Moduli of curves with non-abelian level structure

M. Pikaart  A.J. de Jong^{∗}

October 5, 1998

1 Introduction

Deligne and Mumford introduced the moduli stack _{G}M_{g} parametrizing smooth genus g curves with Teichmüller structure of level G, a finite group. For example, if G = {e}, resp. G ≅ ( /n )^{2g}, this moduli stack is just the moduli stack of smooth curves M_{g}, resp. of smooth curves with abelian level structure (sometimes denoted M_{g}[n]). They also defined \overline{M_{g}}, the moduli stack of stable curves and proved it is proper over Spec( ). Taking the normalization of \overline{M_{g}} × Spec( [1/#G]) in the function field of _{G}M_{g} defines \overline{_{G}M_{g}}, proper over Spec( [1/#G]).

Let Π denote the standard fundamental group of a compact Riemann surface of genus g. The nth powers together with the kth order commutators generate a normal subgroup Π^{(k),n}. (We regard [a, b] as a commutator of order 2.) Let G be the quotient Π/Π^{(k),n}. We show that if k ≥ 1, n ≥ 3 the coarse moduli scheme \overline{_{G}M_{g}} for \overline{_{G}M_{g}} exists, and we actually have \overline{_{G}M_{g}} ≅ \overline{_{G}M_{g}}. The following theorem is our main result.

Theorem 3.1.1 Suppose k ∈ {1, 2, 3} and n ≥ 3. The structural morphism \overline{_{G}M_{g}} → Spec( [1/n]) is smooth if and only if

• k = 1 and g = 2,

• k = 2 and n is odd,

• k = 3 and n is odd or n is divisible by 4.

Furthermore, if k ≥ 4, n ≥ 3 and n relatively prime to 6, then \overline{_{G}M_{g}} → Spec( [1/n]) is smooth.

∗The research of A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

Looijenga constructed a smooth and compact cover \overline{M_{g}[_{2}^{n}]}^{an}, n ≥ 3 of \overline{M_{g}}^{an} using Prym level structures. He proves that these coverings are universally ramified along the boundary of M_{g}^{an}, see [L]. This holds also for our coverings as can be seen from 3.1.3. These coverings can be applied to the construction of the Chow rings of the stacks \overline{M_{g}}[n]: for example it is clear that the specialization maps are ring homomorphisms.

The paper is organized as follows. Section 2 deals with the definition of the moduli problem. The formulation is in terms of the relative fundamental group of a family of curves. Its final proposition states that \overline{_{G}M_{g}} → Spec( [1/n]) is smooth if and only if the associated compact analytic space \overline{_{G}M_{g}}^{an} is is a manifold. Section 3 contains a precise description of the monodromy along the boundary of \overline{_{G}M_{g}}^{an} for general groups G and the statement of the main result. Sections 4 and 5 are of a topological nature. We describe the monodromy on the relative fundamental group of the universal curve along the boundary in terms of Dehn twists. We have to describe the situation in some detail in order to understand how this monodromy acts on the finite quotients Π/Π^{(k),n}. These considerations prove the main theorem, using Section 6, which contains the necessary computations for a free group on three generators.

The authors were stimulated by the article [L]. We thank Prof. Looijenga for numerous discussions explaining his and other results. We thank Prof. Oort, who remarked that it should be possible to do everything algebraically and drew our attention to the article [AMO].

1.1 Notations and conventions

1. Throughout the paper g is a fixed natural number at least 2.

2. G is a finite group.

3. is a set of primes, contains the primes dividing #G.

4. The stack of stable curves of genus g is denoted \overline{M_{g}}, the open substack of smooth curves M_{g}. Stacks are denoted by script letters. For definitions and results concerning stacks we refer to [DM].

5. Suppose Γ is a (pro-finite) group. A characteristic subgroup of Γ is a normal subgroup fixed by any automorphism of Γ. A characteristic quotient is one whose kernel is a characteristic subgroup. If Γ is profinite and topologically finitely generated then it is the direct limit of its finite characteristic quotients.

6. Let Γ denote a (finitely generated) group. For a, b ∈ Γ we put [a, b] = a^{−1}b^{−1}ab, so that ab = ba[a, b]. We define the lower central series Γ^{(k)} of Γ by Γ^{(1)} = Γ and Γ^{(k+1)} = [Γ^{(k)}, Γ]. The subgroup of Γ generated by nth-powers is denoted Γ^{n}. We write Γ^{(k),n} to indicate the subgroup generated by Γ^{(k)} and nth-powers, Γ^{(k),n} = Γ^{(k)} · Γ^{n}. Any group homomorphism maps commutators to commutators and nth-powers to nth-powers, hence preserves these subgroups. In particular, the subgroups Γ^{(k),n} are characteristic subgroups of Γ.

7. Let Π = Π_{g} denote the standard fundamental group Π = π_{1}(S) of a compact Riemann surface S of genus g.

2 Definition of the moduli problem

In this section we recall the definition of the moduli problem of Teichmüller level structures, see [DM, Section 5]. Furthermore, we prove that the Deligne-Mumford compactification of the associated stack is smooth if and only if the corresponding analytic orbifold is smooth.

2.1 The relative fundamental group

In this section we define the relative fundamental group for a proper smooth morphism f : X → S with connected fibres and endowed with a section s : S → X. In order to motivate the definition in the algebraic case (and since we need it also) we first do the analytic case.

2.1.1 The analytic case

Here f : X → S is a proper smooth morphism of analytic spaces with connected fibres. In addition we are given a section s : S → X of f. We define a locally constant sheaf of groups π_{1}(X/S, s) over S such that for all points p ∈ S we have an isomorphism of groups

π_{1}(X/S, s)_{p} ≅ π_{1}(X_{p}, s(p)),

of the fibre of the sheaf at p with the topological fundamental group of the fibre X_{p} of f at p with base point s(p).

To construct π_{1}(X/S, s) we choose for any point p of S a connected open neighbourhood U_{p} ⊂ S and a topological isomorphism

ϕ_{p} : f^{−1}(U_{p}) ≅ X_{p} × U_{p}.

Such can be found compatible with f and the projection to U_{p}, inducing the identity on X_{p} and such that ϕ_{p} ∘ s equals q ↦ (s(p), q). Over U_{p} we take π_{1}(X/S, s) constant with fibre π_{1}(X_{p}, s(p)).

To glue these we note that given two points p_{1}, p_{2} ∈ S there is for any q ∈ U_{p_{1}} ∩ U_{p_{2}} an identification

X_{p_{1}} = X_{p_{1}} × {q}⟶^{ϕ_{p_{1}}^{−1}}X_{q}⟶^{ϕ_{p_{2}}}X_{p_{2}} × {q} = X_{p_{2}}.

This identification is compatible with base points s(p_{i}) and depends continuously on q ∈ U_{p_{1}} ∩ U_{p_{2}}. This means that the induced isomorphism

π_{1}(X_{p_{1}}, s(p_{1})) ≅ π_{1}(X_{p_{2}}, s(p_{2}))

is constant on the connected components of U_{p_{1}} ∩ U_{p_{2}}. Hence we get the desired gluing. We leave to the reader the trivial verification that these gluings satisfy the desired cocycle condition on U_{p_{1}} ∩ U_{p_{2}} ∩ U_{p_{3}}.

If we choose other ϕ_{p}, say ϕ′_{p}, then for q ∈ U_{p} the map X_{q} → X_{p} × {q} → X_{q}, using first ϕ_{p}^{−1} then ϕ′_{p}, is homotopic to the identity. Hence the resulting sheaves π_{1}(X/S, s) are canonically isomorphic. A similar argument deals with the shrinking of the neighbourhoods U_{p}.

Proposition 2.1.2 The construction given above defines a locally constant sheaf of groups π_{1}(X/S, s) over S, characterized by the following properties:

1. For any point p ∈ S there is given an isomorphism π_{1}(X/S, s)_{p} ≅ π_{1}(X_{p}, s(p)).

2. The monodromy action of π_{1}(S, p) on the fibre of the locally constant sheaf

ρ : π_{1}(S, p) → Aut(π_{1}(X/S, s)_{p})

agrees, via the isomorphism of 1), with the action of π_{1}(S, p) on π_{1}(X_{p}, s(p)) deduced from the split exact sequence

1 ⟶ π_{1}(X_{p}, s(p)) ⟶ π_{1}(X, s(p))⟶ ⟵^{s_{∗}}π_{1}(S, p) ⟶ 1.

