ON MODULI OF POINTED REAL CURVES OF GENUS ZERO I

ÖZGÜR CEYHAN

Abstract. We introduce the moduli space of pointed real curves of genus 0 \overline{M}_{(2k,l)}^{ℝ} and give its natural stratification. The strata of \overline{M}_{(2k,l)}^{ℝ} correspond to real curves with different degeneration types and may be encoded by trees with certain decorations. By using this stratification, we calculate the first Steifel-Whitney class of \overline{M}_{(2k,l)}^{ℝ} and construct the orientation double cover M̃_{(2k,l)} of \overline{M}_{(2k,l)}^{ℝ}.

Contents

1. Introduction 2 Acknowledgment 3 2. Preliminaries: Pointed Complex Curves and Their Moduli 3 2.1. Pointed Curves and Their Trees 3 2.2. Deformations of Pointed Curves 5 2.3. Stratification of the Moduli Space \overline{M}_{n} 6 2.4. Forgetful morphism 6 2.5. Automorphisms of \overline{M}_{n} 7 3. Moduli of Pointed Real Curves of Genus 0 8 3.1. Real Structures on \overline{M}_{n} 8 3.2. σ-Invariant Pointed Curves and σ-Equivariant Families of Pointed

Curves 9

3.3. The Moduli Space of Pointed Real Curves \overline{M}_{(2k,l)}^{ℝ} 10 4. Stratification of \overline{M}_{(2k,l)}^{ℝ} 11 4.1. Spaces of ℤ_{2}-Equivariant Point Configurations in ℂℙ^{1} 11 4.2. O/U-Planar Trees: One-vertex Case. 13 4.3. Pointed Real Curves and O-Planar Trees 16 4.4. O-Planar Trees and Their Configuration Spaces 17 4.5. Boundaries of the Strata 18 5. The First Steifel-Whitney Class of \overline{M}_{(2k,l)}^{ℝ} 23 5.1. Orientations of Top-Dimensional Strata 23 5.2. Induced Orientations on Codimension One Strata 24 5.3. Conventions 27 5.4. The First Steifel-Whitney Class 27 6. The Orientation Double Covering of \overline{M}_{(2k,l)}^{ℝ} 30 6.1. Construction of M̃_{(2k,l)} 30 6.2. Combinatorial Types of Strata of M̃_{(2k,l)} 33 References 34

Date: October 24, 2005.

The author is partially supported by the Israeli Science Foundation grant 86/01 and by the Collège Doctoral Européen.

1. Introduction

The moduli space \overline{M}_{n} of stable n-pointed (complex) curves of genus 0 has been extensively studied as one of the fundamental models of moduli problems in algebraic geometry (see [Ke, Knu, KM1, KM2, M1]). The moduli space \overline{M}_{n} carries a set of antiholomorphic involutions. The fixed point sets of these involutions are the moduli spaces of pointed real curves of genus 0. These moduli spaces parameterize the isomorphism classes of pointed curves of genus 0 with a real structure. For each of these spaces, a different set of labelled points stay in the real parts of the curves and the different pairs of labelled points are conjugated by the real structures of the curves. The moduli spaces of pointed real curves have recently attracted attention in various contexts such as multiple ζ-motives [GM], representations of quantum groups [EHKR, HK] and Welschinger invariants [W1, W2].

The aim of this note is to explore the topological properties of the moduli spaces of pointed real curves of genus 0. Hence, we first introduce a natural combinatorial stratification of the moduli spaces of pointed real curves of genus 0 through the stratification of \overline{M}_{n}. Each stratum is determined by the degeneration type of the real curve. They are identified with spaces of real point configurations in the projective line. The degeneration types of the pointed real curves are encoded by trees with corresponding decorations. Secondly, we calculate the first Steifel-Whitney classes of the moduli spaces in terms of their stratifications. The moduli spaces of pointed real curves are not orientable for n ≥ 5. We construct the orientation double covers of the moduli spaces for the non-orientable cases. The double covering which is constructed in this work significantly differs from the ‘double covering’ in the recent literature on open Gromov-Witten invariants and moduli spaces of pseudoholomorphic discs (see [FOOO, Liu]): Our double covering has no boundaries.

Section 2 contains a brief overview of facts about the moduli space \overline{M}_{n} and its intersection theory. In Section 3, we introduce real structures on \overline{M}_{n} and real parts \overline{M}_{(2k,l)}^{ℝ} as moduli spaces of (2k, l)-pointed real curves. The following section, the stratification of \overline{M}_{(2k,l)}^{ℝ} is given according to the degeneration types of pointed real curves of genus 0. In Section 5, the first Steifel-Whitney class of \overline{M}_{(2k,l)}^{ℝ} is calculated by using the stratification given in Section 4. Then in Section 6, we construct the orientation double coverings M̃_{(2k,l)} → \overline{M}_{(2k,l)}^{ℝ}.

Acknowledgment. I take this opportunity to express my deep gratitude to my supervisors V. Kharlamov and M. Polyak for their assistance and guidance in this work. I also wish to thank to Yu. I. Manin for his interest and suggestions which have been invaluable for this work.

I am thankful to E. Ha, A. Mellit, D. Radnell and A. Wand for their comments and suggestions.

Thanks are also due to Max-Planck-Institut für Mathematik, Israel Institute of Technology and Institut de Recherhe Mathématique Avancée de Strasbourg for their hospitality.

2. Preliminaries: Pointed Complex Curves and Their Moduli

This section reviews the basic facts on pointed complex curves of genus 0 and their moduli space. Moduli spaces of genus 0 curves with labelled points are stratified according to degeneration types of curves. The group of holomorphic automorphisms Aut_{♯}(\overline{M}_{n}) respecting the stratification is discussed.

In this paper, the genus of the curves is 0 except when the contrary is stated explicitly. Therefore, we omit mentioning the genus of the curves.

2.1. Pointed Curves and Their Trees.

Definition 2.1. An n-pointed curve (Σ; p) is a connected complex algebraic curve Σ with distinct, smooth, labelled points p = (p_{1}, ⋯, p_{n}) ⊂ Σ, satisfying the following conditions:

• Σ has only nodal singularities.

• The arithmetic genus of Σ is equal to 0. A family of n-pointed curves over a complex manifold S is a proper, holomorphic map π_{S} : U_{S} → S with n sections p_{1}, ⋯, p_{n} such that each geometric fiber (Σ(s), p(s)) is an n-pointed curve.

Two such curves (Σ; p) and (Σ′, p′) are isomorphic if there exists a biholomorphic equivalence Φ : Σ → Σ′ mapping p_{s} to p′_{s}.

An n-pointed curve is stable if its automorphism group is trivial (i.e, on each irreducible component, the number of singular points plus the number of labelled points is at least three).

2.1.1. Graphs.

Definition 2.2. A graph Γ is a collection of finite sets of vertices V_{Γ} and flags (or half edges) F_{Γ} with a boundary map ∂_{Γ} : F_{Γ} → V_{Γ} and an involution j_{Γ} : F_{Γ} → F_{Γ} (j_{Γ}^{2} = id). We call E_{Γ} = {(f_{1}, f_{2}) ∈ F_{Γ}^{2} | f_{1} = j_{Γ}f_{2} & f_{1} ≠ f_{2}} the set of edges, and T_{Γ} = {f ∈ F_{Γ} | f = j_{Γ}f} the set of tails. For a vertex v ∈ V_{Γ}, let F_{Γ}(v) = ∂^{−1}(v) and |v| = |F_{Γ}(v)| be the valency of v.

We think of a graph Γ in terms of its following geometric realization ||Γ||: Consider the disjoint union of closed intervals ⨆_{f_{i}∈F_{Γ}}[0, 1] × f_{i} and identify (0, f_{i}) with (0, f_{j}) if ∂_{Γ}f_{i} = ∂_{Γ}f_{j}, and identify (t, f_{i}) with (1 − t, j_{Γ}f_{i}) for t ∈ [0, 1] and f_{i} ≠ f_{j}. The geometric realization of Γ has a piecewise linear structure.

Definition 2.3. A tree τ is a graph whose geometric realization is connected and simply-connected. If |v| > 2 for all v ∈ V_{τ} then such a tree is called stable.

We associate a subtree τ_{v} for each vertex v ∈ V_{τ} which is given by V_{τ_{v}} = {v}, F_{τ_{v}} = F_{τ}(v), j_{τ_{v}} = id, and ∂_{τ_{v}} = ∂_{τ}.

Definition 2.4. Let τ and γ be trees with n tails. A morphism between these trees ϕ : τ → γ is a pair of maps ϕ_{F} : F_{γ} → F_{τ} and ϕ_{V} : V_{τ} → V_{γ} satisfying the following conditions:

• ϕ_{F} is injective and ϕ_{V} is surjective.

• The following diagram commutes

F_{τ} ⟶^{∂_{τ}} V_{τ}
ϕ_{F} ↑  ↓^{ϕ}_{V}
F_{γ} ⟶^{∂_{γ}} V_{γ}.

• ϕ_{F} ∘ j_{γ} = j_{τ} ∘ ϕ_{F}.

• ϕ_{T} is a bijection.

An isomorphism ϕ : τ → γ is a morphism where ϕ_{F} and ϕ_{V} are bijections. We denote the isomorphic trees by τ ≈ γ.

Each morphism induces a piecewise linear map on geometric realizations.

Lemma 2.5. Let τ and γ be stable trees with n tails. Any isomorphism ϕ : τ → γ is uniquely defined by its restriction on tails ϕ_{T} : T_{γ} → T_{τ}.

Proof. Let ϕ, φ : τ → γ be two isomorphisms such that their restriction on tails are the same. Consider the path _{t_{1}}P_{t_{2}} in ||τ|| that connects a pair of tails t_{1}, t_{2}. The automorphism φ^{−1} ∘ ϕ of τ maps _{t_{1}}P_{t_{2}} to itself; otherwise, the union of the t_{1}P_{t_{2}} and its image φ^{−1} ∘ ϕ(_{t_{1}}P_{t_{2}}) gives a loop in ||τ||, which contradicts simplyconnectedness. Moreover, the restriction of φ^{−1} ∘ ϕ to the path _{t_{1}}P_{t_{2}} is identity map since it preserves distances of vertices to tails. This follows from the compatibility of the automorphism φ^{−1} ∘ ϕ with ∂_{τ} and j_{τ}.

The geometric realization ||τ|| of τ can be covered by paths that connects pairs of tails of τ. We conclude that the automorphism φ^{−1} ∘ ϕ is identity since it is identity every such path. □

There are only finitely many isomorphisms classes of stable trees whose set of tails is T_{τ} = {1, ⋯, n}. We call the isomorphism classes of such trees n-trees. We denote the set of all n-trees by T ree.

2.1.2. Dual Trees of Pointed Curves. Let (Σ; p) be an n-pointed curve and Σ̂ → Σ be its normalization. Let (Σ̂_{v}; p̂_{v}) be the following |v|-pointed stable curve: Σ̂_{v} is a component of Σ̂, and the labelled points p̂_{v} be the set of points consisting of the preimages of special (i.e. labelled and nodal) points on Σ_{v} under the normalization map. The points p̂_{v} = (p_{f_{1}}, ⋯, p_{f_{|v|}}) on Σ̂_{v} are ordered by the flags f_{∗} ∈ F_{τ}(v).

Definition 2.6. The dual tree τ of an n-pointed curve (Σ; p) is the tree consisting of following data:

• V_{τ} is the set of components of Σ̂.

• F_{τ} is the set consisting of the preimages of special points.

• ∂_{τ} : f ↦ v if and only if p_{f} ∈ Σ̂_{v}.

• j_{τ} : f ↦ f if and only if p_{f} is a labelled point, and j_{τ} : f_{1} ↦ f_{2} if and only if p_{f_{1}} ∈ Σ̂_{v_{1}} and p_{f_{2}} ∈ Σ̂_{v_{2}} are the preimages of the nodal point Σ_{v_{1}} ∩ Σ_{v_{2}}.

Lemma 2.7. Let Φ be an isomorphism between the n-pointed stable curves (Σ; p) and (Σ′, p′).

(i) Φ induces an isomorphism ϕ between their dual trees τ, γ.

(ii) Φ is uniquely defined by its restriction on labelled points.

Proof. (i) The result follows from the decomposition of Φ into its restriction to each irreducible component and the Def. 2.4.

(ii) Due to Lemma 2.5, the isomorphism ϕ : τ → γ is determined by the restriction of Φ to the labelled points. The isomorphism ϕ determines which component of Σ is mapped to which component of Σ′ as well as the restriction of Φ to the special points. Each component of Σ is rational and has at least three special points. Therefore, the restriction of Φ to a component uniquely given by the images of the three special points. □

2.2. Deformations of Pointed Curves. Let τ be the dual tree of (Σ; p) and Σ̂ → Σ be the normalization. Let (Σ̂_{v}; p̂_{v}) be the following |v|-pointed stable curve corresponding to the irreducible component Σ_{v} of (Σ; p). Let Ω_{Σ}^{1} be the sheaf of Kähler differentials.

The infinitesimal deformations of a nodal curve Σ with divisor D_{p} = p_{1} + ⋯ + p_{n} is canonically identified with the complex vector space

Ext_{O_{Σ}}^{1}(Ω_{Σ}^{1}(D_{p}), O_{Σ}), (1)

and the obstruction lies in

Ext_{O_{Σ}}^{2}(Ω_{Σ}^{1}(D_{p}), O_{Σ}).

In this case, it is known that there is no obstructions (see, for example [Liu]).

The space of infinitesimal deformations is the tangent space of the space of deformations at (Σ; p). It can be written explicitly in the following form:

⨁_{v∈V_{τ}} H^{1}(Σ̂_{v}, TΣ̂_{v}(−Dp̂_{v})) ⊕ ⨁_{(f_{e},f^{e})∈E_{τ}} T_{p_{fe}} Σ̂ ⊗ T_{p_{f^{e}}} Σ̂. (2)

The first part corresponds to the equisingular deformations of Σ with the divisor Dp̂_{v} = ∑_{f_{i}∈F_{τ} (v)} p_{i}, and the second part corresponds to the smoothing of nodal points p_{e} of the edges e = (f_{e}, f^{e}) (see [HM]).

2.2.1. Combinatorics of Degenerations. Let (Σ; p) be an n-pointed curve with the dual tree τ. Consider the deformation of a nodal point of (Σ; p). Such a deformation of (Σ; p) gives a contraction of an edge of τ : Let e = (f_{e}, f^{e}) ∈ E_{γ} be the edge corresponding to the nodal point and ∂_{τ}(e) = {v_{e}, v^{e}}, and consider the equivalence relation ∼ on the set of vertices, defined by: v ∼ v for all v ∈ V_{γ} \ {v_{e}, v^{e}}, and v_{e} ∼ v^{e}. Then, there is a tree τ/e whose vertices are V_{τ}/ ∼ and whose flags are F_{τ} \{f_{e}, f^{e}}. The boundary map and involution of τ/e are the restrictions of ∂_{τ} and j_{τ}.

