic p-gonal Riemann surface is a closed Riemann surface which can be realized as a cyclic p-fold covering space of the Riemann sphere (see [9] and [16]). We will denote by ℳ_{g}^{p} the subset of cyclic pgonal Riemann surfaces in ℳ_{g}. The complexification of a smooth cyclic p-gonal real algebraic curve gives rise to a cyclic p-gonal Riemann surface X_{g} with a symmetry, we shall call such a surface X_{g} a real cyclic p-gonal Riemann surface.

Let ℳ_{g}^{pℛ} be the locus of real cyclic p-gonal Riemann surfaces in the moduli space of Riemann surfaces of genus g. In Section 3 we proved that ℳ_{g}^{2ℛ} is connected, and now we want to show that the situation is essentially different for ℳ_{g}^{pℛ} , p > 2. The set of cyclic p-gonal Riemann surfaces is actually disconnected in general. Let P^{n}(F_{p}) denote the n-dimensional projective space over the finite field with p elements, and define the following subset D_{p}^{r} of P^{r−1}(F_{p}):

D_{p}^{r} = {m = (m_{1}, …, m_{r}) | ∑_{1}^{r} m_{i} = 0, Π_{1}^{r}m_{i} ≠ 0}.

The symmetric group ∑_{r} acts on D_{p}^{r} in the natural way and we write D_{p}^{(r)} for D_{p}^{r}/ ∑_{r}. Let m = (m_{1}, …, m_{r}) ∈ \overline{m} ∈ D_{p}^{(r)} . Consider the abstract Fuchsian group Γ with signature s(Γ) = (0, [p , …^{r} , p]) and epimorphism φ_{m}: Γ → F_{p} (F_{p} as an additive group) defined by φ_{m}(x_{i}) = m_{i}. The group ker φ_{m} is a surface Fuchsian group that uniformizes a cyclic p-gonal Riemann surface. In fact, the map π: ℋ / ker φ_{m} → ℋ /Γ is a cyclic covering map of the Riemann sphere. There is a natural decomposition ℳ_{g}^{p} = ⋃ ℳ_{g}^{p}(\overline{m}), where ℳ_{g}^{p}(\overline{m}) is the set of Riemann surfaces uniformized by pairs (Γ, φ_{t}), with t ∈ \overline{m} in D_{p}^{(r)} .

Let p > 2. In general the action of ∑_{r} on D_{p}^{r} is not transitive. So, for r = 2p, the elements m_{1} = (1 , …,^{2p} 1) and m_{2} = (1, p − 1, …^{p} , 1, p − 1) belong to distinct classes under the action of ∑_{r}.

Theorem 5.1 (Theorem 2 in [14]). The union ℳ_{g}^{p} is the disjoint union of the sets ℳ_{g}^{p}(\overline{m}), where \overline{m} ranges on the whole set D_{p}^{(r)} .

The integers p, g and r in the statement are related by the Riemann–Hurwitz formula 2g = (p − 1)(r − 2).

We shall see that any connected component of ℳ_{g}^{p} contains real, cyclic pgonal Riemann surfaces. As a consequence, we find that ℳ_{g}^{pℛ} is not connected in general.

Theorem 5.2. Let ℳ_{g}^{p} = ∐ ℳ_{g}^{p}(\overline{m}). Each connected component ℳ_{g}^{p}(\overline{m})

contains a real, cyclic p-gonal Riemann surface.

Proof. Let now Δ be an NEC group with signature

s(Δ) = (0; +; [−]; {(p …^{r} , p)})

and canonical presentation Δ = ⟨e, c_{0}, c_{1}, …, c_{r} | c_{0}^{2} = c_{i}^{2} = (c_{i−1}c_{i})^{p} = ec_{0}e^{−1}c_{r} = 1, 1 ≤ i ≤ r⟩. Consider the epimorphism θ_{m}: Δ → D_{p} = ⟨σ_{1}, σ_{2} | σ_{i}^{2} = (σ_{1}σ_{2})^{p} = 1, i = 1, 2⟩ defined recursively by θ_{m}(c_{0}) = σ_{2} , θ_{m}(c_{i}) = θ_{m}(c_{i−1})(σ_{1}σ_{2})^{m_{i}}. Hence ℋ / ker θ_{m} is a real Riemann surface since ∑ m_{i} = 0. For instance, σ_{2} represents a symmetry of ℋ / ker θ_{m}. Finally, ℋ / ker θ_{m} belongs to ℳ_{g}^{p} since θ_{m}^{−1}(⟨σ_{1}σ_{2}⟩) is a Fuchsian group with signature (0, [p …^{r} , p]) and ker θ_{m} = ker θ_{m}/θ_{m}^{−1}(⟨σ_{1}σ_{2}⟩).

The surface ℋ / ker θ_{m} constructed above has a dihedral group D_{p} of automorphisms with all the symmetries conformally conjugate since p is an odd prime. Using [17], and [18], we find that the species of any of the symmetries in D_{p} is +1, for all \overline{m} ∈ D_{p}^{(r)} . Hence each connected component of ℳ_{g}^{p} contains a real, cyclic p-gonal Riemann surface with a symmetry with species +1.