3. The construction of π_{1}(X/S, s) commutes with arbitrary base change S′ → S.

Proof. Suppose γ ∈ π_{1}(S, p) and α ∈ π_{1}(X_{p}, s(p)). It is well known and easy to prove that s_{∗}(γ)αs_{∗}(γ^{−1}) is equal to the horizontal transportation of the loop α over γ. Clearly this describes the monodromy representation for the sheaf π_{1}(X/S, s). The proof of the other assertions is left to the reader. □

Remark 2.1.3 Suppose s′ : S → X is a second section of f. In general π_{1}(X/S, s) is not isomorphic to π_{1}(X/S, s′). However, locally on S, say over U ⊂ S, we can choose a homotopy H between s and s′. This will induce an identification

i_{H} : π_{1}(X/S, s)|_{U} ≅ π_{1}(X/S, s′)|_{U}.

This is unique up to an inner automorphism of π_{1}(X/S, s). Indeed, if H′ is another such homotopy, then combining H and H′ gives a familly of loops in X over U, with base points s(u), i.e., a section of π_{1}(X/S, s) over U. The map (i_{H′})^{−1} ∘ i_{H} is equal to conjugation with this section.

2.1.4 The algebraic case

Here we consider a proper smooth morphism of schemes f : X → S with connected geometric fibres. As before we have a section s : S → X of the morphism f. Further, we assume given a set of primes such that all residue characteristics of S are not in .

We recall some general notations concerning algebraic fundamental groups. We refer to [SGA 1] and [Mu] for more details. If Y is a scheme, then Ét(Y) denotes the category of finite étale coverings Y′ → Y. If p̄ is a geometric point of Y then we denote by F_{Y,p̄} the fundamental functor (or fibre functor) F_{Y,p̄} : Ét(Y) ⟶ Set which associates to Y′ the set of geometric points p̄ → Y′ lying over p̄ → Y. By definition we have π_{1}(Y, p̄) = Aut(F_{Y,p̄}); this is the fundamental group of Y with base point p̄. To compare the fundamental groups with base points p̄, resp. q̄ we use a path from p̄ to q̄, i.e., an isomorphism of fibre functors α : F_{Y,p̄} ⟶ F_{Y,q̄}. Obviously, α gives an isomorphism α_{∗} : π_{1}(Y, p̄) ⟶ π_{1}(Y, q̄). We note that it is independent of the choice of α up to conjugation. A morphism of schemes h : Y → Z, defines a functor h^{∗} : Ét(Z) → Ét(Y), which satisfies F_{Y,p̄} = F_{Z,h(p̄)} ∘ h^{∗}. Therefore we get h_{∗} on loops and on paths.

A slight modification of the above gives π_{1}^{}(Y, p̄), the algebraic fundamental group classifying Galois coverings of degree in . Formally it can be defined as

π_{1}^{}(Y, p̄) = lim_{⟵} G,

where the limit is taken over all surjections π_{1}(Y, p̄) → G onto finite groups G whose orders have only prime factors from .

In the sequel we will use the following results from [SGA 1]: The sequence

π_{1}(Xp̄, s(p̄)) ⟶ π_{1}(X, s(p̄)) ⟶ π_{1}(S, p̄) ⟶ 1

is exact. If we take the pushout of this sequence with the surjection π_{1}(Xp̄, s(p̄)) → π_{1}^{}(Xp̄, p̄) then the sequence also becomes left exact

1 ⟶ π_{1}^{}(Xp̄, p̄) ⟶ π′_{1}(X, s(p̄))⟶ ⟵^{s_{∗}}π_{1}(S, p̄) ⟶ 1.  (∗)

See [SGA 1, Exposé XII 4.3, 4.4]. As before the section s defines a splitting s_{∗} : π_{1}(S, p̄) → π_{1}(X, s(p̄)).

Proposition 2.1.5 ([SGA 1, Exposé XII 4.5]) There is a pro-object in the category of locally constant sheaves of groups on S_{ét}, denoted π_{1}^{}(X/S, s), determined up to unique isomorphism by the following properties:

1. For any geometric point p̄ of S there is given an isomorphism

π_{1}^{}(X/S, s)p̄ ⟶ π_{1}^{}(Xp̄, s(p̄)).

2. The monodromy presentation

ρ : π_{1}(S, p̄) ⟶ Aut(π_{1}^{}(X/S, s)p̄)

equals, via 1), the action of π_{1}(S, p̄) on π_{1}^{}(Xp̄, s(p̄)) deduced from (*).

3. The construction of π_{1}^{}(X/S, s) commutes with arbitrary base change S′ → S.

Proof. Of course we may assume that S is connected. Take a geometric point p̄ of S. First we note that, since π_{1}^{}(Xp̄, s(p̄)) is topologically finitely generated, it is the direct limit of its characteristic finite quotients:

π_{1}^{}(Xp̄, s(p̄)) = lim_{\overleftarrow{ω}} G_{ω}

The action of π_{1}(S, p̄) on π_{1}^{}(Xp̄, s(p̄)) deduced from (*) gives an action ρ_{ω} on each G_{ω}. This defines a finite locally constant étale sheaf F_{ω} on S_{ét} whose fibre in p̄ is given by G_{ω} and monodromy action equal to ρ_{ω}. We put

π_{1}^{}(X/S, s) = lim_{\overleftarrow{ω}} F_{ω}.

This immediately gives 1) and 2) for our chosen point p̄. Part 3) is also clear if there exists a lift of p̄ to a geometric point of S′, see [SGA 1].

Thus it suffices to prove 1) and 2) for a second geometric point q̄ of S. Note that by definition

π_{1}(X, s(p̄)) = Aut(F_{S,p̄} ∘ s^{∗}).

Hence if we choose an isomorphism α : F_{S,p̄} → F_{S,q̄} then we get a commutative diagram

π_{1}(X, s(p̄)) ⟶ ⟵^{s_{∗}} π_{1}(S, p̄)
↓^{α_{∗}}  ↓^{α_{∗}}
π_{1}(X, s(q̄)) ⟶ ⟵^{s_{∗}} π_{1}(S, q̄).

By (*) this induces an isomorphism

α^{}: π_{1}^{}(Xp̄, s(p̄)) ⟶ π_{1}^{}(Xq̄, s(q̄)).

This is an isomorphism compatible with the actions of π_{1}(S, p̄) and π_{1}(S, q̄), compared via α_{∗}.

The fibre of π_{1}^{}(X/S, s) at q̄ is by definition

F_{S,q̄}(π_{1}^{}(X/S, s))⟶^{α^{−1}}F_{S,p̄}(π_{1}^{}(X/S, s)) = π_{1}^{}(Xp̄, s(p̄)).

If we use α^{}to identify this with π_{1}^{}(Xq̄, s(q̄)) then we see by the above that the monodromy action on this exactly corresponds to the action deduced from (*) (with q̄ in stead of p̄). We leave to the reader the verification that another choice of α gives the same identification

π_{1}^{}(X/S, s)q̄ ⟶ π_{1}^{}(Xq̄, s(q̄)).
□

Suppose s′ : S → X is a second section of f. Take a geometric point p̄ of S. We write ip̄ for the morphism Xp̄ → X. We say that a path β on X connecting s(p̄) to s′(p̄) lies in Xp̄ if there exists a path β̃ in Xp̄ such that i_{p̄,∗}(β̃) = β. We remark that any path β connecting s(p̄) to s′(p̄) lies in Xp̄ if and only if f_{∗}(β) = 1 (in π_{1}(S, p̄)). This is easily seen using the first exact sequence above.

Let us take such a β lying in Xp̄. It gives rise to a commutative diagram

1 ⟶ π_{1}^{}(Xp̄, s(p̄)) ⟶ π′_{1}(X, s(p̄)) ⟶ π_{1}(S, p̄) ⟶ 1
↓≅  ↓^{β_{∗}}  ↓^{id}
1 ⟶ π_{1}^{}(Xp̄, s′(p̄)) ⟶ π′_{1}(X, s′(p̄)) ⟶ π_{1}(S, p̄) ⟶ 1

The isomorphism π_{1}^{}(Xp̄, s(p̄)) ≅ π_{1}^{}(Xp̄, s′(p̄)) determined in this way is unique up to inner conjugation. We constructed π_{1}^{}(X/S, s), resp. π_{1}^{}(X/S, s′) as the limit of sheaves F_{ω}, resp. F′_{ω} corresponding to characteristic quotients π_{1}^{}(Xp̄, s(p̄)) → G_{ω}, resp. π_{1}^{}(X/S, s′) → G_{ω}. Note that the isomorphisms G_{ω} → G′_{ω} induced from the above are also unique up to inner conjugation.