We denote by τ > γ, if τ is obtained by contracting some edges of γ.

2.3. Stratification of the Moduli Space \overline{M}_{n}. The moduli space \overline{M}_{n} is the space of isomorphism classes of n-pointed stable curves. This space is stratified according to degeneration types of n-pointed stable curves which are given by n-trees. The principle stratum M_{n} corresponds to the one-vertex n-tree and it is the quotient of the product (ℂℙ^{1})^{n} minus the diagonals Δ = ⋃_{k<l}{(p_{1}, ⋯, p_{n})|p_{k} = p_{l}} by Aut(ℂℙ^{1}) = P SL_{2}(ℂ).

Theorem(Knudsen & Keel, [Knu, Ke]). (i) For any n ≥ 3, \overline{M}_{n} is a smooth projective algebraic variety of (real) dimension 2n − 6.

(ii) Any family of n-pointed stable curves over S is induced by a unique morphism κ : S → \overline{M}_{n}. The universal family of curves \overline{U}_{n} of \overline{M}_{n} is isomorphic to \overline{M}_{n+1}.

(iii) For any n-tree, there exists a quasi-projective subvariety D_{τ} ⊂ \overline{M}_{n} parameterizing the curves whose dual tree is given by τ. D_{τ} is isomorphic to ∏_{v∈V_{τ}} M_{|v|}. Its (real) codimension is 2|E_{τ}|.

(iv) \overline{M}_{n} is stratified by pairwise disjoint subvarieties D_{τ}. The closure of any stratum D_{τ} is stratified by {D_{γ} | τ > γ}.

Example 2.8. (i) From the definition it follows that \overline{M}_{n} is empty for n < 3. \overline{M}_{3} is simply a point, and its universal curve \overline{U}_{3} is ℂℙ^{1} endowed with three points.

(ii) The moduli space \overline{M}_{4} is ℂℙ^{1} with three points. These points D_{τ_{1}}, D_{τ_{2}} and D_{τ_{3}} correspond to the curves with two irreducible components, and M_{4} is the complement of these three points (see Fig. 1). The universal family \overline{U}_{4} is a del Pezzo surface of degree 5 which is obtained by blowing up three points of ℂℙ^{1} × ℂℙ^{1}.

(iii) The moduli space \overline{M}_{5} is isomorphic to \overline{U}_{4}. It has ten divisors and each of these divisors contains three codimension two strata. The corresponding 5-trees are shown in Fig. 1.

2.4. Forgetful morphism. We say that (Σ; p_{1}, ⋯, p_{n−1}) is obtained by forgetting the labelled point p_{n} of the n-pointed curve (Σ; p_{1}, ⋯, p_{n}). However, the resulting pointed curve may well be unstable. This happens when the component Σ_{v} of Σ

Figure 1. All strata of \overline{M}_{n} for n = 3, 4, and 5.

supporting p_{n} has only two additional special points. In this case, we contract this component to its intersection point(s) with the components adjacent to Σ_{v}. With this stabilization we extend this map to whole space and obtain π_{n} : \overline{M}_{n} → \overline{M}_{n−1}. The forgetfull morphism π_{n} is a fibration whose fiber over (Σ; p) ∈ \overline{M}_{n−1} is Σ. In other words, π_{n} : \overline{M}_{n} → \overline{M}_{n−1} is the universal family of curves. For details, see [Ke, Knu].

2.5. Automorphisms of \overline{M}_{n}. The open stratum M_{n} of the moduli space \overline{M}_{n} can be identified with the orbit space ((ℂℙ^{1})^{n} \ Δ)/P SL_{2}(ℂ). The latter orbit space may be viewed as the configuration space of (n − 3) ordered distinct points of ℂℙ^{1} \ {0, 1, ∞}:

M_{n} ≅ {p = (z_{1}, ⋯, z_{n}) ∈ ℂ^{n−3} | z_{i} ≠ z_{j} ∀i ≠ j, z_{n−2} = 0, z_{n−1} = 1, z_{n} = ∞}.

where z_{i} := [z_{i} : 1] are the coordinates of labelled points p_{i} in an affine chart of ℂℙ^{1}.

Let ψ = (ψ_{1}, …, ψ_{n−3}) : M_{n} → M_{n} be a non-constant holomorphic map. In [K], Kaliman discovered the following fact:

Theorem (Kaliman, [K]). For n ≥ 4, every non-constant holomorphic endomorphism ψ = (ψ_{1}, ⋯, ψ_{n−3}) of M_{n} is an automorphism and its components ψ_{r} are of the form

ψ_{r}(p) = \frac{z_{ϱ(r)} − z_{ϱ(n−2)}}{z_{ϱ(r)} − z_{ϱ(n)}} / \frac{z_{ϱ(n−2)} − z_{ϱ(n−1)}}{z_{ϱ(n)} − z_{ϱ(n−1)}} , 1 ≤ r ≤ n − 3

where ϱ ∈ S_{n} is a permutation not depending on r.

Kaliman’s theorem implies the following corollary.

Theorem (Kaliman & Lin, [K, Lin]). Every holomorphic automorphism of M_{n} is produced by a certain permutation ϱ ∈ S_{n}. Hence, Aut(M_{n}) ≅ S_{n}.

On the other hand, the permutation S_{n} acts on the compactification \overline{M}_{n} of M_{n} via relabeling: for ϱ ∈ S_{n}, there is an holomorphic map ψ_{ϱ} which is given by

ψ_{ϱ} : (Σ; p) ↦ (Σ; ϱ(p)) := (Σ; p_{ϱ(1)}, ⋯, p_{ϱ(n)}). (3)

Therefore, the permutation action given in (3) forms a subgroup of holomorphic automorphisms Aut(\overline{M}_{n}).

Let Aut_{♯}(\overline{M}_{n}) be the group of holomorphic automorphisms of \overline{M}_{n} that respect the stratification: ψ ∈ Aut_{♯}(\overline{M}_{n}) maps D_{τ} onto D_{γ} where dimD_{τ} = dimD_{γ}. Kaliman’s theorem leads us to the following immediate corollary.

Theorem 1. The group Aut_{♯}(\overline{M}_{n}) is S_{n}.

Proof. Let ψ ∈ Aut_{♯}(\overline{M}_{n}). The restriction of ψ to the open stratum gives the permutation action on M_{n} since the automorphism group of the open stratum M_{n} contains only permutations ψ_{ϱ}. The unicity theorem of holomorphic maps implies that ψ = ψ_{ϱ} since they coincide on the open stratum ψ|_{M_{n}} = ψ_{ϱ}|_{M_{n}}. □

Remark 2.9. Note that the whole group of holomorphic automorphisms Aut(\overline{M}_{n}) is not necessarily isomorphic to S_{n}. For example, the automorphism groups of \overline{M}_{4} is P SL_{2}(ℂ). According to our knowledge, there is no systematic exposition of Aut(\overline{M}_{n}) for n > 5.

3. Moduli of Pointed Real Curves of Genus 0

A real structure on complex variety X is an anti-holomorphic involution c_{X} : X → X. The fixed point set ℝX = Fix(c_{X}) of the involution is called real part of the variety (or of the real structure).

In this section, the moduli spaces of pointed real curves of genus 0 are introduced as the fixed point sets of real structures on \overline{M}_{n}.

3.1. Real Structures on \overline{M}_{n}. The moduli space \overline{M}_{n} comes equipped with a natural real structure. The involution c : (Σ; p) ↦ (\overline{Σ}; p) acts on \overline{M}_{n}. Here a complex curve Σ is regarded as a pair Σ = (C, J), where C is the underlying 2-dimensional manifold and J is a complex structure on it, and \overline{Σ} = (C, −J) is its complex conjugated pair.^{1}

Lemma 3.1. The map c is a real structure on \overline{M}_{n}.

Proof. The differentiability of c follows form the Kodaira-Spencer construction of infinitesimal deformations. We need to show that the differential of c is anti-linear at each (Σ; p) ∈ \overline{M}_{n}. It is sufficient to show that it is anti-linear without taking the quotient with respect to P SL_{2}(ℂ).

The infinitesimal deformations of a nodal curve Σ with divisor D_{p} = p_{1} + ⋯ + p_{n} is canonically identified with the complex vector space Ext_{O_{Σ}}^{1}(Ω_{Σ}^{1}(D_{p}), O_{Σ}), (see Section 2.2). By reversing the complex structure on Σ, we reverse the complex structure on the tangent space Ext_{O_{Σ}}^{1}(Ω_{Σ}(D_{p}), O_{Σ}) at (Σ; p). The differential of the map (Σ; p) ↦ (\overline{Σ}; p)

Ext_{O_{Σ}}^{1}(Ω_{Σ}(D_{p}), O_{Σ}) → Ext_{O\overline{Σ}}^{1}(Ω\overline{Σ}(D_{p}), O\overline{Σ}), v ↦ v̄

1There is some notational ambiguity. The bar over \overline{M}_{n} and that over \overline{Σ} refer to different structures on underlying manifolds: the first one refers to the compactification of \overline{M}_{n} and second refer to the manifold with reverse complex structure. Both of these notations are widely used, we use the bar for both cases. The context should make it clear which structure is referred to.

is clearly anti-linear. □

The group of holomorphic automorphism Aut_{♯}(\overline{M}_{n}) respecting the stratification is the permutation group S_{n} (see Theorem 1). The permutation group S_{n} acts on \overline{M}_{n} via relabeling as given in (3). So that, the set of anti-holomorphic automorphisms that respect the stratification can be obtain by composing the principle real structure (Σ; p) ↦ (\overline{Σ}, p) with a permutation ψ_{ϱ} of ϱ ∈ S_{n}.

For each involution σ ∈ S_{n}, we have an additional real structure on \overline{M}_{n}:

(4)  c_{σ} := c ∘ ψ_{σ} : (Σ; p) ↦ (\overline{Σ}; σ(p)).

Lemma 3.2. Every real structure of \overline{M}_{n} preserving the stratification are of the form (4) where σ ∈ S_{n} is an involution.

Proof. The holomoprhic and anti-holomorphic automorphisms preserving the stratification are in ℤ_{2}-extension of the subgroup of automorphisms preserving the stratification Aut_{♯}(\overline{M}_{n}) ⊂ Aut(\overline{M}_{n}) by the principle real structure c. Therefore, every real structure preserving the stratification of \overline{M}_{n} are given by a certain involution σ ∈ S_{n} and are of the form (4). □

3.2. σ-Invariant Pointed Curves and σ-Equivariant Families of Pointed Curves.

Definition 3.3. An n-pointed stable curve (Σ; p) is called σ-invariant if it admits a real structure conj : Σ → Σ such that conj(p_{i}) = p_{σ(i)} for all i ∈ {1, ⋯, n}.

A family of n-pointed stable curves π_{S} : U_{S} → S is called σ-equivariant if a pair of maps c_{S} : s ↦ c_{S}(s), c_{U} : s × (Σ(s), p(s)) ↦ c_{S}(s) × (\overline{Σ}(s), σ(p(s))) act as real structures respectively on S and U, and the following diagram commutes

U_{S} ⟶^{c_{U}} U_{S}
π_{S} ↓  ↓^{π_{S}}
S ⟶^{c_{S}} S.

Notice that, if (Σ; p) is σ-invariant, then the real structure conj is uniquely determined by its restriction on the labelled points due to Lemma 2.7.

Lemma 3.4. Let π : U_{S} → S be a σ-equivariant family. Each (Σ; p) ∈ ℝS is σ-invariant.

Proof. If c_{S}((Σ; p)) = (Σ; p) then there exists a unique biholomorphic equivalence conj : Σ → \overline{Σ} (in other words, anti-holomorphic conj : Σ → Σ) such that conj(p_{i}) = p_{σ(i)}. The restriction of conj on labelled points is an involution. By applying Lemma 2.7 to conj^{2}, we determine that conj is an involution on Σ. □

3.3. The Moduli Space of Pointed Real Curves \overline{M}_{(2k,l)}^{ℝ}. Let Fix(σ) be the fixed point set of the action of σ on the labeling set {1, ⋯, n}, and let Perm(σ) be its complement.

Let ϱ ∈ S_{n} and ψ_{ϱ} be the corresponding automorphism of \overline{M}_{n}. The conjugation of real structure c_{σ} with ψ_{ϱ} provide a conjugate real structure c_{σ′} = ψ_{ϱ} ∘ c_{σ} ∘ ψ_{ϱ^{−1}}. The conjugacy classes of real structures are determined by the cardinalities |Fix(σ)| and |Perm(σ)|. Therefore, from now on, we only consider c_{σ} where

σ = ( \begin{matrix}1 & ⋯ & k & k + 1 & ⋯ & 2k & 2k + 1 & ⋯ & n \\ k + 1 & ⋯ & 2k & 1 & ⋯ & k & 2k + 1 & ⋯ & n \end{matrix} ) . (5)

Definition 3.5. For σ as above in (5), σ-invariant curves are called (2k, l)-pointed real curves.

The fixed point set Fix(c_{σ}) is called the moduli space of (2k, l)-pointed real curves and denoted by \overline{M}_{(2k,l)}^{ℝ}.

Theorem 2. (i) For any n ≥ 3, \overline{M}_{(2k,l)}^{ℝ} is a smooth projective manifold of real dimension n − 3.

(ii) The universal family of curves π : \overline{U}_{n} → \overline{M}_{n} is a σ-equivariant family.

(iii) Any σ-equivariant family of n-pointed stable curves over π_{S} : U_{S} → S is induced by a unique pair of real morphisms

U_{S} ⟶^{κ̂} \overline{U}_{n}
π_{S} ↓  ↓^{π}
S ⟶^{κ} \overline{M}_{n}.

Proof. (i) The smoothness of the real part of c_{σ} is a consequence of the implicit function theorem, and dim_{ℝ} \overline{M}_{(2k,l)}^{ℝ} = dim_{ℂ} \overline{M}_{n} = n − 3 since the real part \overline{M}_{(2k,l)}^{ℝ} is not empty. (ii) The fiber over (Σ; p) ∈ \overline{M}_{n} is π^{−1}((Σ; p)) = Σ. We define real structures on \overline{M}_{n} and \overline{U}_{n} as follows;

c_{σ} : (Σ; p)  ↦ (\overline{Σ}, σ(p)),
ĉ_{σ} : z ∈ π^{−1}((Σ; p)) ↦ z ∈ π^{−1}((\overline{Σ}, σ(p))),

The real structures c_{σ}, ĉ_{σ} satisfy the conditions of σ-equivariant families in Definition 3.3. (iii) Due to Knudsen’s theorem (see Section 2.3), each of the morphisms κ : S → \overline{M}_{n} and κ̂ : U_{S} → \overline{U}_{n} are unique. Therefore, they are the same as c_{σ} ∘ κ ∘ c_{S} : S → \overline{M}_{n} and ĉ_{σ} ∘ κ̂ ∘ c_{U} : U_{S} → \overline{U}_{n}. Hence, the morphisms κ, κ̂ are real. □

Corollary 3. Let 𝔐_{σ} be the contravariant functor that sends a real variety (S, c_{S}) to the σ-equivariant family of curves over S. The real variety (\overline{M}_{n}, c_{σ}) represents the moduli functor 𝔐_{σ}.