As a particular case we have:

Theorem 5.3. The locus ℳ_{(p−1)^{2}}^{pℛ} of a real, cyclic p-gonal Riemann surface in ℳ_{(p−1)^{2}} is disconnected for prime integers p > 2.

Gross and Harris proved Theorem 5.3 in the case p = 3 (see [16]).

To prove that ℳ_{g}^{pℛ} is disconnected we used the fact that ℳ_{g}^{p} is disconnected. The final paragraph of this work is to remark that, in general, given a connected component ℳ_{g}^{p}(\overline{m}) of ℳ_{g}^{p}, the set ℳ_{g}^{p}(\overline{m}) ∩ ℳ_{g}^{pℛ} of real p-gonal Riemann surfaces contained in ℳ_{g}^{p}(\overline{m}) is disconnected in general. The reason is that the representations (Γ, φ_{m}) of cyclic p-gonal Riemann surfaces admit an action of ∑_{r}, while the representations θ_{m}: Δ → D_{p}, s(Δ) = (0; +; [−]; {(p …^{r} , p)}), of real, cyclic p-gonal Riemann surfaces admit only actions of cyclic groups.

Example 5.4. ℳ_{16}^{5}( \overline{(1, 1, 1, 1, 1, 4, 4, 4, 4, 4)} ) ∩ ℳ_{16}^{5ℛ} is not connected.

Let now Δ be an NEC group with signature s(Δ) = (0; +; [−]; {(5^{10})}) and presentation Δ = ⟨e, c_{0}, c_{1}, …, c_{r} | c_{0}^{2} = c_{i}^{2} = (c_{i−1}c_{i})^{5} = ec_{0}e^{−1}c_{r} = 1, 1 ≤ i ≤ 10⟩. We construct epimorphisms

θ_{1}, θ_{2}: Δ → D_{5} = ⟨σ_{1}, σ_{2} | σ_{i}^{2} = (σ_{1}σ_{2})^{5} = 1, i = 1, 2⟩ as follows:

θ_{1} :

\begin{aligned}θ_{1}(c_{0}) & = σ_{2} \\ θ_{1}(c_{4}) & = σ_{2} \\ θ_{1}(c_{8}) & = σ_{2}(σ_{1}σ_{2})^{2} \end{aligned} \begin{aligned}θ_{1}(c_{1}) & = σ_{2}(σ_{1}σ_{2}) \\ θ_{1}(c_{5}) & = σ_{2}(σ_{1}σ_{2}) \\ θ_{1}(c_{9}) & = σ_{2}(σ_{1}σ_{2}) \end{aligned} \begin{aligned}θ_{1}(c_{2}) & = σ_{2} \\ θ_{1}(c_{6}) & = σ_{2} \\ θ_{1}(c_{10}) & = σ_{2}. \end{aligned} \begin{aligned}θ_{1}(c_{3}) & = σ_{2}(σ_{1}σ_{2}) \\ θ_{1}(c_{7}) & = σ_{2}(σ_{1}σ_{2}) \end{aligned}

θ_{2} :

\begin{aligned}θ_{2}(c_{0}) & = σ_{2} \\ θ_{2}(c_{4}) & = σ_{2} \\ θ_{2}(c_{8}) & = σ_{2}(σ_{1}σ_{2})^{2} \end{aligned} \begin{aligned}θ_{2}(c_{1}) & = σ_{2}(σ_{1}σ_{2}) \\ θ_{2}(c_{5}) & = σ_{2}(σ_{1}σ_{2})^{4} \\ θ_{2}(c_{9}) & = σ_{2}(σ_{1}σ_{2}) \end{aligned} \begin{aligned}θ_{2}(c_{2}) & = σ_{2} \\ θ_{2}(c_{6}) & = σ_{2} \\ θ_{2}(c_{10}) & = σ_{2}. \end{aligned} \begin{aligned}θ_{2}(c_{3}) & = σ_{2}(σ_{1}σ_{2}) \\ θ_{2}(c_{7}) & = σ_{2}(σ_{1}σ_{2}) \end{aligned}

On the locus of real Riemann surfaces 355

We define the surfaces X_{1} = ℋ / ker θ_{1} and X_{2} = ℋ / ker θ_{2} . The transposition (5, 6) ∈ ∑_{10} maps the representation φ_{1}: Γ_{1} = θ_{1}^{−1}(⟨σ_{1}σ_{2}⟩) → F_{5} to the representation φ_{2}: Γ_{2} = θ_{2}^{−1}(⟨σ_{1}σ_{2}⟩) → F_{p}. Then the surfaces X_{1} and X_{2} belong to the same connected component of ℳ_{16}^{5} .