Let S_{ω} → S be the finite étale covering of S trivializing the action of π_{1}(S, p̄) on both G_{ω} and G′_{ω}. We can use the above to get an isomorphism of (constant) sheaves of groups

F_{ω}|_{S_{ω}} ⟶ F′_{ω}|_{S_{ω}}.

We claim this isomorphism is unique up to inner conjugation. This is clear if we only change β, but what happens if we change p̄ to q̄? Take α : F_{S,p̄} → F_{S,q̄} as in the proof of Proposition 2.1.5. What we have to check is that s′_{∗}(α) ∘ β ∘ s_{∗}(α^{−1}) is a path connecting s(q̄) to s′(q̄) lying in Xq̄. But this is clear since f_{∗}(s′_{∗}(α) ∘ β ∘ s_{∗}(α^{−1})) = α ∘ α^{−1} = 1.

Corollary 2.1.6 The construction above defines locally in the étale topology on S identifications of the finite quotients of the sheaves π_{1}^{}(X/S, s) and π_{1}^{}(X/S, s′). These identifications are unique up to inner conjugation and agree via 1) of Proposition 2.1.5 with the usual identifications of π_{1}^{}(Xp̄, s(p̄)) and π_{1}^{}(Xp̄, s′(p̄)).

2.2 Exterior homomorphisms

In this section we define the sheaf of exterior homomorphisms of the relative fundamental group of X over S into a fixed finite group G. As in [DM, 5.5] this sheaf will be denoted Hom^{ext}(π_{1}(X/S), G) and will be a finite locally constant sheaf of sets on S (or S_{ét}). We note that π_{1}(X/S) has not been defined.

2.2.1 The analytic case

Here f : X → S is a proper smooth morphism of analytic spaces with connected fibres.

If f has a section s then we can look at the locally constant sheaf

F = Hom(π_{1}(X/S, s), G)

on S. It has finite fibres since π_{1}(X_{p}, s(p)) is finitely generated for all p in S. There is a natural action of the sheaf π_{1}(X/S, s) on the sheaf F given by conjugation. We define the sheaf of exterior homomorphisms as the quotient of F by this action

Hom^{ext}(π_{1}(X/S), G) := F/π_{1}(X/S, s).

It is clear from Remark 2.1.3 that the right hand side does not depend on the chosen section s.

In general, we choose an open covering S = ⋃ U_{i} such that X → S has a section over each U_{i}. The sheaf Hom^{ext}(π_{1}(X/S), G) is then defined by gluing the sheaves constructed above. The fibres are described by the formula:

Hom^{ext}(π_{1}(X/S), G)_{p} = Hom(π_{1}(X_{p}, q), G)/π_{1}(X_{p}, q),

where q is any point of the fibre X_{p}. Note that the monodromy action of γ ∈ π_{1}(S, p) on this is given by horizontal transport of loops in π_{1}(X_{p}, q).

2.2.2 The algebraic case

Here we assume that S is a scheme over Spec( [1/#G]). The morphism f : X → S is still assumed proper smooth with connected geometric fibres.

The construction of Hom^{ext}(π_{1}(X/S), G) in this case is exactly the same as for the analytic case. We use π_{1}^{}(X/S, s), where is the set of primes dividing #G, and we use that sections of f exist locally in the étale topology on S. We use also Corollary 2.1.6. The geometric fibres of the resulting sheaf can be described as follows:

Hom^{ext}(π_{1}(X/S), G)p̄ = Hom(π_{1}^{}(Xp̄, q̄), G)/π_{1}^{}(Xp̄, q̄) = Hom(π_{1}(X_{p}, q), G)/π_{1}(X_{p}, q),

the last equality holds in view of our definition of . This justifies dropping from the notations.

2.2.3 Comparison

We note that if S is a scheme of finite type over there is a canonical homomorphism π_{1}(X^{an}/S^{an}, s) → π_{1}^{}(X/S, s)^{an}, identifying the relevant finite quotient sheaves. Clearly this gives rise to an identification of sheaves of exterior homomorphisms.

2.3 Teichmüller level structures

In this section the morphism f : X → S will be a familly of smooth projective curves of genus g. The abstract finite group G will be fixed. In both the analytic case and the algebraic case we make the following definition.

Definition 2.3.1 [DM, 5.6] A Teichmüller structure α of level G on X → S is a surjective exterior homomorphism

α ∈ Γ(S, Hom^{ext}(π_{1}(X/S), G)).

Thus locally on S (resp. S_{ét}) α corresponds to a surjective homomorphism π_{1}(X/S, s) → G.

We want to consider the moduli spaces parametrizing smooth stable curves of genus g with a Teichmüller structure of level G. However, as usual, it is more convenient to work with stacks. Thus let_{G}M_{g} denote the stack whose category of sections over the scheme S (lying over Spec( [1/#G])) is the category of smooth stable curves X → S of genus g endowed with a Teichmüller structure of level G, see [DM, Section 5]. The construction of Hom^{ext}(π_{1}(X/S), G) shows that the stack_{G}M_{g} is representable finite étale over M_{g}[1/#G]. Thus_{G}M_{g} is a separated algebraic stack, smooth over Spec( [1/#G]).

In a similar way we define the analytic stack _{G}M_{g}^{an} classifying complex curves with a Teichmüller structure of level G. By comparing the stacks _{G}M_{g} and GM_{g}^{an} with the stack M_{g} and using the known result for M_{g} one derives easily the following result.

Theorem 2.3.2 With notations as above.

1. A coarse moduli scheme _{G}M_{g} for _{G}M_{g} exists. It is separated of finite type over Spec( [1/#G]).

2. A coarse analytic moduli space _{G}M_{g}^{an} for _{G}M_{g}^{an} exists. It is canonically isomorphic to the analytic space associated to the complex variety _{G}M_{g} ⊗ .

Suppose we have a surjection G → G′. There is a natural forgetful morphism GM_{g} → _{G′}M_{g}. This morphism is representable finite étale. Hence, if _{G′}M_{g} is a scheme (i.e., isomorphic to _{G′}M_{g}), then so is _{G}M_{g}. In this case the morphism G′M_{g} → _{G}M_{g} is finite étale (perhaps of degree 0). Finally, we note that _{G}M_{g} might be empty, this being the case if G is not isomorphic to a quotient of the fundamental group of a Riemann surface of genus g.

2.3.3 Abelian level structures

In this section we treat the algebraic case. Let f : X → S be a smooth stable curve of genus g over S. Fix a natural number m. An abelian structure structure of level m on X over S is defined as an isomorphism of étale sheaves over S

( /m )_{S}^{2g} ⟶ R^{1}f_{∗}( /m ).

Such a level structure can only exist if the base scheme S lies over Spec( [1/m]); let us assume this is the case.

Note that the sheaf Hom^{ext}(π_{1}(X/S), /m ) is isomorphic to the subsheaf of primitive elements in R^{1}f_{∗}( /m ). Thus we see that the moduli stack of curves with an abelian level m structure is isomorphic to the stack _{G}M_{g} with G = ( /m )^{2g}. In particular, if m ≥ 3, then _{G}M_{g} is a fine moduli scheme smooth over Spec( [1/m]). The following is deduced from the above.

Proposition 2.3.4 If the finite group G allows a surjection onto ( /m )^{2g} for some m ≥ 3 then the coarse moduli scheme _{G}M_{g} is a fine moduli scheme.

2.3.5 Compactifications

In order to get compact moduli spaces we just take the normalization with respect to the Deligne-Mumford compactification. In this subsection the finite group G will be fixed, of order n = #G.

Consider the Deligne-Mumford compactification M_{g} ⊂ \overline{M_{g}}. We define \overline{_{G}M_{g}} as the normalization of \overline{M_{g}}[1/n] with respect to_{G}M_{g}. Similarly we define \overline{_{G}M_{g}}^{an} as the normalization of \overline{M_{g}}^{an} with respect to_{G}M_{g}^{an}. As per convention we denote \overline{_{G}M_{g}} (resp. \overline{_{G}M_{g}}^{an}) the associated coarse moduli scheme (resp. analytic moduli space). The morphism

\overline{_{G}M_{g}} ⟶ Spec( [1/n])

is proper, since \overline{M_{g}}[1/n] is proper over Spec( [1/n]). To see whether this morphism is smooth we have the following criterion.