Proof. The statement is immediate consequence of the definition of moduli functor and Theorem 2. □

4. Stratification of \overline{M}_{(2k,l)}^{ℝ}

A stratification for \overline{M}_{(2k,l)}^{ℝ} can be obtained by the stratification of \overline{M}_{n} given in Section 2.3.

Lemma 4.1. Let τ and \overline{τ} be the dual trees of (Σ; p) and (\overline{Σ}; σ(p)) respectively.

(i) If τ and \overline{τ} are not isomorphic, then the restriction of c_{σ} on the union of complex strata D_{τ} ⋃ D\overline{τ} gives a real structure with empty real part.

(ii) If τ and \overline{τ} are isomorphic, then the restriction of c_{σ} on D_{τ} gives a real structure on it and the real part ℝD_{τ} is the intersection of \overline{M}_{(2k,l)}^{ℝ} with D_{τ}.

Proof. (i) Since τ and \overline{τ} are not isomorphic, D_{τ} and D\overline{τ} are disjoint complex strata. The restriction of c_{σ} on D_{τ} ⋃ D\overline{τ} swaps the strata, therefore the real part of this real structure is empty.

(ii) Since τ and \overline{τ} are isomorphic, the n-pointed curves (Σ; p) and (\overline{Σ}; σ(p)) are in D_{τ}. Therefore, D_{τ} = D\overline{τ} and the restriction of c_{σ} on D_{τ} is a real structure. The real part ℝD_{τ} of the σ-equivariant family D_{τ} is \overline{M}_{(2k,l)}^{ℝ} ⋂ D_{τ} since \overline{M}_{(2k,l)}^{ℝ} = Fixc_{σ}. □

A tree τ is called σ-invariant if it is isomorphic to τ̄, and the set of σ-invariant n-trees are denoted by T ree(σ).

Theorem 4. \overline{M}_{(2k,l)}^{ℝ} is stratified by real analytic subsets ℝD_{τ} where τ ∈ T ree(σ).

Proof. The statement is a direct consequence of Lemma 4.1. □

Although the notion of σ-invariant trees leads us a combinatorial stratification of \overline{M}_{(2k,l)}^{ℝ} given in Theorem 4, it does not give a stratification in terms of connected strata. For a σ-invariant τ, the real part of the strata ℝD_{τ} has many connected components. In the next subsection, we refine this stratification by using the spaces of ℤ_{2}-equivariant point configurations in projective line ℂℙ^{1}.

4.1. Spaces of ℤ_{2}-Equivariant Point Configurations in ℂℙ^{1}. Let z := [z : 1] be the affine coordinate on ℂℙ^{1}. Consider the upper half-plane ℍ^{+} = {z ∈ ℂℙ^{1} | ℑ(z) > 0} (resp. lower half plane ℍ^{−} = {z ∈ ℂℙ^{1} | ℑ(z) < 0}) as a half of the ℂℙ^{1} with respect to z ↦ z̄, and the real part ℝℙ^{1} as its boundary. Denote by ℍ the compactified disc ℍ^{+} ∪ ℝℙ^{1}.

4.1.1. Irreducible (2k, l)-Pointed Real Curves and Configuration Spaces. Let (Σ; p) be an irreducible (2k, l)-pointed real curve. As a real curve, Σ is isomorphic to ℂℙ^{1} with real structure which is either z ↦ z̄ or z ↦ −1/z̄ (see, for example [M2]). The real structure z ↦ −1/z̄ has empty real part. Notice that a (2k, l)-pointed curve with empty real part is possible only when Fix(σ) = ∅.

We will consider the configuration spaces of real curves with non-empty and empty real parts as separate cases.

Case I. Pointed Real Curves with Non-empty Real Part. Each finite subset p of ℂℙ^{1} which is invariant under the real structure z ↦ z̄ inherits additional structures:

I. An oriented cyclic ordering on Fix(σ): For any point p ∈ (p ⋂ ℝℙ^{1}) there is unique q ∈ (p ⋂ ℝℙ^{1}) which follows the point p in the positive direction of ℝℙ^{1} (the direction in which the coordinate x := [x : 1] on ℝℙ^{1} increases). The elements of p are labelled, therefore the cyclic ordering can be seen as a linear ordering on (p ⋂ ℝℙ^{1}) \ {p_{n}}. This linear ordering gives an oriented cyclic ordering on Fix(σ) = {2k +1, ⋯, n} which we denote by {i_{1}} < ⋯ < {i_{l−1}} < {i_{l}} where i_{l} = n.

II. A 2-partition on Perm(σ): The subset p ⋂(ℂℙ^{1}\ℝℙ^{1}) of p admits a partition into two disjoint subsets {p_{i} ∈ ℍ^{±}}. This partition gives an ordered 2partition Perm^{±} := {i | p_{i} ∈ ℍ^{±}} of Perm(σ). The subsets Perm^{±} are swapped by the permutation σ.

The set of data o := {(ℂℙ^{1}, z ↦ z̄); Perm^{±}; Fix(σ) = {{i_{1}} < ⋯ < {i_{l}}}} is called oriented combinatorial type of the ℤ_{2}-equivariant point configuration p on (ℂℙ^{1}, z ↦ z̄).

The oriented combinatorial types of equivariant point configurations on (ℂℙ^{1}, z ↦ z̄) enumerate the connected components of the space { \atop Conf}_{(2k,l)} of k distinct pairs of conjugate points on ℍ^{+} ⋃ ℍ^{−} and l distinct points on ℝℙ^{1}:

{ \atop Conf}_{(2k,l)} := \{(p_{1}, ⋯, p_{2k}; q_{1}, ⋯, q_{l}) | p_{i} ∈ ℂℙ^{1} \ ℝℙ^{1}, p_{i} = p_{j} ⇔ i = j,
p_{i} = p̄_{j} ⇔ i = σ(j) & q_{i} ∈ ℝℙ^{1}, q_{i} = q_{j} ⇔ i = j}.

The number of connected components is 2^{k}(l − 1)!.^{2} They are all diffeomorphic to each other; natural diffeomorphisms are given by σ-invariant relabeling.

Let z := [z : 1] be the affine coordinate on ℂℙ^{1} and x := [x : 1] be affine coordinate on ℝℙ^{1}. The action of SL_{2}(ℝ) on ℍ is given by

SL_{2}(ℝ) × ℍ → ℍ, (Λ, z) ↦ Λ(z) = \frac{az + b}{cz + d}, Λ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℝ)

in affine coordinates. It induces an isomorphism SL_{2}(ℝ)/ ± I → Aut(ℍ). The automorphism group Aut(ℍ) acts on { \atop Conf}_{(2k,l)}

Λ : (z_{1}, ⋯, z_{2k}; x_{1}, ⋯, x_{l}) ↦ (Λ(z_{1}), ⋯, Λ(z_{2k}); Λ(x_{1}), ⋯, Λ(x_{l})).

It preserves each of the connected components of { \atop Conf}_{(2k,l)}. This action is free when 2k + l ≥ 3 and it commutes with diffeomorphisms given by σ-invariant relabelings. Therefore, the quotient space C̃_{(2k,l)} := { \atop Conf}_{(2k,l)}/Aut(ℍ) is a manifold of dimension 2k + l − 3 whose connected components are pairwise diffeomorphic.

2Here we use the convention n! = 1 whenever n ≤ 0.

In addition to the automorphisms considered above, there is a diffeomorphism −𝕀 of { \atop Conf}_{(2k,l)} which is given in affine coordinates as follows.

−𝕀 : (z_{1}, ⋯, z_{2k}; x_{1}, ⋯, x_{l}) ↦ (−z_{1}, ⋯, −z_{2k}; −x_{1}, ⋯, −x_{l}).

Consider the quotient space Conf_{(2k,l)} = { \atop Conf}_{(2k,l)}/(−𝕀). Note that, −𝕀 interchanges components with reverse combinatorial types. Namely, the combinatorial type \overline{o} of −𝕀(p) is obtained from the combinatorial type o of p by reversing the cyclic ordering on Fix(σ) and swapping Perm^{+} and Perm^{−}. The equivalence classes of oriented combinatorial types with respect to the action of −𝕀 are called un-oriented combinatorial types of ℤ_{2}-equivariant point configurations on (ℂℙ^{1}, z ↦ \overline{z}). The un-oriented combinatorial types enumerate the connected components of Conf_{(2k,l)}.

The diffeomorphism −𝕀 commutes with each σ-invariant relabeling and normalize the action of Aut(ℍ). Therefore, the quotient space C_{(2k,l)} := Conf_{(2k,l)}/Aut(ℍ) is a manifold of dimension 2k + l − 3, its connected components are diffeomorphic to the components of C̃_{(2k,l)}, and, moreover, the quotient map C̃_{(2k,l)} → C_{(2k,l)} is a trivial double covering. Case II. Pointed Real Curves with Empty Real Part. Let (Σ; p) be an irreducible (2k, 0)-pointed curve and let ℝΣ = ∅. Such a pointed real curve is isomorphic to (ℂℙ^{1}, p) with real structure conj : z ↦ −1/z̄.

The group of automorphisms of ℂℙ^{1} which commutes with conj is

Aut(ℂℙ^{1}, conj) := \{( \begin{matrix}a & b \\ −b̄ & ā \end{matrix} ) ∈ SL_{2}(ℂ)\} .

Thus, the group Aut(ℂℙ^{1}, conj) acts naturally on the space

Conf_{(2k,0)}^{∅} := {(z_{1}, ⋯, z_{2k}) | z_{i} = −1/z̄_{i+k}}

of ℤ_{2}-equivariant point configurations on (ℂℙ^{1}, z ↦ −1/z̄). For k ≥ 2, the action is free and the quotient B_{(2k,0)} := Conf_{(2k,0)}^{∅}/Aut(ℂℙ^{1}, conj) is a 2k − 3 dimensional connected manifold. For ℤ_{2}-equivariant point configurations on (ℂℙ^{1}, z ↦ −1/\overline{z}), the data {(ℂℙ^{1}, z ↦ −1/z̄)} is called the combinatorial type of the ℤ_{2}-equivariant point configurations.

4.2. O/U-Planar Trees: One-vertex Case. An oriented planar (o-planar) structure on the one-vertex n-tree τ is one of the two possible set of data

o := \{ \begin{matrix}\{(ℂℙ^{1}, z ↦ z̄); a σ − equivariant two − partition Perm^{±} of Perm(σ); \\ an oriented cyclic ordering on Fix(σ)\}, \\ \{(ℂℙ^{1}, z ↦ −1/z̄)\}. \end{matrix}

We denote the o-planar trees by (τ, o).

An un-oriented planar (u-planar) structure u on the one-vertex n-tree τ is a pair of reverse o-planar structure \{o, \overline{o}\}. We denote the u-planar trees by (τ, u).

Remark 4.2. For the case (ℂℙ^{1}, z ↦ −1/z̄), the o-planar and u-planar structures have the same set of data. However, this convention above is suitable for our presentation.

As it is shown in the Section 4.1.1, each connected component of C_{(2k,l)} for l > 0 (resp. C_{(2k,0)} ⋃ B_{(2k,0)} for l = 0) is associated to a unique u-planar tree since the group Aut(ℍ) is connected and the un-oriented combinatorial types of ℤ_{2}-equivariant point configurations are encoded by the same set of data. We denote the connected components of C_{(2k,l)} (and C_{(2k,0)} ⋃ B_{(2k,0)}) by C_{(τ,u)}. Similarly, each connected component of C̃_{(2k,l)} is associated to a unique o-planar tree. We denote the connected components of C̃_{(2k,l)} by C_{(τ,o)}.

4.2.1. A Normal Position of ℤ_{2}-Equivariant Point Configurations on ℂℙ^{1}. By using the automorphisms we can make the following choices for the representatives of the points in C̃_{(2k,l)} and B_{(2k,0)}. Case I. Pointed Real Curves with Non-empty Real Part. Every element in C̃_{(2k,l)} is represented by (ℂℙ^{1}, p) with p ∈ { \atop Conf}_{(2k,l)}. Calibrating the choice by Aut(ℍ), consider an isomorphism (ℂℙ^{1}, p) ↦ (ℂℙ^{1}, p′) which puts the labelled points in the following normal position p′ ∈ ℂℙ^{1}.

A. In the case l ≥ 3, the three consecutive labelled points (p′_{i_{l−1}}, p′_{n}, p′_{i_{1}}) in ℝℙ^{1} are put in the position x′_{i_{l−1}} = 1, x′_{n} = ∞, x′_{i_{1}} = 0. We then obtain

p′ = (z_{1}, ⋯, z_{k}, \overline{z}_{k+1}, ⋯, \overline{z}_{2k}, x_{2k+1}, ⋯, x_{2k+l−1}, ∞).

B. In the case l = 1, 2, the three labelled points {p_{k}, p_{2k}, p_{n}} are put in the position \{±\sqrt{−1}, ∞\}. Then,

p′ = \{ \begin{matrix}(z_{1}, ⋯, z_{k−1}, ϵ\sqrt{−1}, \overline{z}_{k+1}, ⋯, \overline{z}_{2k−1}, −ϵ\sqrt{−1}, x_{2k+1}, ∞) & if l = 2, \\ (z_{1}, ⋯, z_{k−1}, ϵ\sqrt{−1}, \overline{z}_{k+1}, ⋯, \overline{z}_{2k−1}, −ϵ\sqrt{−1}, ∞) & if l = 1 \end{matrix}

where ϵ = ±. C. In the case l = 0,the labelled points \{p_{k}, p_{2k}\} are fixed at \{±\sqrt{−1}\} and p_{i} where \{i\} = \{k −1, 2k −1\} ⋂ Perm^{+} is placed on the interval ]0, \sqrt{−1}[⊂ ℍ^{+}. Then,

p′ = (z_{1}, ⋯, z_{k−2}, ϵ_{1}λ\sqrt{−1}, ϵ_{2}\sqrt{−1}, \overline{z}_{k+1}, ⋯, \overline{z}_{2k−2}, −ϵ_{1}λ\sqrt{−1}, −ϵ_{1}\sqrt{−1}).

where λ ∈]0, 1[ and ϵ_{i} = ±, i = 1, 2.

Remark 4.3 (A′.). In case of k > 1 and l > 3, we can consider the alternative map which puts the labelled points p′ into the following normal positions. In this case, three labelled points \{p_{k}, p_{2k}, p_{n}\} can be put at \{z_{k}, z_{2k}, x_{n}\} = \{±\sqrt{−1}, ∞\} by the action of Aut(ℍ). We then obtain

p′ = (z_{1}, ⋯, z_{k−1}, ϵ\sqrt{−1}, \overline{z}_{k+1}, ⋯, \overline{z}_{2k−1}, −ϵ\sqrt{−1}, x_{2k+1}, ⋯, x_{2k+l−1}, ∞)

where ϵ = ±.