Finally we see that X_{1} and X_{2} lie in different components of ℳ_{16}^{5ℛ} . Assume that X_{1} and X_{2} are in the same component of ℳ_{16}^{5ℛ} . Let τ_{1} be a symmetry of X_{1} contained in D_{5} . Since the action of D_{5} on X_{1} is not topologically equivalent to the action of D_{5} on X_{2} , then there exists a Riemann surface X admitting an action of D_{5} which is topologically equivalent to the action of D_{5} on X_{1} and such that X has a symmetry τ ∉ D_{5} , not conformally equivalent to τ_{1} in Aut(X). So the order of ττ_{1} is even and τ induces a symmetry of X/⟨τ_{1}⟩. Now, let α be an automorphism of order 5 in D_{5} . Since α and τα^{j}τ are conjugate in Aut(X) for 1 ≤ j ≤ 4, then the markings 1, 4, 1, 4, 1, 4, 1, 1, 4, 4 corresponding to the components of m ∈ \overline{(1, 1, 1, 1, 1, 4, 4, 4, 4, 4)} are situated symmetrically with respect to τ in Fix(τ_{1}) giving the rotation indices of α. This does not happen for the representation θ_{1}: Δ → D_{5} .

References

[1] Accola, R.D.M.: On cyclic trigonal Riemann surfaces I. - Trans. Amer. Math. Soc. 283, 1984, 423–449.

[2] Alling, N.L., and N. Greenleaf: Foundations of the Theory of Klein Surfaces. - Lecture Notes in Math. 219, Springer-Verlag, Berlin, 1971.

[3] Bujalance, E., and A.F. Costa: A combinatorial approach to the symmetries of M and M − 1 Riemann surfaces. - London Math. Soc. Lecture Notes 173, 1992, 16–25.

[4] Bujalance, E., and A.F. Costa: On symmetries of p-hyperellitic Riemann surfaces. - Math. Ann. 308, 1997, 31–45.

[5] Bujalance, E., A.F. Costa, and D. Singerman: Applications of Hoare’s theorem to symmetries of Riemann surfaces. - Ann. Acad. Sci. Fenn. Math. 18, 1993, 307–322.

[6] Bujalance, E., J.J. Etayo, J.M. Gamboa, and G. Gromadzki: Automorphism Groups of Compact Bordered Klein Surfaces. - Lecture Notes in Math. 1439, Springer-Verlag, Berlin, 1990.

[7] Buser, P., and M. Seppälä: Real structures of Teichmüller spaces, Dehn twist, and moduli spaces of curves. - Math. Z. 232, 1999, 547–558.

[8] Buser, P., M. Seppälä, and R. Silhol: Triangulations and moduli spaces of Riemann surfaces with group actions. - Manuscripta Math. 88, 1995, 209–224.

[9] Cornalba, M.: Systèmes pluricanoniques sur l’espace des modules des courbes et diviseurs de courbes k -gonales (d’après Harris et Mumford). - Astérisque 121–122, 1985, 7–24.

[10] Costa, A.F., and M. Izquierdo: On the locus of real algebraic curves. - Atti Sem. Mat. Fis. Univ. Modena 49, 2001, 91–107.

[11] Earle, C.J.: On moduli of closed Riemann surfaces with symmetries. - In: Advances in the Theory of Riemann Surfaces. Ann. of Math. Stud. 66, Princeton, 1971, 119–130.

[12] Frediani, P.: Real algebraic functions, real algebraic curves and their moduli spaces. - Ph.D. Thesis, University of Pisa, 1998.

356 Antonio F. Costa and Milagros Izquierdo

[13] Frediani, P.: Monodromies of generic real algebraic functions. - Topology Appl. 99, 1999, 67–97.

[14] González Díez, G.: On prime Galois coverings of the Riemann sphere. - Ann. Mat. Pura Appl. 168, 1995, 1–15.

[15] Gromazki, G.: On a Harnack–Natanzon theorem for the family of real forms of Riemann surfaces. - J. Pure Appl. Algebra 121, 1997, 253–269.

[16] Gross, B.H., and J. Harris: Real algebraic curves. - Ann. Sci. École Norm. Sup. 14, 1981, 157–182.

[17] Hoare, A.H.M.: Subgroups of NEC groups and finite permutation groups. - Quart. J. Math. Oxford 41, 1990, 45–59.

[18] Hoare, A.H.M., and D. Singerman: The orientability of subgroups of plane groups. - In: Groups, St Andrews, 1981, 221–227.

[19] Izquierdo, M., and D. Singerman: Pairs of symmetries of Riemann surfaces. - Ann. Acad. Sci. Fenn. Math. 23, 1998, 3–24.

[20] Jones, G., and D. Singerman: Complex Functions. - Cambridge University Press, Cambridge, 1988.

[21] Klein, F.: Über Realitätverhältnisse bei der einem beliebigen Geschlechte zugehörigen Normalkurve der φ. - Math. Ann. 42, 1892, 1–29.

[22] Seppälä, M.: Teichmüller spaces of Klein surfaces. - Ann. Acad. Sci. Fenn. A I Math. Dissertationes 15, 1978, 1–37.

[23] Seppälä, M.: Real algebraic curves in the moduli space of complex curves. - Comp. Math. 74, 1990, 259–283.

[24] Seppälä, M., and T. Sorvali: Affine coordinates for Teichmüller spaces. - Math. Ann. 284, 1989, 165–176.

Received 16 July 2001