Proposition 2.3.6 For finite groups G which allow a surjection G → ( /m )^{2g} for some m ≥ 3 the following statements are equivalent:

1. The morphism \overline{_{G}M_{g}} ⟶ Spec( [1/n]) is smooth.

2. The analytic space \overline{_{G}M_{g}}^{an} is a (nonsingular) complex manifold.

Proof. We use that \overline{_{G}M_{g}}^{an} is isomorphic to the analytic space associated to the variety \overline{_{G}M_{g}} ⊗ . Thus it suffices to show that \overline{_{G}M_{g}} ⊗ nonsingular implies \overline{_{G}M_{g}} ⊗ _{ p̄} nonsingular, where p > 0 and p does not divide n. The argument will be based on the fact that the morphism φ : \overline{_{G}M_{g}} → \overline{M_{g}}[1/n] is tamely ramified along the boundary.

To see this we need a description of the complete local rings of \overline{_{G}M_{g}} in points on the boundary. Suppose that C is a stable curve of genus g over an algebraically closed field k of characteristic p. The singular points of C are P_{1}, …, P_{ℓ}. Let C → Spf(W (k)[[t_{1}, …, t_{3g−3}]]) be the universal deformation of C. We choose the parameters t_{i} such that t_{i} = 0 defines the locus where P_{i} survives as a singular point, for i = 1, …, ℓ. Put A = W (k)[[t_{1}, …, t_{3g−3}]]. Since C is (uniquely) algebraizable we have a morphism Spec(A) → \overline{M_{g}} and

Spec(A[1/t_{1} … t_{ℓ}]) ⟶ M_{g}.

We consider the fibre product

Spec(A[1/t_{1} … t_{ℓ}]) ×_{M_{g}} _{G}M_{g}.

This is finite étale over A[1/t_{1} … t_{ℓ}] hence affine. The normalization of this over Spec(A) is Spec(B); here B is a product of complete local rings finite over A, ramified only over t_{1} … t_{ℓ} = 0. Thus we have

Spec(B[1/t_{1} … t_{ℓ}]) = Spec(A[1/t_{1} … t_{ℓ}]) ×_{M_{g}} _{G}M_{g}.

We claim that the morphism Spec(B) → \overline{_{G}M_{g}} identifies complete local rings at the points of \overline{_{G}M_{g}} lying over [C] ∈ \overline{M_{g}}(k). By general theory we know that the completion of \overline{M_{g}} at [C] is Spf(A)/Aut(C). A formal argument gives that the completion of \overline{_{G}M_{g}} along φ^{−1}([C]) is isomorphic to Spf(B)/Aut(C). Thus it suffices to show that the action of Aut(C) on φ^{−1}([C]) is free. By comparing levels, it suffices to show this for G = ( /m )^{2g}; this is the content of [Del, Proposition 3.5]. See also references in remark below.

We use Abhyankar’s lemma which asserts that any normal local domain A′, finite generically étale over A and ramified only along t_{1} … t_{ℓ} = 0 is contained in A[t_{i}^{1/n_{i}}] for some n_{1}, …, n_{ℓ} relatively prime to p. In addition, it is easily seen that A′ is formally smooth over W (k) if and only if A′ is actually equal to A[t_{i}^{1/n_{i}}] for some n_{1}, …, n_{ℓ}.

Let C_{K} denote the lift of C to K = Q(W (k)) given by setting t_{1} = … = t_{3g−3} = 0. The homomorphism A → K[[t_{1}, …, t_{3g−3}]] is such that

B ⊗_{A} K[[t_{1}, …, t_{3g−3}]]

describes the complete local rings of \overline{_{G}M_{g}} along φ^{−1}([C_{K}]). The result follows by comparing B to this ring. □

Remark 2.3.7 The arguments above actually show that in the situation of the proposition the stacks \overline{_{G}M_{g}} are schemes, i.e., \overline{_{G}M_{g}} ≅ \overline{_{G}M_{g}}. Thus we get a stable curve over \overline{_{G}M_{g}} from the morphism \overline{_{G}M_{g}} ≅ \overline{_{G}M_{g}} → \overline{M_{g}}; this also follows from the existence of such a stable curve in the case of abelian level structures, see [P, Thm. 10.9], [GO, page 12] and [Mo, Bemerkung 1].

3 Monodromy along the boundary

In this section we study the moduli spaces \overline{_{G}M_{g}} along the boundary. To do this it suffices to understand the monodromy along the boundary on the relative fundamental group of the universal curve over M_{g}^{an}.

3.1 The results

Let us formulate the main result. Let Π denote the standard fundamental group of a compact Riemann surface of genus g. We fix natural numbers k, n with k ≥ 1 and n ≥ 3. We will consider moduli of curves of genus g with a Teichmüller structure of level G, where

G = Π/Π^{(k+1),n}.

(For notations see 1.1.) By a result of Labute [Lab], the quotients Π^{(k)}/Π^{(k+1)} are finitely generated free abelian groups. Thus G has a filtration whose successive quotients are finite abelian groups of exponent n. Any prime dividing #G also divides n. Further, there is a surjection G → ( /n )^{2g}. By Section 2 we get a moduli scheme

\overline{_{G}M_{g}} ⟶ Spec( [1/n])

whose interior _{G}M_{g} classifies smooth genus g curves C with a surjection π_{1}(C) → G given up to inner automorphisms.

Theorem 3.1.1 Suppose k ∈ {1, 2, 3} and n ≥ 3. The structural morphism \overline{_{G}M_{g}} → Spec( [1/n]) is smooth if and only if

• k = 1 and g = 2,

• k = 2 and n is odd,

• k = 3 and n is odd or n is divisible by 4.

Furthermore, if k ≥ 4, n ≥ 3 and n relatively prime to 6, then \overline{_{G}M_{g}} → Spec( [1/n]) is smooth.

Remark 3.1.2 The abelian case of this theorem, i.e., the case k = 1, has been proven by Mostafa [Mo] and Van Geemen-Oort [GO].

A smooth and compact cover \overline{M_{g}[_{2}^{n}]}^{an} → \overline{M_{g}}^{an} with n even and n ≥ 6 has been constructed by Looijenga using Prym level structures, see [L]. This has been extended to higher level structures \overline{M_{g}[_{q}^{n}]}^{an} by one of us (Pikaart). Using an analog of Proposition 2.3.6 this result may be extended to an open set of Spec( ), at least if q ≥ 3.

We note that \overline{_{G}M_{g}} → \overline{M_{g}}[1/n] is a Galois cover with group Aut(G)/Inn(G) if g ≥ 3 (if g = 2 one has to divide out an additional {±1}). This follows since for any surjection Π → G the kernel is equal to Π^{(k),n}. However, \overline{_{G}M_{g}} ⊗ need not be connected, see [DM, 5.13] for a description of the set of connected components.

To prove the theorem, it suffices to consider \overline{_{G}M_{g}}^{an}, see Proposition 2.3.6. We already know that _{G}M_{g}^{an} is smooth. As in the proof of Proposition 2.3.6 we describe analytic neighbourhoods of points in the boundary.

Let C be a complex stable curve of genus g with singular points {P_{1}, …, P_{ℓ}}. Let Γ = Γ(C) be its dual graph; an edge for each point P_{j}, a vertex for an irreducible component of C. Let π : (C, C) → (B, 0) be a local universal deformation of C, where B ⊂ ^{3g−s} is a polydisc neighbourhood of 0. The coordinates z_{i} are chosen such that z_{j} = 0, 1 ≤ j ≤ ℓ parametrizes curves where the singular point P_{j} subsists. The discriminant locus Δ ⊂ B of π is thus given by z_{1} … z_{ℓ} = 0. Put U = B \ Δ, let x ∈ U and choose y ∈ C_{x} = π^{−1}(x). The fundamental group of U is an abelian group, freely generated by simple loops around the divisors z_{j} = 0, thus naturally isomorphic to the free abelian group on the edges of Γ, i.e., π_{1}(U, x) ≅ ⨁_{e∈Edges(Γ)} e. The map C|_{U} → U is a locally trivial fibration, hence we have the exact sequence

1 ⟶ π_{1}(C_{x}, y) ⟶ π_{1}(C|_{U}, y) ⟶ π_{1}(U, x) ⟶ 1.

(Use that π_{2}(U) = (0).) This short exact sequence provides us with the monodromy representation

ρ : π_{1}(U, x) ⟶ Out(π_{1}(C_{x}, y)).