Case II. Pointed Real Curves with Empty Real Part. Every element of B_{(2k,0)}, k ≥ 2, is represented by (ℂℙ^{1}, p) with p ∈ Conf_{(2k,0)}^{∅}. Calibrating the choice by Aut(ℂℙ^{1}, conj), consider an isomorphism (ℂℙ^{1}, p) ↦ (ℂℙ^{1}, p′) which puts the labelled points of (Σ; p) in the following normal position p′ ∈ ℂℙ^{1}.

D.

p = (z_{1}, ⋯, z_{k−2}, λ, 1, \frac{−1}{\overline{z}_{k+1}} , ⋯, \frac{−1}{\overline{z}_{2k−2}} , −\frac{1}{λ}, −1), where λ ∈] − 1, 1[.

4.2.2. Every ℤ_{2}-equivariant point configuration defines a (2k, l)-pointed real curve. Hence, we define

(6)  Ξ : ⨆_{(τ,u)} C_{(τ,u)} → M_{(2k,l)}^{ℝ}

which maps ℤ_{2}-equivariant point configurations to the corresponding isomorphisms classes of irreducible (2k, l)-pointed curves.

Lemma 4.4. (i) The map Ξ is a diffeomorphism.

(ii) The configuration space C_{(τ,u)} is diffeomorphic to • ((ℍ^{+})^{k} \ Δ) × □^{l−3} when l > 2,

• ((ℍ^{+} \ \{\sqrt{−1}\})^{k−1} \ Δ) × □^{l−1} when l = 1, 2,

• ((ℍ^{+} \\{\sqrt{−1}, \sqrt{−1}/2\})^{k−2} \Δ)×□^{1} when l = 0 and the type of real structure is (ℂℙ^{1}, z ↦ z̄),

• ((ℂℙ^{1} \ {−1, −1/2, 1/2, 1})^{k−2} \ (Δ ∪ Δ^{c})) × □^{1} when l = 0 and the type of

the real structure is (ℂℙ^{1}, z ↦ −1/z̄). Here, Δ is the union of all diagonals z_{i} ≠ z_{j}(i ≠ j), Δ^{c} is the union of all crossdiagonals z_{i} ≠ −\frac{1}{z̄_{j}} (i ≠ j), and □^{l} is the l-dimensional open simplex.

Proof. (i) The map Ξ is clearly smooth. It is surjective since any (2k, l)-pointed irreducible curve is isomorphic either to (ℂℙ^{1}, z ↦ z̄) or (ℂℙ^{1}, z ↦ −1/z̄) with a ℤ_{2}- equivariant point configuration p on it. It is injective since the group of holomorphic automorphisms commuting with the real structure z ↦ z̄ is generated by Aut(ℍ) and −𝕀, and the group of holomorphic automorphisms commuting with the real structure z ↦ −1/z̄ is Aut(ℂℙ^{1}, conj). These automorphism are taken into account during construction of the configuration spaces.

(ii) As it is shown in Section 4.1.1 C_{(τ,u)} is the quotient C_{(τ,o)} ⨆ C_{(τ,ō)}/(−𝕀). The spaces C_{(τ,u)} and C_{(τ,o)} are clearly diffeomorphic. To replace C_{(τ,u)} by C_{(τ,o)}, we choose an o-planar representative for each one-vertex u-planar tree (τ, u) among (τ, o), (τ, ō) as follows.

• l ≥ 3 case: Let (τ, o) be the representative of (τ, u) for which p_{2k+1} < p_{n−1} < p_{n} with respect to the cyclic ordering on Fix(σ).

• l = 0, 1, 2 case: Let (τ, o) be the representative of (τ, u) for which k ∈ Perm^{+}. We put the ℤ_{2}-equivariant point configurations into a normal position as in 4.2.1.

The parameterizations stated in (ii) for l > 0 cases follow from (A) and (B) of 4.2.1. In the case of l = 0 and (ℂℙ^{1}, z ↦ z̄), according to (C) the configuration space C_{(τ,o)} is a locally trivial fibration over □^{1} =]0, 1[ whose fibers over λ ∈ □^{1} are (ℍ^{+} \ \{\sqrt{−1}, λ\sqrt{−1}\})^{k−2} \ Δ. Similarly, in the case of l = 0 and (ℂℙ^{1}, z ↦ −1/z̄), according to (D) the configuration space C_{(τ,u)} is a locally trivial fibration over □^{1} =] − 1, 1[ whose fibers over λ ∈ □^{1} are (ℂℙ^{1} \ {−1, −1/λ, λ, 1})^{k−2} \ Δ. Since the bases of these locally trivial fibrations are contractible, they are trivial fibrations, and the result follows. □

4.3. Pointed Real Curves and O-Planar Trees. This section extends the notions of o/u-planar structures to the all σ-invariant trees.

4.3.1. Notations. Let (Σ; p) ∈ ℝD_{τ} for some τ ∈ T ree(σ) and c_{Σ} : Σ → Σ be the real structure on Σ. We denote the set of real components {v ∈ V_{τ} | c_{Σ}(Σ_{v}) = Σ_{v}} of Σ by V_{τ}^{ℝ}. If this set is empty, then ℝΣ is an isolated real node; we call the edge of τ representing the isolated real node special invariant edge.

Two vertices v, v̄ ∈ V_{τ} \ V_{τ}^{ℝ} are said to be conjugate if the real structure c_{Σ} maps the components Σ_{v} and Σv̄ onto each other. Similarly, we call the flags f, f̄ ∈ F_{τ} \ {f_{∗} ∈ F_{τ} | p_{f_{∗}} ∈ ℝΣ} conjugate if c_{Σ} swaps the corresponding special points p_{f}, pf̄.

4.3.2. O/U-Planar Trees: General Case. Let τ ∈ T ree(σ). Each (2k, l)-pointed curve (Σ; p) ∈ ℝD_{τ} with ℝΣ ≠ ∅ admits additional structures:

I′. If V_{τ}^{ℝ} ≠ ∅, one can fix an oriented combinatorial type of point configurations on each real component Σ_{v}. Namely, for v ∈ V_{τ}^{ℝ} with ℝΣ ≠ ∅, an oriented combinatorial type o_{v} is given by an oriented cyclic ordering on the set of invariant flags F_{τ}^{ℝ}(v) = {f | p_{f} ∈ ℝΣ_{v}} and an ordered 2-partition F_{τ}^{±}(v) = {i | p_{i} ∈ ℍ^{±}} of F_{τ}(v) \ F_{τ}^{ℝ}(v). In the case of V_{τ}^{ℝ} = {v^{∗}} and ℝΣ = ∅, the o-planar structure is simply given by the type of the real structure i.e, ℝΣ_{v^{∗}} = ∅ (as in 4.1.1).

II′. If V_{τ}^{ℝ} = ∅, then one can fix an ordering of the flags of the special invariant

edge: f_{e} ↦ ±, f^{e} ↦ ∓. This additional structures motivate the following definition.

An o-planar structure on τ is the following set of data,

o := \{ \begin{matrix}\{(τ_{v}, o_{v}) | v ∈ V_{τ}^{ℝ}\} & when V_{τ}^{ℝ} ≠ ∅, \\ \{ordering \{f_{e}, f^{e}\} → \{±\}\}, & when V_{τ}^{ℝ} = ∅, \end{matrix}

where (τ_{v}, o_{v}) is an o-planar structure on the one-vertex tree τ_{v} for v ∈ V_{τ}^{ℝ}, and e = (f_{e}, f^{e}) is the special invariant edge of τ when V_{τ}^{ℝ} = ∅.

Similarly, an u-planar structure on τ is the following set of data,

u := \{ \begin{matrix}\{(τ_{v}, u_{v}) | v ∈ V_{τ}^{ℝ}\} & when V_{τ}^{ℝ} ≠ ∅, \\ \{∅\} & when V_{τ}^{ℝ} = ∅, \end{matrix}

where (τ_{v}, u_{v}) is an u-planar structure on the one-vertex tree τ_{v} for v ∈ V_{τ}^{ℝ}. We denote u-planar trees by (τ, u).

Remark 4.5. The real structures on real components are not independent; either all are conjugate to z ↦ z̄ or there is unique real component with a real structure conjugate to z ↦ −1/z̄. In the latter case we denote this unique real component by Σ_{v^{∗}}.

4.3.3. Notations. We associate the subsets of vertices V_{τ}^{±} and flags F_{τ}^{±} to every oplanar tree (τ, o) when V_{τ}^{ℝ} ≠ ∅ as follows. Let v_{1} ∈ V_{τ} \ V_{τ}^{ℝ} and let v_{2} ∈ V_{τ}^{ℝ} be the closest invariant vertex to v_{1} in the geometric realization ||τ|| of τ. Let f ∈ F_{τ}(v_{2}) be in the shortest path connecting the vertices v_{1} and v_{2}. The vertex v_{1} ∈ V_{τ}^{±} whenever the flag f ∈ F_{τ}^{±}(v_{2}). The subset of flags F_{τ}^{±} is defined as ∂_{τ}^{−1}(V_{τ}^{±}).

Similarly, we associate the subsets of vertices V_{τ}^{±} and flags F_{τ}^{±} to every o-planar tree (τ, o) when ℝΣ is isolated real node: Let f_{e} ↦ + and ∂_{τ}(f_{e}) = v_{e}. The vertex v ∈ V_{τ} is an element of the subset V_{τ}^{+} whenever it is closer to v_{e} than v^{e} in ||τ||. The complement V_{τ} \ V_{τ}^{+} is denoted by V_{τ}^{−}. The subset of flags F_{τ}^{±} is defined as ∂_{τ}^{−1}(V_{τ}^{±}).

4.4. O-Planar Trees and Their Configuration Spaces. We associate a space to each o-planar tree as follows.

C_{(τ,o)} := \{ \begin{matrix}∏_{v∈V_{τ}^{ℝ}} C_{(τ_{v},o_{v})} × ∏_{v∈V_{τ}^{+}} M_{|v|} & when V_{τ}^{ℝ} ≠ ∅ and (ℂℙ^{1}, z ↦ z̄) \\ C_{(τ_{v^{∗}},o_{v^{∗}})} × ∏_{\{v,v̄\}⊂V_{τ} \V_{τ}^{ℝ}} M_{|v|} & when V_{τ}^{ℝ} ≠ ∅ and (ℂℙ^{1}, z ↦ −1/z̄) \\ ∏_{v∈V_{τ}^{+}} M_{|v|} & when V_{τ}^{ℝ} = ∅ \end{matrix}

Here the product for z ↦ −1/\overline{z} case runs over the un-ordered pairs of conjugate vertices belonging to V_{τ} \ V_{τ}^{ℝ} i.e, {v, v̄} = {v̄, v}.

For each u-planar tree (τ, u), we first choose an o-planar representative and then put C_{(τ,u)} = C_{(τ,o)}. Note that, so defined space C_{(τ,u)} does not depend on the o-planar representatives up to an isomorphism.

Lemma 4.6. Let τ ∈ T ree(σ). The real part ℝD_{τ} is diffeomorphic to ⨆_{(τ,u)} C_{(τ,u)} where the disjoint union is taken over all possible u-planar structures of τ.

Proof. The complex strata D_{τ} is diffeomorphic to the product ∏_{v∈V_{τ}} \overline{M}_{|v|}. The real structure c_{σ} : D_{τ} → D_{τ} maps the factor \overline{M}_{|v|} onto \overline{M}_{|v̄|} when v and v̄ are conjugate vertices, and maps the factor \overline{M}_{|v|} onto itself when v ∈ V_{τ}^{ℝ}. Therefore the real part ℝD_{τ} of c_{σ} is given by

∏_{v∈V_{τ}^{ℝ}} C_{(2k_{v},l_{v})} × ∏_{\{v,v̄\}⊂V_{τ} \V_{τ}^{ℝ}} \overline{M}_{|v|} when |V_{τ}^{ℝ}| > 1,
(C_{(|v^{∗}|,0)} ⨆ B_{(|v^{∗}|,0)}) × ∏_{\{v,v̄\}⊂V_{τ} \V_{τ}^{ℝ}} \overline{M}_{|v|} when|V_{τ}^{ℝ}| = 1,
∏_{\{v,v̄\}⊂V_{τ} \V_{τ}^{ℝ}} \overline{M}_{|v|} when|V_{τ}^{ℝ}| = 0,

where k_{v} = |F_{τ}^{+}(v)| and l_{v} = |F_{τ}^{ℝ}(v)|. The decompositions of C_{(2k,l)} and C_{(2k,0)} ⨆ B_{(2k,0)} into their connected components are given in Lemma 4.4. □

Theorem 5. \overline{M}_{(2k,l)}^{ℝ} is stratified by C_{(τ,u)}.

Proof. The moduli space \overline{M}_{(2k,l)}^{ℝ} can be stratified by ℝD_{τ} due to Theorem 4. The claim directly follows from the stratification of open strata ℝD_{τ} given in Lemma 4.6. □

Remark 4.7. Above stratification gives a cell decomposition in the case of \overline{M}_{(0,l)}^{ℝ}. This cell decomposition has been examined in [De, DJS, GM, Ka]. In [De], Devadoss gave the cell decomposition of \overline{M}_{(0,l)}^{ℝ} and also calculated the Euler characteristic of this space. The hyperbolicity and the fundamental group of \overline{M}_{(0,l)}^{ℝ} are discussed in [DJS]. A double covering of \overline{M}_{(0,l)}^{ℝ} is given in [Ka].

4.5. Boundaries of the Strata. In this section we investigate the adjacency of the strata C_{(τ,u)} in \overline{M}_{(2k,l)}^{ℝ}. We start with considering the complex situation in order to introduce natural coordinates near each codimension one stratum C_{(τ,u)}.

4.5.1. Intermezzo: Coordinates at the Codimension One Strata. Let γ be a 2-vertex n-tree with V_{γ} = {v_{e}, v^{e}}, F_{γ}(v^{e}) = {i_{1},⋯,i_{s}, f^{e}} and F_{γ}(v_{e}) = {f_{e}, i_{s+1}, ⋯, i_{n−1}, n}. Let (z, w) := [z : 1] × [w : 1] be affine coordinates on ℂℙ^{1} × ℂℙ^{1}.