The points P_{j} determine non-trivial distinct isotopy classes of circles on C_{x}, which have pairwise disjoint representatives c_{j}. The fundamental group of U is also naturally isomorphic to the free abelian group on these circles, π_{1}(U, x) ≅ ⨁_{i=1}^{l} c_{i}. Under this identification we have that

ρ(c_{i}) = D_{c_{i}},

where D_{c_{i}} is the exterior automorphism of π_{1}(C_{x}) given by a Dehn twist (also written D_{c_{j}}) around the circle c_{i} (see [Deh],[Lam]).

We will describe of a neighbourhood of a point in \overline{_{G}M_{g}}^{an} lying above [C]. Let Z be the fibre product

Z = U ×_{M_{g}^{an}} _{G}M_{g}^{an}.

The normalization of B in the function field of Z is denoted Z̄. Note that Z → U is a finite topological covering space given by the set

S = Hom-surj(π_{1}(C_{x}, y), G)/π_{1}(C_{x}, y)

with π_{1}(U, x)-action defined via ρ. As in the proof of Proposition 2.3.6 there is an action of Aut(C) on Z̄ and Z̄/Aut(C) defines a neighbourhood of φ^{−1}([C]) ⊂ \overline{_{G}M_{g}}^{an}. As in that proof we get that \overline{_{G}M_{g}}^{an} is smooth along φ^{−1}([C]) if and only if Z̄ is smooth. (Here we use again that Aut(C) acts freely on φ^{−1}([C]).) Finally there is the following criterion: Z̄ is smooth if and only if for all s ∈ S we have

Stab(s) = ⨁_{e∈Edges(Γ)} n_{e} e

for certain n_{e} ∈ . Notice that if m ∈ , then mF^{0} + mF^{1} (F^{i} as below) is of this form, but 2mF^{0} + mF^{1} is not.

We remark that the arguments above go through for arbitrary finite groups G, with a surjection onto ( /n )^{2g}. In order to describe the stabilizers in our more special situation we introduce the following notation:

A^{(k),n} = Ker (Aut(Π) → Aut(Π/Π^{(k+1),n})) ,
I^{(k),n} = Ker (Inn(Π) → Inn(Π/Π^{(k+1),n})) ,
O^{(k),n} = Ker (Out(Π) → Out(Π/Π^{(k+1),n})) .

Oda et al. consider a variant with n = 0. By choosing an isomorphism π_{1}(C_{x}, y) ≅ Π we may view ρ as a map into Out(Π). It is clear that Stab(s) = ρ^{−1}(O^{(k),n}) for any s ∈ S (use Remark 3.1.2). Therefore, Theorem 3.1.1 follows from Theorem 3.1.3 below.

We will describe a decreasing filtration F^{i} on ⨁_{e∈Edges(Γ)} e. An edge e such that Γ\e is disconnected is called a bridge. A bridge b is said to bound a genus one curve if one of the two components of C_{x}\{ the circle corresponding to b} has genus one. A pair of distinct edges {e, f} is called a cut pair if neither e nor f is a bridge and Γ\{e, f} is disconnected. A subset E of the edges of Γ is called a maximal cut system if E contains at least one cut pair, any two elements of E form a cut pair and no element of E forms a cut pair with an element outside E. Let B be the set of bridges of Γ and let B_{1} be the subset of B consisting of bridges which bound genus one curves. Let {E_{i}}_{i∈I} denote the maximal cut systems.Set

D_{i} := Ker( ⨁_{e∈E_{i}} e →^{deg} ).

We define a decreasing filtration F^{i} on ⨁_{e∈Edges(Γ)} e, as follows:

F^{0} = ⨁_{e∈Edges(Γ)} e,
F^{1} = ⨁_{i∈I} D_{i} ⨁ (⨁_{b∈B} b) ,
F^{2} = ⨁_{b∈B} b,
F^{3} = (0).

Furthermore, set F_{1}^{2} := ⨁_{b∈B_{1}} b. This refines the filtration into F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ F_{1}^{2} ⊃ F^{3} = (0). Here is the main result of this article.

Theorem 3.1.3 Notations as above. For n, l ∈ , define n_{l} := n/gcd(l, n).

1. If k = 1, then  ρ^{−1}(O^{(k),n}) = nF^{0} + F^{1}
2. If k = 2, then  ρ^{−1}(O^{(k),n}) = nF^{0} + n_{2}F^{1} + F^{2}
3. If k = 3 and 2||n, then  ρ^{−1}(O^{(k),n}) = nF^{0} + \frac{1}{2}nF^{1} + n_{2}F^{2} + n_{6}F_{1}^{2}
If k = 3 and n is odd or 4|n, then ρ^{−1}(O^{(k),n}) = nF^{0} + nF^{1} + n_{2}F^{2} + n_{6}F_{1}^{2}
4. If k ≥ 4 and (n, 6) = 1, then  ρ^{−1}(O^{(k),n}) = nF^{0}

Remark 3.1.4 The case k = 1 has been proven by Brylinski ([B]). The case k ≥ 4 follows from 3, the easy inclusions of Section 4 and the inclusions O^{(k),n} ⊂ O^{(l),n} if k ≥ l.

If we take n = 0 then Theorem 3.1.3 reduces to the non-pointed case of [AMO, Main Theorem].

4 Description of Dehn twists and easy inclusions

In this section we will prove the inclusions “⊃” from Theorem 3.1.3.

Let Γ be the graph of a stable curve and (S, {c_{i}}) the smooth model with a set of circles as described in Subsection 3.1. We will describe the Dehn twist associated to a bridge, cut pair or circle separatedly.

4.1 Bridges

Let b be a bridge on S, let g be the genus of S. Modulo a homeomorphism the situation looks as follows:

Fig. 1

Cutting S along b yields the decomposition S = S_{1} ∪ S_{2}. Let g_{i} be the genus of S_{i}. Choose a base point p in S_{1} and standard generators α_{±i}, 1 ≤ i ≤ g, for π := π_{1}(S, p) such that α_{±i} are in S_{1} if i ≤ g_{1} and α_{±i} for i ≥ g_{1} + 1 hits b exactly twice. We set v = [α_{1}, α_{−1}] ⋯ [α_{g_{1}}, α_{−g_{1}}], it is freely homotopic to b for a suitable orientation of b. We list the effect of the Dehn twist D_{b} on the standard generators:

\begin{matrix}i ∈ \{±1, …, ±g_{1}\} & D_{b} : α_{i} ↦ α_{i}, \\ i ∈ \{±(g_{1} + 1), …, ±g\} & D_{b} : α_{i} ↦ v^{−1}α_{i}v \end{matrix}

This gives for the mth-powers:

\begin{matrix}i ∈ \{±1, …, ±g_{1}\} & D_{b}^{m} : α_{i} ↦ α_{i}, \\ i ∈ \{±(g_{1} + 1), …, ±g\} & D_{b}^{m} : α_{i} ↦ v^{−m}α_{i}v^{m} \end{matrix}

We see that D_{b}(α_{±i})α_{±i}^{−1} ∈ π^{(3)} for all i. This proves that ρ(b) ∈ O^{(2)} and thus by linearity ρ(F^{2}) ⊂ O^{(2)}.

Suppose b bounds a genus one curve, say S_{2} has genus one. In this case v = [α_{−g}, α_{g}] and we only have to consider

D_{b}^{m}(α_{±g})α_{±g}^{−1} = [[α_{g}, α_{−g}]^{m}, α_{±g}] ≡ [[α_{g}, α_{−g}], α_{±g}]^{m} mod π^{(4)}.

To prove this element lies in π^{(4),n} we define f : F (x, y, z) → π by x ↦ α_{g}, y ↦ α_{−g} and z ↦ 1. Then f([[x, y], x]) = [[α_{g}, α_{−g}], α_{g}] and f([[x, y], y]) = [[α_{g}, α_{−g}], α_{−g}]. From Lemma 6.3 we see that n_{6}|m implies that the element above lies in π^{(4),n}. Thus ρ(n_{6}F_{1}^{2}) ⊂ O^{(3),n}.

Suppose b does not bound a genus one curve. In that case we have

D_{b}^{m}(α_{±i})α_{±i}^{−1} = [v^{m}, α_{±i}^{−1}] ≡ [v, α_{±i}]^{−m}
≡ [[α_{1}, α_{−1}], α_{±i}]^{−m} ⋯ [[α_{g_{1}}, α_{−g_{1}}], α_{±i}]^{−m}

for i > g_{1} in π/π^{(4)}. Define g_{j} : F (x, y, z) → π by x ↦ α_{j}, y ↦ α_{−j} and z ↦ α_{±i}. Then g_{j}([[x, y], z]^{m}) = [[α_{j}, α_{−j}], α_{±i}]^{m}. From Lemma 6.3 we see that n_{2}|m implies ρ(mb) ∈ O^{(3),n}, and thus ρ(n_{2}F^{2}) ⊂ O^{(3),n}.

4.2 Edges which are not bridges

Let c be a circle on S which is not a bridge. Modulo a homeomorphism the situation looks as follows:

Fig. 2

We choose a point p in S and standard generators α_{±i} for π = π_{1}(S, p) such that α_{−g} is the only one intersecting c and α_{g} is freely homotopic to c. The action of the Dehn twist becomes: D_{c}(α_{i}) = α_{i} if i ≠ −g and D_{c}(α_{−g}) = α_{g}α_{−g}. Thus D_{c}^{m}(α_{i})α_{i}^{−1} = 1 or α_{g}^{m}. Evidently, if n|m then ρ(mc) ∈ O^{(k),n} for all k. Together with the results of 4.1 this gives ρ(nF^{0}) ⊂ O^{(k),n} for all k.

4.3 Cut systems

Let e_{1}, e_{2} be a cut pair on S. Modulo a homeomorphism the situation is as follows:

Fig. 3

Cutting S along e_{1} and e_{2} yields the decomposition S = S_{1} ∪ S_{2}. Let g_{1} be the genus of S_{1}. Choose a base point p in S_{1}. We take standard generators α_{±i} for π_{1}(S, p), such that for i ∈ {±1, …, ±g_{1}, g_{1} + 1}, α_{i} is in S_{1}, for i ∈ {±(g_{1} + 2), …, ±g}, α_{i} enters S_{2} via e_{1} and leaves S_{2} also via e_{1} and α_{−(g_{1}+1)} enters S_{2} via e_{1} and leaves S_{2} via e_{2}. Furthermore, we want α_{g_{1}+1} to be freely homotopic to e_{2}, for a suitable orientation of e_{2}. It follows that after suitable orientation of e_{1} the loop [α_{1}, α_{−1}] ⋯ [α_{g_{1}}, α_{−g_{1}}]α_{g_{1}+1}^{−1} is freely homotopic to e_{1}. Let us write v = [α_{1}, α_{−1}] ⋯ [α_{g_{1}}, α_{−g_{1}}]. We list the effect of the Dehn twists D_{e_{i}} on these generators:

\begin{matrix}i ∈ \{±1, …, ±g_{1}, g_{1} + 1\} & \{ \begin{matrix}D_{e_{1}} : α_{i} ↦ α_{i} \\ D_{e_{2}} : α_{i} ↦ α_{i} \end{matrix} \\ i = −(g_{1} + 1) & \{ \begin{matrix}D_{e_{1}} : α_{i} ↦ α_{i}vα_{g_{1}+1}^{−1} \\ D_{e_{2}} : α_{i} ↦ α_{g_{1}+1}^{−1}α_{i} \end{matrix} \\ i ∈ \{±(g_{1} + 2), …, ±g\} & \{ \begin{matrix}D_{e_{1}} : α_{i} ↦ α_{g_{1}+1}v^{−1}α_{i}vα_{g_{1}+1}^{−1} \\ D_{e_{2}} : α_{i} ↦ α_{i} \end{matrix} \end{matrix}

Thus we get the following formulae for D_{e_{2}}D_{e_{1}}^{−1}:

\begin{matrix}|i| ≤ g_{1} & D_{e_{2}}D_{e_{1}}^{−1}(α_{i}) = α_{i}, \\ i = g_{1} + 1 & D_{e_{2}}D_{e_{1}}^{−1}(α_{i}) = α_{i}, \\ i = −(g_{1} + 1) & D_{e_{2}}D_{e_{1}}^{−1}(α_{i}) = α_{g_{1}+1}^{−1}α_{i}α_{g_{1}+1}v^{−1}α_{i}^{−1}α_{i} = [α_{g_{1}+1}, α_{i}^{−1}]v^{−1}[v^{−1}, α_{i}^{−1}]α_{i}, \\ |i| ≥ g_{1} + 2 & D_{e_{2}}D_{e_{1}}^{−1}(α_{i}) = vα_{g_{1}+1}^{−1}α_{i}α_{g_{1}+1}v^{−1}α_{i}^{−1}α_{i} = v[α_{g_{1}+1}, α_{i}^{−1}]v^{−1}[v^{−1}, α^{−1}]α_{i} \end{matrix}

and for the mth powers:

\begin{matrix}i ∈ \{±1, …, ±g_{1}, g_{1} + 1\} & (D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i}) = α_{i}, \\ i = −(g_{1} + 1) & (D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i}) = α_{g_{1}+1}^{−m}α_{i}(α_{g_{1}+1}v^{−1})^{m}, \\ i ∈ \{±(g_{1} + 2), …, ±g\} & (D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i}) = (vα_{g_{1}+1}^{−1})^{m}α_{i}(α_{g_{1}+1}v^{−1})^{m}. \end{matrix}

This proves that for a cut pair {e_{1}, e_{2}} we have ρ(e_{2} − e_{1}) ∈ O^{(1)}. It follows from this and 4.1 that ρ(F^{1}) ⊂ O^{(1)}, because F^{1} is generated by elements of the form e_{2} − e_{1} for a cut pair {e_{1}, e_{2}} and the elements b, b ∈ B.

To finish the proof of the inclusions “⊃” we have to show that ρ(me_{2} −me_{1}) ∈ O^{(2),n} if n_{2}|m and ρ(me_{2} − me_{1}) ∈ O^{(3),n} if n|m, or \frac{1}{2}n|m in case 2||m. We have to show that the divisibility conditions imply (D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i})α_{i}^{−1} ∈ π^{(3),n}, respectively π^{(4),n}. We will make computations in π modulo π^{(4)}. In the case that i ∈ {±1, …, ±g_{1}, g_{1} + 1} there is nothing to prove. We have (α_{g_{1}+1}v^{−1})^{m} ≡ α_{g_{1}+1}^{m}v^{−m}[v^{−1}, α_{g_{1}+1}]^{\frac{1}{2} m(m−1)} mod π^{(4)}, as one can prove by induction. From equality 5 of Subsection 6 we have

α_{g_{1}+1}^{−m}α_{i}α_{g_{1}+1}^{m}α_{i}^{−1} ≡ [α_{g_{1}+1}, α_{i}^{−1}]^{m}[[α_{g_{1}+1}, α_{i}^{−1}], α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}.

Thus for i = −(g_{1} + 1) we get

(D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i})α_{i}^{−1}
≡ α_{g_{1}+1}^{−m}α_{i}(α_{g_{1}+1}v^{−1})^{m}α_{i}^{−1}
≡ α_{g_{1}+1}^{−m}α_{i}α_{g_{1}+1}^{m}v^{−m}[v^{−1}, α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}α_{i}^{−1}
≡ [α_{g_{1}+1}, α_{i}^{−1}]^{m}[[α_{g_{1}+1}, α_{i}^{−1}], α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}v^{−m}[v, α_{i}]^{m}[v, α_{g_{1}+1}]^{− \frac{1}{2} m(m−1)}
≡ [α_{g_{1}+1}, α_{i}^{−1}]^{m}[[α_{g_{1}+1}, α_{i}^{−1}], α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}[α_{1}, α_{−1}]^{m} ⋯ [α_{g_{1}}, α_{−g_{1}}]^{m}[[α_{1}, α_{−1}], α_{i}]^{m} ⋯

[[α_{g_{1}}, α_{−g_{1}}], α_{i}]^{m}[[α_{1}, α_{−1}], α_{g_{1}+1}]^{− \frac{1}{2} m(m−1)} ⋯ [[α_{g_{1}}, α_{−g_{1}}], α_{g_{1}+1}]^{− \frac{1}{2} m(m−1)}.

In this product there are, apart from factors in π^{(3)}, a number of terms of the form [x, y]^{m}, x, y ∈ π. By the divisibility conditions all these are in π^{(4),n}, since this is true in the case of a free group ⟨x, y, z⟩ by Lemma 6.3. Thus it is clear that the whole product lies in π^{(3),n} for n_{2}|m.

The other terms in the product are of the form [[x, y], z]^{\frac{1}{2} m(m−1)}, where x, y, z ∈ π. It follows from Lemma 6.3, by mapping G = ⟨x, y, z⟩ into π that these terms are in π^{(4),n} as soon as n_{2}|\frac{1}{2}m(m − 1). This is equivalent to the condition n|m or \frac{1}{2}n|m in case 2||n.