Consider a neighborhood V ⊂ D_{γ} of a nodal n-pointed curve (Σ^{o}, p^{o}) ∈ D_{γ}. Any (Σ; p) ∈ V can be identified with a nodal curve {(z − z_{f_{e}}) · (w − w_{f^{e}}) = 0} in ℂℙ^{1} × ℂℙ^{1} with special points p_{v_{e}} = (a_{f_{e}}, a_{i_{s+1}}, ⋯, a_{i_{n−1}}, a_{n}) ⊂ {w − w_{f^{e}} = 0} and p_{v^{e}} = (b_{f^{e}}, b_{i_{1}}, ⋯, b_{i_{s}}) ⊂ {z − z_{f_{e}} = 0}. In order to determine a nodal curve in ℂℙ^{1} ×ℂℙ^{1} and the position of its special points uniquely defined by (Σ; p), we make the following choice. Firstly, we fix three labelled points a_{i_{s+1}}, a_{i_{n−1}}, a_{n} on the line {w − w_{f^{e}} = 0} whenever |v_{e}| > 3, and three special points a_{f_{e}}, a_{i_{n−1}}, a_{n} whenever |v_{e}| = 3. Secondly, we fix three special points b_{f^{e}}, b_{i_{1}}, b_{i_{s}} on {z − z_{f_{e}} = 0}. Finally, we choose: a_{i_{s+1}} = (0, 0), a_{i_{n−1}} = (1, 0), a_{n} = (∞, 0) for |v_{e}| > 3; a_{f_{e}} = (0, 0), a_{i_{n−1}} = (1, 0), a_{n} = (∞, 0) for |v_{e}| = 3; and b_{i_{1}} = (z_{f_{e}}, 1), b_{i_{s}} = (z_{f_{e}}, ∞), b_{f^{e}} = (z_{f_{e}}, 0). Then the components z and w of the special points provide a coordinate system in V; in particular, for |v_{e}| > 3 such a coordinate system is formed by z_{f^{e}}, z_{i_{∗}} with i_{∗} = i_{s+2}, ⋯, i_{n−2}, and w_{j_{∗}} with j_{∗} = i_{2}, ⋯, i_{s−1}.

We now consider a family of n-pointed curves over V times the ϵ-ball B_{ϵ} = {|t| < ϵ}. It is given by a family curves {(z − z_{f_{e}}) · w + t = 0 | t ∈ B_{ϵ}} in ℂℙ^{1} × ℂℙ^{1}. The labelled points (z_{i}, w_{i}), i = 1, ⋯ n on these curves are chosen in the following way. If |v_{e}| > 3, we put (z_{i_{1}}, w_{i_{1}}) = (z_{f_{e}} − t, 1), (z_{i_{s}}, w_{i_{s}}) = (z_{f_{e}}, ∞), (z_{i_{s+1},w_{i_{s+1}}}) = (0, t/z_{f_{e}}), (z_{i_{n−1}}, w_{i_{n−1}}) = (1, −t/(1 − z_{f_{e}})) and (z_{n}, w_{n}) = (∞, 0). Similarly, for |v_{e}| = 3, (z_{i_{1}}, w_{i_{1}}) = (−t, 1), (z_{i_{n−2}}, w_{i_{n−2}}) = (0, ∞), (z_{i_{n−1}}, w_{i_{n−1}}) = (1, −t) and (z_{n}, w_{n}) = (∞, 0). The other labelled points are taken in an arbitrary position. The component z of the special points and the parameter t provide a coordinate system in V × B_{ϵ}.

Due to Knudsen’s theorem there exists a unique κ : V × B_{ϵ} → \overline{M}_{n} which gives the family of n-pointed curves given above.

Lemma 4.8. det(dκ) ≠ 0 at (Σ^{o}, p^{o}) ∈ D_{γ}. Hence, κ gives a local isomorphism.

Proof. The parameter t gives a regular function on κ(V × B_{ϵ}) which is 0 along D_{γ} ⋂ κ(V × B_{ϵ}). The differential dκ(v⃗) = v⃗ for v⃗ ∈ T_{(Σ^{o},p^{o})}V since the restriction of κ on V × {0} is identity map. On the other hand, dκ(∂_{t}) ≠ 0 since the curves are non-isomorphic for different values of parameter t. □

Remark 4.9. Due to Lemma 4.8, the coordinates on V × B_{ϵ} provides a coordinates system at (Σ^{o}, p^{o}) ∈ D_{γ}. There is a natural coordinate projection ρ : V × B_{ϵ} → V that is given by the projections of labelled points of the curves {(z −z_{f_{e}})·w +t = 0} onto {w = 0} axes.

For a σ-invariant γ and c_{σ}-invariant V, above coordinates and the local isomorphism κ are equivariant with respect to a suitable real structure on ℂℙ^{1} × ℂℙ^{1}. Therefore, the real part ℝV ×] − ϵ, ϵ[ of V × B_{ϵ} provides a neighborhood for a (Σ^{o}, p^{o}) in ℝD_{γ} with a set of coordinates on it.

4.5.2. Contraction Morphisms for O-Planar Trees. Let (γ, ô) be an o-planar tree and ϕ : γ → τ be a morphism of n-trees contracting an invariant set of edges E_{con} = E_{γ} \ϕ_{E}(E_{τ}). In such a situation, we associate a particular o-planar structure o on τ, as it is described below in separate cases A and B, and speak on a contraction morphism φ : (γ, ô) → (τ, o). In all the cases, except A-(2), the o-planar structure o is uniquely defined by ô.

A. Let E_{con} = {e = (f_{e}, f^{e})} and e be an invariant edge.

(1) If ∂_{γ}(e) = {v_{e}, v^{e}} ⊂ V_{γ}^{ℝ}, we then convert the o-planar structures at the vertices v_{e} and v^{e} given by

\begin{aligned}ô_{v_{e}} & = & \{ℝΣ_{v_{e}} ≠ ∅; F_{γ}^{±}(v_{e}); F_{γ}^{ℝ}(v_{e}) = \{\{i_{1}\} < ⋯ < \{i_{m}\} < \{f_{e}\}\}\} \\ ô_{v^{e}} & = & \{ℝΣ_{v^{e}} ≠ ∅; F_{γ}^{±}(v^{e}); F_{γ}^{ℝ}(v^{e}) = \{\{i′_{1}\} < ⋯ < \{i′_{m′}\} < \{f^{e}\}\}\}. \end{aligned}

to an o-planar structure at vertex v = ϕ_{V} ({v_{e}, v^{e}}) of (τ, o) defining it by

\begin{aligned}o_{v} & = & \{ℝΣ_{v} ≠ ∅; F_{τ}^{±}(v) = F_{γ}^{±}(v_{e}) ⋃ F_{γ}^{±}(v^{e}); \\ & & F_{τ}^{ℝ}(v) = \{\{i_{1}\} < ⋯ < \{i_{m}\} < \{i′_{1}\} < ⋯ < \{i′_{m′}\}\}\}. \end{aligned}

The o-planar structures are kept unchanged at all other invariant vertices.

(2) If e is a special invariant edge, then we convert the o-planar structure ô = {f_{e} ↦ +, f^{e} ↦ −} of γ into an o-planar structure at the vertex v = ϕ_{V} ({v_{e}, v^{e}}) of τ defining it by

o_{v} = {ℝΣ_{v} ≠ ∅; F_{τ}^{+}(v) = F_{γ}(v_{e}) \ {f_{e}}, F_{τ}^{−}(v) = F_{γ}(v^{e}) \ {f^{e}}; F_{τ}^{ℝ}(v) = ∅}.

or by

o_{v} = {ℝΣ_{v} = ∅}.

B. Let E_{con} = {e_{i} = (f_{e_{i}}, f^{e_{i}}) | i = 1, 2} where f_{e_{i}}, i = 1, 2 and f^{e_{i}}, i = 1, 2 are conjugate pairs of flags.

(1) If ∂_{γ}(e_{i}) = {v̂, v^{e_{i}}}, and v̂ ∈ V_{γ}^{ℝ}, v^{e_{i}} ∉ V_{γ}^{ℝ}, then we convert the o-planar structure at the vertex v̂ given by

ov̂ = {ℝΣv̂ ≠ ∅; F_{γ}^{±}(v̂); F_{γ}^{ℝ}(v̂) = {{i_{1}} < ⋯ < {i_{m}}}}.

to an o-planar structure at vertex v = ϕ({v̂, v^{e_{1}}, v^{e_{2}}}) of τ defining it by

o_{v} = {ℝΣ ≠ ∅; F_{τ}^{+}(v) = F_{γ}^{+}(v̂) ⋃ F_{γ}^{+}(v^{e_{1}}) \ {f_{e_{1}}, f^{e_{1}}},
F_{τ}^{−}(v) = F_{γ}^{−}(v̂) ⋃ F_{γ}^{−}(v^{e_{2}}) \ {f_{e_{2}}, f^{e_{2}}};
F_{τ}^{ℝ}(v) = {{i_{1}} < ⋯ < {i_{m}}}}.

(2) If E_{con} = {e_{i} = (f_{e_{i}}, f^{e_{i}}) | i = 1, 2} and ∂_{γ}(e_{i}) ⋂ V_{γ}^{ℝ} = ∅, then we define the o-planar structure at each vertex v in τ to be the same o-planar structure with the o-planar structure at same vertex v of (γ, ô).

Remark 4.10. Let (γ, ô) be an o-planar tree with ℝΣ ≠ ∅, and let φ_{e} : (γ, ô) → (τ, o) be the contraction of an edge e ∈ E_{γ}. If the o-planar tree (τ, o) and the u-planar tree (γ, û) underlying (γ, ô) are given, then the o-planar structure ô can be reconstructed. For this reason, when a stratum C_{(γ,û)} contained in the boundary of \overline{C}_{(τ,u)} is given, we denote the corresponding o-planar structure ô by δ(o).

Proposition 4.11. A stratum C_{(γ,û)} is contained in the boundary of \overline{C}_{(τ,u)} if and only if the u-planar structures u, û can be lifted to o-planar structures o, ô such a way that (τ, o) is obtained by contracting an invariant set of edges of (γ, ô).

Proof. We need to consider the statement only for the strata of codimension one and two. These cases correspond to the contraction morphisms from two/three-vertex oplanar (sub)trees to one-vertex o-planar (sub)trees given in A and B. For a stratum of higher codimension, the statement can be proved by applying the elementary contractions A and B inductively. Here, we consider only the case (A)-(1). The proof for other cases is the same.

We first assume that (τ, o) is obtained by contracting the edge e of (γ, ô), where (γ, ô) is an o-planar two-vertex tree with V_{γ} = V_{γ}^{ℝ} = {v_{e}, v^{e}}. An element (Σ; p) ∈ C_{(γ,ô)} can be represented by the nodal curve {(z − z_{f_{e}}) · w = 0} in ℂℙ^{1} × ℂℙ^{1} with special points a_{f} = (z_{f}, 0) and b_{f} = (z_{f_{e}}, w_{f}) such that

a_{f} ∈ {w = 0 & ℑ(z) > 0}  for f ∈ F_{γ}^{+}(v_{e})
af̄ ∈ {w = 0 & ℑ(z) < 0}  for f̄ ∈ F_{γ}^{−}(v_{e})
{a_{i_{1}} < ⋯ < a_{i_{m}}} ⊂ {w = 0 & ℑ(z) = 0} for i_{∗} ∈ F_{γ}^{ℝ}(v_{e})

on the axis w = 0, and b_{∗} = (z_{f_{e}}, w_{∗})

b_{f} ∈ {z − z_{f_{e}} = 0 & ℑ(w) > 0}  for f ∈ F_{γ}^{+}(v^{e})
bf̄ ∈ {z − z_{f_{e}} = 0 & ℑ(w) < 0}  for f̄ ∈ F_{γ}^{−}(v^{e})
{b_{i′_{1}} < ⋯ < b_{i′_{m′}}} ⊂ {z − z_{f_{e}} = 0 & ℑ(w) = 0} for i′_{∗} ∈ F_{γ}^{ℝ}(v^{e})

When we include the curve {(z−z_{f_{e}})·w = 0} into the family {(z−z_{f_{e}})·w+t = 0}, the complex orientation defined on the irreducible components w = 0 and z −z_{f_{e}} = 0 by the halves ℑ(z) > 0 and, respectively, ℑ(w) > 0 extends continuously to a complex orientation of {(z − z_{f_{e}}) · w + t = 0} with t ∈ [0, ϵ[ defined by, say, ℑ(z) > 0. As a result, the curves {(z − z_{f_{e}}) · w + t = 0} with t ∈ [0, ϵ[ acquire an o-planar structure given by

(z_{f}, w_{f}) ∈ {(z − z_{f_{e}}) · w + t = 0 & ℑ(z) > 0} for f ∈ F_{γ}^{+}(v_{e}) ⋃ F_{γ}^{+}(v^{e})
(z\overline{f}, w\overline{f}) ∈ \{(z − z_{f_{e}}) · w + t = 0 & ℑ(z) < 0\} for f̄ ∈ F_{γ}^{−}(v_{e}) ⋃ ∈ F_{γ}^{−}(v^{e})
(z_{f}, w_{f}) ∈ {(z − z_{f_{e}}) · w + t = 0 & ℑ(z) = 0} for f ∈ F_{γ}^{ℝ}(v_{e}) ⋃ F_{γ}^{ℝ}(v^{e})

where the points on the real part of the curves {(z − z_{f_{e}}) · w + t = 0} are cyclicly ordered by

z_{i_{1}} < ⋯ < z_{i_{m}} < z_{i′_{1}} < ⋯ < z_{i′_{m′}}.

This is exactly the o-planar structure (τ, o) defined in (A)-(1) of 4.5.2.

Now assume that C_{(γ,û)}, where (γ, û) is an u-planar tree with V_{γ} = V_{γ}^{ℝ} = {v_{e}, v^{e}}, is contained in the boundary of \overline{C}_{(τ,u)}. There are four different o-planar representatives of (γ, û), and any two non reverse to each other representatives ô_{1}, ô_{2} provide by contraction two different o-planar structures (τ, o_{i}), i = 1, 2. By the already proved part of the statement, C_{(γ,û)} is contained in the boundary of \overline{C}_{(τ,o_{i})} for each i = 1, 2. It remains to notice that any codimension one stratum is adjacent to at most two main strata. □

Example 4.12. (i) The first nontrivial example is \overline{M}_{4}. There are 3 types of real structures: c_{id}, c_{σ_{1}}, c_{σ_{2}}, where σ_{1} = (1 2) transposes labelled points p_{1}, and p_{2} and σ_{2} = (1 3)(2 4) transposes also labelled points p_{3} and p_{4}. These then give \overline{M}_{(2k,l)}^{ℝ}, where (2k, l) = (0, 4), (2, 2), and (4, 0) respectively.

In the case σ = id, M_{(0,4)}^{ℝ} is the space of configuration of 4 distinct points on ℝℙ^{1} up to the action of P SL_{2}(ℝ). The 4-pointed curve (Σ; p) ∈ M_{(0,4)}^{ℝ} can be identified with (0, x_{2}, 1, ∞) where x_{2} ∈ ℝℙ^{1} \ {0, 1, ∞} by the P SL_{2}(ℝ) action. Hence M_{(0,4)}^{ℝ} = ℝℙ^{1} \ \{0, 1, ∞\} and its compactification is \overline{M}_{(0,4)}^{ℝ} = ℝℙ^{1}. The three intervals of M_{(0,4)}^{ℝ} are the three configuration spaces C_{(τ,u_{i})} and the three points are the configuration spaces C_{(γ_{i},u_{i})}. The u-planar trees (τ, u_{i}) and (γ_{i}, u_{i}) are given in Fig. 2.

Figure 2. All strata of \overline{M}_{(0,4)}.

In the case σ = σ_{1}, M_{(2,2)}^{ℝ} is the space of distinct configurations of 2 points in ℝℙ^{1} and a pair of complex conjugate points in ℂℙ^{1}. (Σ; p) ∈ M_{(2,2)}^{ℝ} is identified with (\sqrt{−1}, −\sqrt{−1}, x_{3}, ∞) ∈ C_{(τ,u)}, −∞ < x_{3} < ∞. Hence, M_{(2,2)}^{ℝ} = ℝℙ^{1} \ \{∞\} and its compactification is \overline{M}_{(2,2)}^{ℝ} = ℝℙ^{1}. The interval M_{(2,2)}^{ℝ} is C_{(τ,u)} and the point at its closure is C_{(γ,u)}.

In the case σ = σ_{2}, the space M_{(4,0)}^{ℝ} has different pieces parameterizing real curves with non-empty and empty real parts: The subspace of M_{(4,0)}^{ℝ} parameterizing the (4, 0)-pointed real curve with ℝΣ ≠ ∅ is identified with (λ\sqrt{−1}, \sqrt{−1}, −λ\sqrt{−1}, −\sqrt{−1}) where λ ∈] − 1, 1[\{0}. The subspace of M_{(4,0)}^{ℝ} parameterizing the real curves with ℝΣ = ∅ is (λ, 1, −λ, −1), where λ ∈] − 1, 1[. Note that, the pieces parameterizing ℝΣ ≠ ∅ and ℝΣ = ∅ are joined through the boundary points corresponding to curves with isolated real singular points. The compactification \overline{M}_{(4,0)}^{ℝ} is ℝℙ^{1}.

(ii) The moduli space \overline{M}_{5} has 3 different types of real structures that are given by σ =id, (1 2) and (1 3)(2 4).

The space M_{(0,5)}^{ℝ} is identified with the configuration space of 5 distinct points on ℝℙ^{1} modulo P SL_{2}(ℝ). It is (ℝℙ^{1} \ {0, 1, ∞})^{2} \ Δ, where Δ is union of all diagonals. Each connected component of M_{(0,5)}^{ℝ} is isomorphic to a two dimensional simplex. The closure of each cell can be obtained by adding the boundaries given in Sec. 4.5; for an example see Fig. 3a. It gives the compactification of \overline{M}_{(0,5)}^{ℝ} which is a torus with 3 points blown up: the cells corresponding to u-planar trees (τ, u_{1}) and (τ, u_{2}) are glued along the face corresponding to (γ, u) which gives (τ, u_{i}), i = 1, 2 by contracting some edges, see Fig. 3b.

The space M_{(2,3)}^{ℝ} is isomorphic to configurations of a conjugate pair of points on ℂℙ^{1}. The automorphisms allows us to identify such configurations with (z, z̄, 0, 1, ∞) where z ∈ ℂ \ ℝ. Hence, it can be given as ℂℙ^{1} \ ℝℙ^{1}. The \overline{M}_{(2,3)}^{ℝ} is obtained as a sphere with 3 points blown up according to the stratification given in Sec. 4.5.

Finally, elements of M_{(4,1)}^{ℝ} can be identified with (z, \sqrt{−1}, z̄, −\sqrt{−1}, ∞). Hence it can be identified with ℂℙ^{1} \ (ℝℙ^{1} ∪ \{\sqrt{−1}, −\sqrt{−1}\}). Therefore, connected components are isomorphic to ℍ^{+} \ \{\sqrt{−1}\}. The \overline{M}_{(4,1)}^{ℝ} is a sphere with a point blown up.

Figure 3. (a) Stratification of C̃_{(τ,u)}. (b) The stratification of \overline{M}_{(0,5)}^{ℝ}.

In fact, these are all the possible real parts of the del Pezzo surfaces of degree 5 (see [DIK]).

5. The First Steifel-Whitney Class of \overline{M}_{(2k,l)}^{ℝ}

In this section we calculate the first Steifel-Whitney class of \overline{M}_{(2k,l)}^{ℝ} by using the stratification given in Theorem 5.

5.1. Orientations of Top-Dimensional Strata. For a one-vertex o-planar tree (τ, o), the coordinates on C_{(τ,o)}, that are introduced in Sec. 4.2.1, determine an orientation on the stratum C_{(τ,o)}. For instance, let Fix(σ) ≥ 3 and let the o-planar structure on (τ, o) be given by Perm^{±} and the linear ordering x_{i_{1}} = 0 < x_{i_{2}} < ⋯ < x_{i_{l−1}} = 1 < x_{i_{l}} := x_{n} = ∞ on Fix(σ). The coordinates in (A) of 4.2.1 gives the following top-dimensional differential form on C_{(τ,o)}:

ω_{(τ,o)} := (\frac{\sqrt{−1}}{2})^{k} ⋀_{α_{∗}∈Perm^{+}} dz_{α_{∗}} ∧ d\overline{z}_{α_{∗}} ⋀ dx_{i_{2}} ∧ ⋯ ∧ dx_{i_{l−2}}. (7)

The multiplication of top-dimensional forms with a positive valued function Θ : C_{(τ,o)} → ℝ_{>0} defines an equivalence relation on sections of det(T C_{(τ,o)}). An orientation is an equivalence class of nowhere zero top-dimensional form with repsect this equivalence relation. We denote equivalence class of the form ω_{(τ,o)} by [ω_{(τ,o)}].

Similarly, using the coordinates given in (A′), (B), (C) and (D) in 4.2.1 and their ordering, we determine differential forms ω_{(τ,o)} and corresponding orientation [ω_{(τ,o)}] on C_{(τ,o)} for all (τ, o) with |V_{τ}| = 1.

5.1.1. Orientations on Codimension One Strata. Let (γ, o) be a two-vertex o-planar tree. Let V_{γ} = {v_{e}, v^{e}} and e = (f_{e}, f^{e}) be the edge where ∂_{γ}(f_{e}) = v_{e} and ∂_{γ}(f^{e}) = v^{e}. Let v_{e} = ∂_{γ}(n).

By choosing three flags at vertices v_{e}, v^{e} and using the calibrations as in 4.2.1 we obtain a set of coordinates on C_{(γ_{v},o_{v})} for each v ∈ {v_{e}, v^{e}}. More precisely, we use the following choice.

I. Let ℝΣ_{v} ≠ ∅ for v ∈ {v_{e}, v^{e}} and let Fix(σ) ≠ ∅. If |F_{γ}^{ℝ}(v_{e})| ≥ 3 (resp. |F_{γ}^{ℝ}(v^{e})| ≥ 3), then we fix three consecutive labelled points (p_{i_{l−1}}, p_{n}, p_{i_{1}}) on Σ_{v_{e}} (resp. three consecutive special points (p_{i_{q+1}}, p_{i_{q+r}}, p_{f^{e}}) on Σ_{v^{e}} ) at (1, ∞, 0). If |F_{γ}^{ℝ}(v_{e})| < 3 (resp. |F_{γ}^{ℝ}(v^{e})| < 3), then in addition to p_{n} = ∞ (resp. p_{f^{e}} = 0), we fix a conjugate pair of labelled points (p_{α}, p\overline{α}) at (\sqrt{−1}, −\sqrt{−1}) where α is the maximal element in F_{γ}^{+}(v_{e}) (resp. F_{γ}^{+}(v^{e})).

II. Let ℝΣ_{v} ≠ ∅ for v ∈ {v_{e}, v^{e}} and let Fix(σ) = ∅. We put the special point p_{f^{e}} and p_{f_{e}} to 0 on the respective components. We put the conjugate pairs of special points \{p_{n}, p\overline{n}\} (resp. \{p_{α}, p\overline{α}\}) at \{±\sqrt{−1}\} (where α is the maximal element in F_{γ}^{+}(v^{e})).

III. Let ℝΣ_{v} be an isolated real node. We put the special point p_{f^{e}} and p_{f_{e}} to 0 on the respective components. We put the conjugate pairs of special points \{p_{n}, p_{α_{k−1}}\} (resp. \{p\overline{n}, p_{\overline{α}_{k−1}}\}) at \{±\sqrt{−1}\} (where α_{k−1} is the maximal element in F_{γ}^{+}(v_{e}) \ {n}).

The o-planar structures with above calibrations determine a set of coordinates of C_{(γ_{v^{e}},o_{v^{e}})} and C_{(γ_{ve},o_{ve})}. If the o-planar structure is o_{v} = {ℝΣ_{v} ≠ ∅; F_{γ}^{±}(v); F_{γ}^{ℝ} = {{i_{1}} < ⋯ < {i_{l_{v}}}}} at a vertex v ∈ {v_{e}, v^{e}}, then we give the coordinates by

(z_{α_{1}}, ⋯, z_{α_{kv}} , z_{\overline{α}_{1}}, ⋯, z_{\overline{α}_{kv}} , x_{i_{1}}, ⋯, x_{i_{lv}})

where α_{∗} ∈ F_{γ}^{+}(v) and three of the special points are fixed according to (I) or (II). If o-planar structure is o = {f_{e} ↦ ±, f^{e} ↦ ∓} then we give the coordinates by

(z_{α_{1}}, ⋯, z_{α_{kv}})

where ∂_{γ}(α_{1}), ⋯, ∂_{γ}(α_{k_{v}}) ∈ F_{γ}^{+} and three of the special points are fixed according to (III).

These set of coordinates and their ordering determine orientations [ω_{(γ_{v^{e}},o_{v^{e}})}] and [ω_{(γ_{ve},o_{ve})}] as in 5.1, therefore an orientation on C_{(γ,o)}. The orientation determined by these coordinates is

[ω_{(γ_{v^{e}},o_{v^{e}})}] ∧ [ω_{(γ_{ve},o_{ve})}] when V_{γ} = V_{γ}^{ℝ}
[ω_{(γ_{v},o_{v})}]  when V_{γ}^{ℝ} = ∅ and v ∈ V_{γ}^{+}.

5.2. Induced Orientations on Codimension One Strata. Let \overline{C}_{(τ,u)} be a top dimensional stratum and C_{(γ,û)} be a codimesion one stratum contained in the boundary of \overline{C}_{(τ,u)}. We lift the u-planar structures u, û to o-planar representatives o, ô = δ(o) such that (τ, o) is obtained by contracting the edge of (γ, δ(o)) (see Prop. 4.11). Then we pick a point (Σ^{o}, p^{o}) ∈ C_{(γ,δ(o))} and the tubular neighborhood ℝV × [0, ϵ[ of (Σ^{0}, p^{o}) in \overline{C}_{(τ,o)}.

The orientation [ω_{(τ,o)}], introduced in 5.1, induces some orientation on C_{(γ,δ(o))} which we denote by [ω_{(γ,δ(o))}]. It means that the differential form ω_{(γ,δ(o))} satisfies the following equation

(8)  −dt ∧ ω_{(γ,δ(o))} = ω_{(τ,o)},

since traditionally the outward normal direction chosen to be normal vector (see, for example [DFN]), and on the codimension one stratum C_{(γ,δ(o))} of \overline{C}_{(τ,o)} such normal vector is −∂_{t}.

In what follows we compare the induced orientation [ω_{(γ,δ(o))}] with [ω_{(γ_{v^{e}},o_{v^{e}})}] ∧ [ω_{(γ_{ve},o_{ve})}].

Figure 4. Codimension 1 boundaries of \overline{C}_{(τ,ô)} where l − r ≥ 3 & r ≥ 2.

5.2.1. Case I: l ≥ 1.

Lemma 5.1.

[ω_{(γ,δ(o))}] = (−1)^{ℵ} [ω_{(γ_{v^{e}},o_{v^{e}})}] ∧ [ω_{(γ_{ve},o_{ve})}]

where the values of ℵ for separate cases are given in the following table.

ℵ l − r ≥ 3 l − r = 2 l − r = 1
x_{l−1} < x_{f_{e}} < x_{l} x_{f_{e}} < x_{l−1} < x_{l}
r ≥ 2 (q + 1)(r + 1)  0  l + 1  l + 1
r = 1  1  1  1  1
r = 0  q + 1  0  0  0

Proof. We will illustrate the statement only the first case where l − r ≥ 3, r ≥ 2. The calculations for other cases are almost identical.

Let (Σ^{o}, p^{o}) ∈ C_{(γ,δ(o))}. We set Σ_{v_{e}}^{o} to be {w = 0} and Σ_{v^{e}}^{o} to be {z − z_{f_{e}} = 0}. According to the convention in 5.1.1, the consecutive special points (p_{i_{l−1}}, p_{n}, p_{i_{1}}) (resp. (p_{i_{q+1}}, p_{i_{q+r}}, p_{f^{e}})) on the component Σ_{v_{e}} (resp. Σ_{v^{e}}) are fixed at (1, ∞, 0). A tubular neighborhood ℝV × [0, ϵ[ of (Σ^{o}, p^{o}) in \overline{C}_{(τ,o)} can be given by the family {(z − z_{f_{e}}) · w + t = 0} with labelled points which is the deformation of nodal point (Σ^{o}, p^{o}).

The boundary orientation [ω_{(γ_{v^{e}},o_{v^{e}})}] ∧ [ω_{(γ_{ve},o_{ve})}] is given by the following differential forms that is determined by the ordering of the coordinates on ℝV according to 5.1.1.

ω_{(γ_{ve},o_{ve})} = (\frac{\sqrt{−1}}{2})^{|F_{γ}^{+}(v_{e})|} ⋀_{α∈F_{γ}^{+}(v_{e})} dz_{α} ∧ d\overline{z}_{α}∧

dx_{i_{2}} ∧ ⋯ ∧ dx_{i_{q}} ∧ dx_{f_{e}} ∧ { \atop dx_{i_{q+1}}} ∧ ⋯ ∧ { \atop dx_{i_{q+r}}} ∧ dx_{i_{q+r+1}} ∧ ⋯ ∧ dx_{i_{l−2}},

ω_{(γ_{v^{e}},o_{v^{e}})} = (\frac{\sqrt{−1}}{2})^{|F_{γ}^{+}(v^{e})|} ⋀_{β∈F_{γ}^{+}(v^{e})} dw_{β} ∧ d\overline{w}_{β} ⋀ dy_{i_{q+2}} ∧ ⋯ ∧ dy_{i_{q+r−1}}.