For i ≥ g_{1} + 2 we have:

(D_{e_{2}}D_{e_{1}}^{−1})^{m}(α_{i})α_{i}^{−1} = (vα_{g_{1}+1}^{−1})^{m}α_{i}(α_{g_{1}+1}v^{−1})^{m}α_{i}^{−1}
≡ v^{m}α_{g_{1}+1}^{−m}[α_{g_{1}+1}^{−1}, v]^{\frac{1}{2} m(m−1)}α_{i}α_{g_{1}+1}^{m}v^{−m}[v^{−1}, α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}α_{i}^{−1}
≡ [α_{g_{1}+1}, α_{i}^{−1}]^{m}[[α_{g_{1}+1}, α_{i}^{−1}], α_{g_{1}+1}]^{\frac{1}{2} m(m−1)}

The same argument works to show that our divisibility conditions imply this product lies in either π^{(3),n} or π^{(4),n}.

5 Completion of proof

In this section we will use the following argument. Suppose we are given a morphism f : S → S′ of compact connected oriented surfaces and a circle c on S. Assume that f maps a tubular neighbourhood of c isomorphically into S′, denote by c′ the image of c. Then we have: f_{∗} ∘ D_{c} = D_{c′} ∘ f_{∗}.

Let Γ be the graph of a stable curve and (S, {c_{i}}) the smooth model with a set of circles as described in Subsection 3.1. Let H be the set of circles which are not bridges and are not involved in any cut system. Let B be the set of bridges of Γ and let {E_{i}|i = 1, …, s} be the set of maximal cut systems. Choose a numbering E_{i} = {e_{i,0}, …, e_{i,f_{i}}} for each i. Remark that Edges(Γ) = H ∪ ⋃ E_{i} ∪ B.

5.1 The case k = 1

We want to prove the inclusion “⊂” for k = 1. Suppose σ ∈ ⊕ e is a counterexample which involves a minimal number of edges. This means that ρ(σ) lies in O^{(1),n} but not all coefficients of σ are divisible by n; the number of nonzero coefficients is minimal. We write σ = ∑ n_{c}c. We may subtract elements of nF_{0} + F_{1} from σ: we already know that nF_{0} + F_{1} maps into O^{(1),n}. Thus, by minimality, we may suppose that: (i) none of the nonzero coefficients n_{c} is divisible by n, (ii) none of the circles c is a bridge, and (iii) of each cutsystem E_{i} at most one element occurs in σ. Modulo a homeomorphism the situation looks as follows:

Fig. 4

Take a point p near c. It is clear that we can find a loop α which intersects c exactly once and none of the other edges involved in σ. But now we have a contradiction, because by Section 4.2 we know: ρ(σ)(α)α^{−1} = c^{n_{c}} ∉ π^{n} if n does not divide n_{c}. Thus we have proven ρ^{−1}(O^{(1),n}) ⊂ nF^{0} + F^{1}, equality follows from sections 4.2 and 4.3.

5.2 The case k = 2

We want to prove the inclusion “⊂” for k = 2. Let σ = ∑_{b∈B} m_{b}b+∑_{i,j} m_{i,j}(e_{i,j} − e_{i,0})+∑_{i} m_{i}e_{i,0}+∑_{c∈H} n_{c}c be such that ρ(σ) ∈ O^{(2),n}. By the above we know that n|m_{i} and n|n_{c}, c ∈ H. We have to show that n_{2}|m_{i,j} for all possible i, j. Suppose this does not hold and suppose furthermore that σ is a minimal counterexample with respect to the number of edges involved. Arguing as above we may assume that σ = ∑_{i,j} m_{i,j}(e_{i,j} − e_{i,0}), where no (nonzero) m_{i,j} is divisible by n_{2}.

Proposition 5.2.1 There is a maximal cut system which is such that if we cut S along every element of the cut system, one connected component contains all other cut systems.

Proof. (Cf. [AMO], Lemma 5.3.) Take any cut system E. If it does not have the required property, that means that there is more than one connected component of S \ E, say S_{E,1}, …, S_{E,l}, which contain maximal cut systems. Suppose S_{E,j} contains a_{j} maximal cut systems and suppose a_{1} = min_{j}{a_{j}}. Take a maximal cut system F in S_{E,1}. Clearly there is a connected component of S \ F which contains at least ∑_{j=2}^{l} a_{j} + 1 components, namely the one containing E. This component contains more maximal cut systems then the one we started with. Continuing in this way, we arrive at our result. □

Let E_{1} = {e_{1,0}, …, e_{1,f_{1}}} be a maximal cut system such that one of the two components bounded by e_{1,0} and e_{1,f_{1}}, contains all other cut systems involved in σ. We suppose these edges are numbered cyclically: one of the connected components of S \ {e_{1,i}, e_{1,i+1}} contains all other e_{1,j} (see figure 6). If E_{1} contains more than two edges, we proceed as follows. Replace the component of S \ {e_{1,1}, e_{1,f_{1}}} which does not contain e_{1,0} by a cylinder to get an oriented surface S′. There is a continuous map f : S → S′ such that for all elements e_{1,j}, except e_{1,0}, we have that f(e_{1,j}) is homotopic to f(e_{1,1}). As explained above, we get that f_{∗} ∘ ρ(σ) = ρ(f_{∗}σ) ∘ f_{∗}. Thus ρ(σ) ∈ O^{(2),n} implies ρ(f_{∗}σ) ∈ O^{(2),n}. It is also clear that f_{∗}(σ) = m_{1,1}(f(e_{1,1}) − f(e_{1,0})) + ∑_{i>1} m_{i,j}(f(e_{i,j}) − f(e_{i,0})). We are reduced to the case that E_{1} = {e_{1,0}, e_{1,1}} consists of two elements.

The situation now is as in 4.3 (figure 3) where all the cut systems E_{i}, i > 1, lie in the component S_{2}. The generator α_{−(g_{1}+1)} may be chosen such that it does not intersect the circles e_{i,j}, i > 1. Thus we get from the formulae of Section 4.3 that

ρ(σ)(α_{−(g_{1}+1)}) · α_{−(g_{1}+1)}^{−1} = (D_{e_{1,1}}D_{e_{1,0}}^{−1})^{m_{1,1}}(α_{−(g_{1}+1)}) · α_{−(g_{1}+1)}^{−1}
= α_{g_{1}+1}^{−m_{1,1}}α_{−(g_{1}+1)}(α_{g_{1}+1}v^{−1})^{m_{1,1}}α_{−(g_{1}+1)}^{−1}.

We define f : π → G = ⟨x, y, z⟩ by α_{1}, α_{−g} ↦ x , α_{−1}, α_{g} ↦ y, α_{g_{1}+1} ↦ z, other generators ↦ 1. (The defining relation for π is indeed mapped to 1.) The expression above is mapped to z^{−m_{1,1}} · (z · [y, x])^{m_{1,1}}. Modulo G^{(3)} this equals [y, x]^{m_{1,1}}, hence ρ(σ) ∈ O^{(3),n} implies n_{2}|m_{1,1} (see Corollary 6.4). In this way we prove the desired contradiction; the inclusion “⊂” for k = 2 follows.

5.3 The case k = 3

We want to prove the inclusion “⊂” for k = 3. Let σ = ∑_{b∈B} m_{b}b+∑_{i,j} m_{i,j}(e_{i,j} − e_{i,0}) + ∑_{i} m_{i}e_{i,0} + ∑_{c∈H} n_{c}c be such that ρ(σ) ∈ O^{(3),n}. We have to show that n_{2}|m_{i,j} or n|m_{i,j}, for all i, j, depending on n being exactly divisible by 2 or not; and n_{6} respectively n_{2} divides m_{b} depending on b ∈ B_{1} or not. Suppose this does not hold and suppose furthermore that σ is a minimal counterexample with respect to the number of edges involved. Thus σ = ∑_{b∈B} m_{b}b+∑_{i,j} m_{i,j}(e_{i,j} −e_{i,0}) and none of the nonzero m_{b}, m_{i,j} satisfy the divisibility conditions.

Case 1. For some b involved in σ one of the connected components, say S′, of S \ {b} contains no other edges involved in σ. The situation looks as follows.