The coordinate projection ℝV × [0, ϵ[→ ℝV is given by the projections of the labelled points lying on the curves {(z − z_{f_{e}}) · w + t = 0} onto {w = 0} axes (see Remark 4.9). By using the identities w_{β} = −t/(z_{β} − x_{f_{e}}) for β ∈ F_{γ}^{+}(v^{e}) and y_{i} = −t/(x_{i} − x_{f_{e}}) for i ∈ F_{γ}^{ℝ}(v^{e}), we obtain the following equalities:

dt = −dx_{i_{q+1}} + dx_{f_{e}}, dx_{f_{e}} = dx_{i_{q+r}}, and

\begin{matrix}dw_{β} = −\frac{dt}{z_{β}−x_{fe}} + \frac{tdz_{β}}{(z_{β}−x_{fe})^{2}} − \frac{tdx_{fe}}{(z_{β}−x_{fe})^{2}} & for β ∈ F_{γ}^{±}(v^{e}), \\ dy_{i} = −\frac{dt}{x_{i}−x_{fe}} + \frac{tdx_{i}}{(x_{i}−x_{fe})^{2}} − \frac{tdx_{fe}}{(x_{i}−x_{fe})^{2}} & for i = q + 2, ⋯, q + r − 1. \end{matrix}

By simple manuplation −dt ∧ ω_{(γ_{v^{e}},o_{v^{e}})} ∧ ω_{(γ_{v^{e}},o_{v^{e}})} gives us

(−1)^{(r−1)(q−1)} (\frac{\sqrt{−1}}{2})^{|F_{γ}^{+}|} Θ ⋀_{α∈F_{γ}^{+}} dz_{α} ∧ d\overline{z}_{α} ⋀ dx_{i_{2}} ∧ ⋯ ∧ dx_{l−2}

where Θ = ∏_{β∈F_{γ}^{+}(v^{e})} t(z_{β} − x_{f_{e}})^{−2} ∏_{i_{q+2},··· ,i_{q+r−1}} t(x_{i} − x_{f_{e}})^{−2}. Since the function Θ > 0, we obtain that the orientation defined by −dt ∧ ω_{(γ_{v^{e}},o_{v^{e}})} ∧ ω_{(γ_{v^{e}},o_{v^{e}})} is equal to (−1)^{ℵ}[ω_{(τ,o)}]. □

5.2.2. Case II. l = 0. The different cases for boundaries of C_{(τ,o)} and B_{(2k,0)} are treated separately. The proofs are essentially same with the Lemma 5.1. We will not repeat the detailed calculations.

Lemma 5.2. Let (τ, o) be a one-vertex o-planar tree with Fix(σ) = ∅ and ℝΣ ≠ ∅, and let [ω_{(τ,o)}] be the orientation defined by the differential form

(\frac{\sqrt{−1}}{2})^{k−2} ⋀ ; Perm^{+}\(Perm^{+} ⋂\{α_{k−1},α_{k},\overline{α}_{k−1},\overline{α}_{k}\}) dz_{α_{∗}} ∧ d\overline{z}_{α_{∗}} ⋀ dλ

which given by the coordinates in (C) of 4.2.1. Here λ = ℑ(z_{α′}) and α′ ∈ \{α_{k−1}, \overline{α}_{k−1}\}.

(i) Let (γ, δ(o)) be a two-vertex o-planar tree where V_{γ} = V_{γ}^{ℝ} = {v_{e}, v^{e}}, and ∂_{γ}(n) = v_{e}. The induced orientation on codimension one strata C_{(γ,δ(o))} of \overline{C}_{(τ,o)} is

[ω_{(γ,δ(o))}] = −[ω_{(γ_{v^{e}},o_{v^{e}})}] ∧ [ω_{(γ_{ve},o_{ve})}].

(ii) Let (γ, δ(o)) be a two-vertex o-planar tree where V_{γ} = {v_{e}, v^{e}}, V_{γ}^{ℝ} = ∅, and let v be the vertex in V_{γ}^{+}. The induced orientation [ω_{(γ,δ(o))}] on codimension one strata C_{(γ,δ(o))} of \overline{C}_{(τ,o)} is given by

ω_{(γ,o)} = − ⋀ ; α_{∗}∈F_{γ}^{+}(v)\(F_{γ}^{+}(v) ⋂\{α_{k−1},α_{k},\overline{α}_{k−1},\overline{α}_{k}\}) dz_{α_{∗}} ∧ dz̄_{α_{∗}}.

The orientation on the boundary of the strata parametering the (Σ; p) with ℝΣ = ∅ is given similar way:

Lemma 5.3. Let (τ, o) be a one-vertex o-planar tree where o = {ℝΣ = ∅} and let [ω_{(τ,o)}] be the orientation on B_{(2k,0)} defined by

(\frac{\sqrt{−1}}{2})^{k−2} ⋀_{α_{1},··· ,α_{k−2}} dz_{α_{∗}} ∧ d\overline{z}_{α_{∗}} ⋀ dλ

which is given by the coordinates in (D) of 4.2.1. Here λ = ℑ(z_{α_{k−1}}).

Let (γ, δ(o)) be a two-vertex o-planar tree where V_{γ} = {v_{e}, v^{e}}, V_{γ}^{ℝ} = ∅, and let v be the vertex in V_{γ}^{+} The induced orientation [ω_{(γ,δ(o))}] on codimension one strata C_{(γ,δ(o))} of C_{(τ,o)} is given by

ω_{(γ,o)} = − ⋀ ; α_{∗}∈F_{γ}^{+}(v)\(F_{γ}^{+}(v) ⋂\{α_{k−1},α_{k},\overline{α}_{k−1},\overline{α}_{k}\}) dz_{α_{∗}} ∧ dz̄_{α_{∗}}.

5.3. Conventions. Let (τ, o_{⋆}) be the one-vertex o-planar tree given by F_{τ}^{+} = {1, 2, ⋯, k} and F_{τ}^{−} = {k + 1, ⋯, 2k}, and let the cyclic ordering on F_{τ}^{ℝ} be given by {2k + 1} < {2k + 2} < ⋯ < {2k + l := n}. All possible o-planar structures with ℝΣ ≠ ∅ on τ are obtained by the permutation actions on (τ, o_{⋆}) as follows.

Let ϱ ∈ S_{n} and let ϕ be the morphism of n-trees where the restriction over the set of tails ϕ_{T} = ϱ and assume that ϕ commutes with the morphism φ where φ_{T} = σ.

The image of the o-planar tree (τ, o_{⋆}) under ϕ is given by action of the permutation ϱ on the o-planar structure o since τ is an one-vertex tree and all one-vertex trees are isomorphic to each other. The o-planar structure of the image is given by ϱ(o_{⋆}) := {ℝΣ ≠ ∅; ϱ(Perm^{±}); Fix(σ) = {{ϱ(2k + 1)} < ⋯ < {ϱ(2k + l − 1)} < {ϱ(n)}}}. The parity |o| of the o-planar structure o = ϱ(o_{⋆}) is the parity of the permutation ϱ.

5.3.1. To be able to give a unique orientation to each component of M_{(2k,l)}^{ℝ}, we have to choose an o-planar representative for each one-vertex u-planar tree with ℝΣ ≠ ∅.

• l ≥ 3 case: Let (τ, o) be the representative of (τ, u) on which p_{1} < p_{n−1} < p_{n} with respect to the cyclic ordering on F_{τ}^{ℝ}.

• l = 0, 1, 2 case: Let (τ, o) be the representative of (τ, u) on which k ∈ Perm^{+}. We denote the set of o-planar representatives of u-planar trees by UT ree(σ).

5.4. The First Steifel-Whitney Class. The Poincare dual of the first Steifel-Whitney class of T \overline{M}_{(2k,l)}^{ℝ} is given by

(9)  W_{1} = \frac{1}{2} ∑_{(τ,o)∈UT ree(σ)} ( ∑_{(γ,δ(o))} ε_{(γ,δ(o))}[\overline{C}_{(γ,δ(o))}]) mod 2,

where [\overline{C}_{(γ,δ(o))}] are the fundamental cycles of the codimension one strata \overline{C}_{(γ,δ(o))} (relative to their boundary) of \overline{C}_{(τ,o)}. Here summation runs over all codimension one strata of \overline{C}_{(τ,o)} for all (τ, o) ∈ UT ree(σ). The parity ε_{(γ,δ(o))} = ±1 is the sign of the induced orientation given in Lemmata 5.1, 5.2 and 5.3.

Let (γ, δ(o_{1})) and (γ, δ(o_{2})) be a pair of o-planar representatives of a two-vertex u-planar tree (γ, u). Let C_{(γ,δ(o_{i}))}, i = 1, 2 be codimesion one strata of the top dimensional strata \overline{C}_{(τ,o_{i})} for (τ, o_{i}) ∈ UT ree(σ). Note that, if C_{(γ,δ(o_{i}))} carry opposite orientation then [\overline{C}_{(γ,δ(o_{1}))}] and [\overline{C}_{(γ,δ(o_{2}))}] cancel each other, and if not, ([\overline{C}_{(γ,δ(o_{2}))}] + [\overline{C}_{(γ,δ(o_{2}))}])/2 gives ℤ_{2}-fundamental cycles of the corresponding codimension one strata \overline{C}_{(γ,u)} relative to its boundary. In other words, the sum (9) detects where the orientation on M_{(2k,l)}^{ℝ} can not be extended to whole moduli space \overline{M}_{(2k,l)}^{ℝ}.

Proposition 5.4. For l > 0, the Poincare dual of the first Steifel-Whitney class of the tangent bundle T \overline{M}_{(2k,l)}^{ℝ} is given by

(10)  W_{1} = ∑_{(γ,u): |v^{e}|=0 mod 2}[\overline{C}_{(γ,u)}] = ∑_{γ: |v^{e}|=0 mod 2}[ℝ\overline{D}_{γ}] mod 2.

First, consider the following technical lemma;

Lemma 5.5. Let (τ, o), (τ, \overline{o}) be one-vertex o-planar trees with reverse o-planar structures. Let (z_{α_{1}}, ⋯, z_{α_{k}}, \overline{z}_{α_{1}}, ⋯, \overline{z}_{α_{k}}, x_{i_{1}}, ⋯, x_{i_{l}}) be coordinates on C_{(τ,o)}. Then, the coordinate transformation w_{α_{∗}} := \overline{z}_{α_{∗}}, y_{i_{∗}} := 1−x_{i_{l−∗}} gives coordinates on C_{(τ,\overline{o})}, and the orientations can be given as

[ω_{(τ,o)}] = (−1)^{μ}[ω_{(τ,\overline{o})}],

where μ = |F_{τ}^{+}| + (|F_{τ}^{ℝ}| − 2)(|F_{τ}^{ℝ}| − 3)/2.

The proof the Lemma follows from the simple calculation of coordinate change.

Proof. (of the Proposition 5.4) Firstly, we introduce an orientation [ω_{(τ,o)}] on each top dimensional strata \overline{C}_{(τ,o)} that are determined uniquely according to the convention in 5.1.1 for all (τ, o) ∈ UT ree(σ).

Let (γ, δ(o_{1})) and (γ, δ(o_{2})) be a pair of o-planar representatives of a two-vertex u-planar tree (γ, u) such that C_{(γ,δ(o_{i}))}, i = 1, 2 are codimension one strata of \overline{C}_{(τ,o_{i})} for (τ, o_{i}) ∈ UT ree(σ). If so, (γ, δ(o_{1})) produces (γ, δ(o_{2})) by reversing the o-planar structure at vertex v^{e} where ∂_{γ}(n) ≠ v^{e}. These two o-planar structures correspond to the combinatorial types of the deformation {(z − z_{f_{e}}) · w + t = 0} of the central fiber {(z −z_{f_{e}})·w = 0} respectively by positive and negative values of the parameter t.

Next, we calculate ε_{(γ,δ(o_{1}))} [\overline{C}_{(γ,δ(o_{1}))}] + ε_{(γ,δ(o_{2}))} [\overline{C}_{(γ,δ(o_{2}))}]. In above cases, the ratio ε_{(γ,δ(o_{1}))}/ε_{(γ,δ(o_{2}))} is given by the difference of the parities (−1)^{|o_{1}|−|o_{2}|} which is simply |o_{1}| − |o_{2}| = |F_{γ}^{+}(v^{e})| + (|F_{γ}^{ℝ}(v^{e})| − 1)(|F_{γ}^{ℝ}(v^{e})| − 2)/2. Here |F_{γ}^{+}(v^{e})| is the parity of permutation swaps F_{γ}^{±}(v^{e}), and (|F_{γ}^{ℝ}(v^{e})| − 1)(|F_{γ}^{ℝ}(v^{e})| − 2)/2 is the parity of reversing the linear ordering on F_{γ}^{ℝ}(v^{e}). On the other hand, reversing the o-planar structure at vertex v^{e} gives the parity μ(v^{e}) which is calculated in Lemma 5.5. It is easy to check that (|o_{1}| − |o_{2}|) + μ(v^{e}) = |F_{γ}^{ℝ}(v^{e})| mod 2. Hence, we have

ε_{(γ,δ(o_{1}))} [\overline{C}_{(γ,δ(o_{1}))}] + ε_{(γ,δ(o_{2}))} [\overline{C}_{(γ,δ(o_{2}))}] = ε_{(γ,δ(o_{1}))}(1 + (−1)^{|F_{γ}^{ℝ}(v^{e})|}) [\overline{C}_{(γ,δ(o_{1}))}] .

This sum vanishes when |v^{e}| is odd. The coefficient (1 + (−1)^{|F_{γ}^{ℝ}(v^{e})|}) is equal to 2 when |v^{e}| is even. Finally, the equation (9) can be rewritten as

W_{1} = \frac{1}{2} ∑_{(τ,o)∈UT ree(σ)} ∑_{(γ,δ(o)):|v^{e}|=0 mod 2} [\overline{C}_{(γ,δ(o))}] .

since we consider the expression in modulo two. Moreover, ([\overline{C}_{(γ,δ(o_{1}))}]+[\overline{C}_{(γ,δ(o_{2}))}])/2 gives us the ℤ_{2}-fundamental cycle [\overline{C}_{(γ,u)}], and then, the above sum gives W_{1} as it is stated. Since the sum ∑_{(τ,u)}[\overline{C}_{(γ,u)}] is taken over all u-planar structures on γ, we obtain the second sum in the statement. □

Example 5.6. Due to Proposition 5.4, the Poincare dual of the first Steifel-Whitney class W_{1} of \overline{M}_{(0,5)}^{ℝ} can be represented by ∑_{γ}[ℝ\overline{D}_{γ}] = ∑_{(γ,u)}[\overline{C}_{(γ,u)}] where γ are 5trees with |v^{e}| = 0 mod 2. These 5-trees are given in Figure 5a, and the union corresponding strata ⋃_{τ} ℝ\overline{D}_{γ} is given as highlighted diagonals in Figure 6a.

Figure 5. The 5-trees of (a) Steifel-Whitney class of \overline{M}_{(0,5)}^{ℝ} due Prop.
5.4, (b) blown-up locus in \overline{M}_{(0,5)}^{ℝ}

The moduli space \overline{M}_{(0,5)}^{ℝ} is a torus with 3 points blow-up (see Figure 6b). Hence, the Poincare dual of W_{1} can also be represented by the (real part of the) exceptional divisors ∑_{γ′}[ℝ\overline{D}_{γ′}] where the 5-trees γ′ are given in Figure 5b.

These two chains are homologous. The sum these chains give a contractible chain (see Figure 6c) plus three disjoint intervals \overline{M}_{(0,5)}^{ℝ}.

Figure 6

Proposition 5.7. The Poincare dual of the first Steifel-Whitney class of the tangent bundle T \overline{M}_{(2k,0)}^{ℝ} vanishes.

Proof. A similar reasoning with Proposition 5.4 gives that W_{1} = 0. We check consistency of the induced orientations on the codimension one strata. We consider two different types of codimension one strata which parameterize:(i) real curves with two real components; and (ii) real curves with isolated real nodes.

We introduce an orientation [ω_{(τ,o)}] on each top dimensional strata \overline{C}_{(τ,o)}, that are determined uniquely according to the convention in 5.1.1 for all (τ, o) ∈ UT ree(σ) with ℝΣ ≠ ∅, and B_{(2k,0)} with [ω_{(2k,0)}] which is defined by

−(\frac{\sqrt{−1}}{2})^{k−2} ⋀_{α_{1},··· ,α_{k−2}} dz_{α_{∗}}∧d\overline{z}_{α_{∗}} ⋀ dλ = (−1)^{|o|}(\frac{\sqrt{−1}}{2})^{k−2} ⋀_{α_{∗}|ℑ(z_{α∗})>0} dz_{α_{∗}}∧d\overline{z}_{α_{∗}} ⋀ dλ

where λ = ℜ(z_{α_{k−1}}).

(i) Let (γ, δ(o_{i})), i = 1, 2 be two-vertex o-planar trees representing the same (γ, u) with V_{γ} = V_{γ}^{ℝ}. If so, (γ, δ(o_{1})) produces (γ, δ(o_{2})) when we reverse the o-planar structure at the vertex v^{e} ≠ ∂_{γ}(n). These two o-planar structures correspond to the combinatorial types of the deformation {(z − z_{f_{e}}) · w + t = 0} of the central fiber {(z − z_{f_{e}}) · w = 0} respectively by positive and negative values of parameter t. By using the same parity reasoning in the proof of Proposition 5.4, we obtain ε_{(γ,o_{1})} [\overline{C}_{(γ,δ(o_{1}))}]+ε_{(γ,o_{2})} [\overline{C}_{(γ,o_{2})}] = 0, since v^{e} supports only one invariant flag which corresponds to the real node.

(ii) Let (γ, o) be an o-planar tree such that V_{γ}^{ℝ} = ∅ and C_{(γ,o)} be a codimension one strata of \overline{C}_{(τ,o)} and \overline{B}_{(2k,0)}.

The induced orientations on C_{(γ,δ(o))} by orientations of \overline{C}_{(τ,o)} and B_{(2k,0)} are given in Lemmata 5.2 and 5.3. They are respectively given by the following differential forms

(−1)^{|o|+1} ⋀ ; α_{∗}∈F_{γ}^{+}(v)\(F_{γ}^{+}(v) ⋂\{α_{k−1},α_{k},\overline{α}_{k−1},\overline{α}_{k}\}) dz_{α_{∗}} ∧ dz̄_{α_{∗}},

(−1)^{|o|} ⋀ ; α_{∗}∈F_{γ}^{+}(v)\(F_{γ}^{+}(v) ⋂\{α_{k−1},α_{k},\overline{α}_{k−1},\overline{α}_{k}\}) dz_{α_{∗}} ∧ dz̄_{α_{∗}},

where v is the vertex in V_{γ}^{+}. Therefore, the orientations induced from different sites are opposite and the sum (−1)^{|o|} [\overline{C}_{(γ,δ(o))}] + (−1)^{|o|+1} [\overline{C}_{(γ,o)}] vanishes for all such (γ, o). □

6. The Orientation Double Covering of \overline{M}_{(2k,l)}^{ℝ}

In Section 5.4, the first Steifel-Whitney class of \overline{M}_{(2k,l)}^{ℝ} is determined in terms of its strata. As a consequence, it is observed that \overline{M}_{(2k,l)}^{ℝ} is not orientable when l ≠ 0 and n ≥ 5. Therefore, in these cases, the \overline{M}_{(2k,l)}^{ℝ} admits a unique orientation double covering. In this section, we will construct such coverings.

6.1. Construction of M̃_{(2k,l)}. Consider the disjoint union \overline{C}_{(2k,l)} = ⨆_{(τ,o)} \overline{C}_{(τ,o)} where |V_{τ}| = 1 and (τ, o) runs over all possible o-planar structures on τ. The union \overline{C}_{(2k,l)} gives a compactification of C̃_{(2k,l)}. We have already seen that the map C̃_{(2k,l)} → M_{(2k,l)}^{ℝ}, which identifies the o-planar with the corresponding u-planar structure, is trivial double covering. The orientation double cover of \overline{M}_{(2k,l)}^{ℝ} must be a manifold with no boundaries. We need to get rid of the faces by gluing them together in order to obtain the orientation double cover. We use the following simple recipe:

Let (γ, δ(o_{i})), i = 1, 2 be two-vertex o-planar trees with V_{γ} = V_{γ}^{ℝ} = {v_{e}, v^{e}} and let ∂_{γ}(n) = v_{e}. Let (τ, o_{i}), = 1, 2 be one-vertex o-planar trees obtained by contracting the invariant edges of (τ, δ(o_{i})), i = 1, 2 respectively.

We glue the codimension one boundaries \overline{C}_{(γ,δ(o_{i}))}, i = 1, 2 of \overline{C}_{(τ,o_{i})} if

(1) (γ, δ(o_{1})) produces (γ, δ(o_{2})) by reversing the o-planar structure at v^{e}, and |v^{e}| = 1 mod 2,

(2) (γ, δ(o_{1})) produces (γ, δ(o_{2})) by reversing the o-planar structure at v_{e}, and |v^{e}| = 0 mod 2.

We denote the manifold obtained by gluing by M̃_{(2k,l)}.

Theorem 6. M̃_{(2k,l)} is the orientation double cover of \overline{M}_{(2k,l)}^{ℝ}.

Proof. From the construction, it is clear that the map M̃_{(2k,l)} → \overline{M}_{(2k,l)}^{ℝ} forgetting the o-planar structures is two-to-one everywhere.

We are going prove that the first Steifel-Whitney class of M̃_{(2k,l)} vanishes. We orient the configuration space \overline{C}_{(τ,o)} of one-vertex τ as follows: If (τ, o) ∈ UT ree(σ), we orient it by [ω_{(τ,o)}] according to 4.2.1, and if (τ, o) ∉ UT ree(σ) (i.e, (τ, \overline{o}) ∈ UT ree(σ)) we orient it by (−1)^{ς+1}[ω_{(τ,o)}] where ς = k + (l − 2)(l − 3)/2. It needs to be checked that the Poincare dual of first Steifel-Whitney class

W_{1} = \frac{1}{2} ∑_{(τ,o)} ∑_{(γ,δ(o))} ε_{(γ,δ(o))} [\overline{C}_{(γ,δ(o))}]

vanishes. Let (γ, δ(o_{i})), i = 1, 2 be a pair of two-vertex o-planar satistying the condition glueing (1) or (2) given above. Then W_{1} can be rewritten in terms of such pairs:

(11) W_{1} = \frac{1}{2} ∑_{(γ,δ(o_{i})):|v^{e}|=1 mod 2}(ε_{(γ,δ(o_{1}))} [\overline{C}_{(γ,δ(o_{1}))}] + ε_{(γ,δ(o_{2}))} [\overline{C}_{(γ,δ(o_{2}))}])

+\frac{1}{2} ∑_{(γ,δ(o_{i})):|v^{e}|=0 mod 2}(ε_{(γ,δ(o_{1}))} [\overline{C}_{(γ,δ(o_{1}))}] + ε_{(γ,δ(o_{2}))} [\overline{C}_{(γ,δ(o_{2}))}]). (12)

The first part of the sum in (11), where |v^{e}| = 1 mod 2, vanishes due to the parity arguments used in Proposition 5.4.

Similarly, we calculate the difference of the parity of o-planar structures (−1)^{|o_{1}|−|o_{2}|} = ε_{(γ,δ(o_{1}))}/ε_{(γ,δ(o_{2}))} when |v^{e}| = 1 mod 2. Let (γ, δ(ô_{2})) be the o-planar tree produced by reversing the o-planar structure at v^{e} for (γ, δ(o_{1})). Let [ω_{(τ,ô_{2})}] be orientation on \overline{C}_{(τ,ô_{2})}. If we reverse the o-planar structure of (τ, ô_{2}), we obtain the o-planar tree (τ, o_{2}) and the orientation on \overline{C}_{(τ,o_{2})} is given by [ω_{(τ,o_{2})}] = (−1)^{ς}[ω_{(τ,ô_{2})}] due to Lemma 5.5 i.e, the orientation that we have fixed is −[ω_{(τ,ô_{2})}].

Therefore, the second sum turns into

ε_{(γ,δ(o_{1}))} [\overline{C}_{(γ,δ(o_{1}))}] + ε_{(γ,δ(o_{2}))} [\overline{C}_{(γ,δ(o_{2}))}] = ε_{(γ,δ(o_{1}))}([\overline{C}_{(γ,δ(o_{1}))}] − [\overline{C}_{(γ,δ(ô_{2}))}]).

We have already seen in Proposition 5.4, the strata C_{(γ,δ(o_{1}))}, C_{(γ,δ(ô_{2}))} have the same induced orientation. Hence the sum vanishes. □

Remark 6.1. In [Ka], Kapranov constructed a different double covering of \overline{M}_{(0,l)}^{ℝ} without any boundaries. Let \overline{C}_{(0,l)} be the disjoint union ⨆_{(τ,o)} \overline{C}_{(τ,o)} where (τ, o) runs over all possible o-planar structures on τ with |V_{τ}| = 1. He has applied the following recipe to obtain the double covering: Let (γ, δ(o_{i})), i = 1, 2 be two o-planar trees corresponding to codimension 1 boundaries of (τ, o_{i}). The faces \overline{C}_{(γ,δ(o_{i}))}, i = 1, 2 are glued if (γ, δ(o_{1})) produces (γ, δ(o_{2})) by reversing the o-planar structure at vertex v^{e} ≠ ∂_{γ}(n). The first Steifel-Whitney class can be explicitly obtained as above.

W_{1} = \frac{1}{2} ∑_{(τ,o)} ∑_{(γ,δ(o)):|v^{e}|=0 mod 2} [\overline{C}_{(γ,o)}] mod 2.

These spaces are not orientable when l ≥ 5.

Remark 6.2. In [FOOO, Liu], a different ‘orientation double covering’ is considered. It can be given as the disjoint union ⨆_{(τ,o)} \overline{C}_{(τ,o)} where F_{τ}^{+} = \{1, ⋯, n\}, and F_{τ}^{ℝ} carries all possible oriented cyclic ordering. It is a disjoint union of manifolds with corners. The covering map ⨆_{(τ,o)} \overline{C}_{(τ,o)} → \overline{M}_{(2k,l)}^{ℝ} is two-to-one only over the open space M_{(2k,l)}^{ℝ}.

6.1.1. Examples. The construction in Section 6.1 can also be applied when 2k + l = 4 and l ≠ 0, although in such cases, the moduli space \overline{M}_{(2k,l)}^{ℝ} is orientable. For such cases, the above construction excludes the trivial coverings and gives the stratification of M̃_{(0,4)} shown on Figure 7. The picture of M̃_{(2,2)} is similar.

Figure 7. Stratification of M̃_{(0,4)}

In Example 4.12, we obtained that \overline{M}_{(0,5)}, \overline{M}_{(2,3)}, and \overline{M}_{(4,1)} are respectively, a torus with 3 points blown up, a sphere with 3 points blown up, and a sphere with one point blown up. The coverings M̃_{(0,5)}, M̃_{(2,3)} and M̃_{(4,1)} are obtained by taking the two copies of the corresponding moduli space of real curves and replacing the blown up loci by annuli. Therefore, M̃_{(0,5)}, M̃_{(2,3)} and M̃_{(4,1)} are surfaces of genus 4, genus 2 and genus 0, respectively (see Figure 8 which illustrates the case (k, l) = (0, 5)).

Figure 8. Stratification of M̃_{(0,5)}

6.2. Combinatorial Types of Strata of M̃_{(2k,l)}. While constructing M̃_{(2k,l)}, the closure of the each codimension one strata are glued in a consistent way. This identification gives an equivalence relation among the o-planar trees when l ≠ 0.

We define the notion of R-equivalence on the set of such o-planar trees by treating different cases separately. Let (γ_{1}, o_{1}), (γ_{2}, o_{2}) be o-planar trees.

(1) If |V_{γ_{i}}^{ℝ}| = 1, then we say that they are R-equivalent whenever the o-planar structures are the same.

(2) If γ_{i} have an edge corresponding to real node (i.e. E_{γ_{i}}^{ℝ} = {e} and V_{γ_{i}}^{ℝ} = ∂_{γ}(e) = {v^{e}, v_{e}}), we first obtain (γ_{i}(e), ô_{i}) by contracting conjugate pairs of edges until there will be none. We say that (γ_{1}, o_{1}) and (γ_{2}, o_{2}) are Requivalent whenever γ_{1}, γ_{2} are isomorphic and

• (γ_{1}(e), ô_{1}) produces (γ_{2}(e), ô_{2}) by reversing o-planar structure at v^{e} and |v^{e}| is odd,

• (γ_{1}(e), ô_{1}) produces (γ_{2}(e), ô_{2}) by reversing o-planar structure at v_{e} and

|v^{e}| is even.

(3) Otherwise, if γ_{i} have more than one edge corresponding to real nodes (i.e. |E_{γ_{i}}|^{ℝ} > 1), we say that (γ_{1}, o_{1}), (γ_{2}, o_{2}) are R-equivalent whenever γ_{1}, γ_{2} are isomorphic and there exists an edge e ∈ E_{γ_{i}}^{ℝ} such that the o-planar trees (γ_{i}(e), ô_{i}) which are obtained by contracting all edges but e are R-equivalent in the sense of case 2.

We call the maximal set of pairwise R-equivalent o-planar trees by R-equivalence classes of o-planar trees.

Theorem 7. A stratification of the orientation double cover M̃_{(2k,l)} is given by

(13)  M̃_{(2k,l)} = ⨆_{R−equivalance classes of o−planar (γ,o)} C_{(γ,o)}.

Proof. The statement directly follows from the construction of M̃_{(2k,l)} and definition of R-equivalence classes. □

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Max Planck Institute für Mathematik, D-53111/Bonn, Germany

E-mail address: ceyhan@mpim-bonn.mpg.de