Fig. 5

We take a basepoint p in S′ and standard generators α_{±i} which are loops in S′ for i ≤ g(S′) and which are not loops in S′ for i > g(S′). We may choose α_{g(S′)+1} such that it does not intersect any circles involved in σ but b. To see this let S′′ denote the connected component of S \ {c ∈ σ, c ≠ b} containing S′. If g(S′′) > g(S′), then there is a ‘hole’ between b and the boundary of S′′, which we can take to be hole number g(S′) + 1 and take α_{g(S′)+1} accordingly. This is the case for example if S′′ = S (i.e., if σ = m_{b}b) or if S′′ is bounded by another bridge b′. If g(S′′) = g(S′), then S′′ is bounded by a cut pair {e_{1}, e_{2}} (belonging to some maximal cut system E_{i}), the part S′′ \ S′ looks like a pair of pants. In this case we can choose α_{g(S′)+1} to go around a pants leg of S′′ \ S′, for example freely homotopic to e_{2} as in Section 4.3.

It is now clear that ρ(σ)(α_{g(S′)+1}) = D_{b}^{m_{b}}(α_{g(S′)+1}) and by Section 4.1 we get

ξ := ρ(σ)(α_{g(S′)+1}) · α_{g(S′)+1}^{−1} = v^{−m_{b}}α_{g(S′)+1}v^{m_{b}}α_{g(S′)+1}^{−1}

with v = [α_{1}, α_{−1}] · . . . · [α_{g(S′)}, α_{−g(S′)}]. Thus ρ(σ) ∈ O^{(3),n} implies ξ ∈ π^{(4),n}.

If b bounds a curve of genus 1, say g(S′) = 1 (the case g−g(S′) = 1 is similar), then v = [α_{1}, α_{−1}]. We define f : π → G = ⟨x, y, z⟩ by α_{1}, α_{−2} ↦ x, α_{−1}, α_{2} ↦ y and other generators ↦ 1. We see that f(ξ) = [x, y]^{−m_{b}}y[x, y]^{m_{b}}y^{−1} ≡ [[x, y], y]^{−m_{b}} modulo G^{(4)}. Thus ξ ∈ π^{(4),n} implies n_{6}|m_{b} by Lemma 6.3.

If b does not bound a curve of genus 1 (i.e., g(S′) ≥ 2 and g − g(S′) ≥ 2) we define g : π → G = ⟨x, y, z⟩ by α_{1}, α_{−g} ↦ x, α_{−1}, α_{g} ↦ y , α_{g(S′)+1} ↦ z^{−1} and other generators ↦ 1. We get g(ξ) = [x, y]^{−m_{b}}z^{−1}[x, y]^{m_{b}}z ≡ [[x, y], z]^{m_{b}}. Application of Lemma 6.3 gives n_{2}|m_{b} as desired.

Case 2: case 1 does not occur. This means that every connected component of S cut out by one bridge involved in σ contains a maximal cut system. Choose a maximal cut system as in Proposition 5.2.1. Let E_{1} = {e_{1,0}, …, e_{1,f_{1}}} be this cut system. Suppose the edges are numbered cyclically and suppose the connected component cut out by e_{1,0} and e_{1,1} which does not contain any edge of E_{1} contains all other maximal cut systems. Call this component S′. By the choice of E_{1} and the assumption, we get that S \ S′ does not contain any bridges involved in σ. The situation thus looks as shown below:

Arguing as in the proof for k = 2, we arrive at a curve with a cut pair e_{1,0} and e_{1,1} which is a maximal cut system with the property that one of the two components cut out by the cut pair contains all other edges involved in the counter example σ. Considering f : π → G as in Section 5.2, we get, as in case 1, that the element z^{−m_{1,1}} · (z · [y, x])^{m_{1,1}} lies in G^{(4),n}. Note that

z^{−m_{1,1}} · (z · [y, x])^{m_{1,1}} ≡ [y, x]^{m_{1,1}}[[y, x], z]^{\frac{1}{2} m_{1,1}(m_{1,1}−1)}.

Thus we get our divisibility condition on m_{1,1} by Lemma 6.3.

6 Computations for a free group on three generators

Lemma 6.1 Let G be a free group on three generators, G = ⟨x, y, z⟩. Then

G/G^{(4)} ≅ {(i_{1}, …, i_{14}) ∈ ^{14}}

where multiplication on the right-hand side is given as follows:

(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}, i_{7}, i_{8}, i_{9}, i_{10}, i_{11}, i_{12}, i_{13}, i_{14})·
(j_{1}, j_{2}, j_{3}, j_{4}, j_{5}, j_{6}, j_{7}, j_{8}, j_{9}, j_{10}, j_{11}, j_{12}, j_{13}, j_{14}) =

( \begin{matrix}i_{1} + j_{1} \\ i_{2} + j_{2} \\ i_{3} + j_{3} \\ i_{4} + j_{4} − i_{2}j_{1} \\ i_{5} + j_{5} − i_{3}j_{2} \\ i_{6} + j_{6} − i_{3}j_{1} \\ i_{7} + j_{7} + i_{4}j_{1} − \frac{1}{2}(j_{1} − 1)j_{1}i_{2} \\ i_{8} + j_{8} + i_{4}j_{2} − \frac{1}{2}(i_{2} − 1)i_{2}j_{1} − i_{2}j_{1}j_{2} \\ i_{9} + j_{9} + i_{4}j_{3} + i_{6}j_{2} − i_{2}i_{3}j_{1} − i_{2}j_{1}j_{3} − i_{3}j_{1}j_{2} \\ i_{10} + j_{10} + i_{5}j_{1} + i_{6}j_{2} − i_{3}j_{1}j_{2} \\ i_{11} + j_{11} + i_{5}j_{2} − \frac{1}{2}(j_{2} − 1)j_{2}i_{3} \\ i_{12} + j_{12} + i_{5}j_{3} − \frac{1}{2}(i_{3} − 1)i_{3}j_{2} − i_{3}j_{2}j_{3} \\ i_{13} + j_{13} + i_{6}j_{1} − \frac{1}{2}(j_{1} − 1)j_{1}i_{3} \\ i_{14} + j_{14} + i_{6}j_{3} − \frac{1}{2}(i_{3} − 1)i_{3}j_{1} − i_{3}j_{1}j_{3} \end{matrix} )

Proof. We have the exact sequences

1 → G^{(k)}/G^{(k+1)} → G/G^{(k+1)} → G/G^{(k)} → 1,

where G^{(k)}/G^{(k+1)} is a finitely generated free abelian group ([MKS, Th.5.12]) of rank N_{i} := \frac{1}{i} ∑_{d|i} μ(d)3^{i/d} ([MKS, Th.5.11]), where μ is the Möbius-function. Explicitly, we have, with r := [x, y], s := [y, z] and t := [x, z]:

G^{(1)}/G^{(2)} = ⟨x, y, z⟩_{ab} ,
G^{(2)}/G^{(3)} = ⟨r, s, t⟩_{ab} ,
G^{(3)}/G^{(4)} = ⟨[r, x], [r, y], [r, z], [s, x], [s, y], [s, z], [t, x], [t, z]⟩_{ab} ,

by the Jacobi-relation [t, y] ≡ [r, z][s, x] mod G^{(4)}. Now recall that if a ∈ G^{(k)} and b ∈ G^{(l)} then ab = ba[a, b] with [a, b] ∈ G^{(k+l)}, so that

ab ≡ ba mod G^{(k+l)}.  (1)

Furthermore we have the following identities: ([MKS, Th.5.1])

[a, b] = [b, a]^{−1},  (2)
[a, bc] = [a, c] [a, b] [[a, b], c],  (3)
[ab, c] = [a, c] [[a, c], b] [b, c].  (4)

It is clear now that any element a of G/G^{(4)} can be written in the form a = x^{i_{1}} ⋯ [t, z]^{i_{14}} in a unique manner. Note that the last eleven factors commute by (1). We define a map ϕ : G/G^{(4)} → {(i_{1}, …, i_{14}) ∈ ^{14}} by

x^{i_{1}}y^{i_{2}}z^{i_{3}}r^{i_{4}}s^{i_{5}}t^{i_{6}}[r, x]^{i_{7}}[r, y]^{i_{8}}[r, z]^{i_{9}}[s, x]^{i_{10}}[s, y]^{i_{11}}[s, z]^{i_{12}}[t, x]^{i_{13}}[t, z]^{i_{14}} ↦ (i_{1}, …, i_{14}).

Before we start the computation, notice that modulo G^{(4)} we have

[a^{i}, b^{j}] ≡ [a, b]^{ij}[[a, b], a]^{\frac{1}{2} ij(i−1)}[[a, b], b]^{\frac{1}{2} ij(j−1)},  (5)

as one proves easily by induction. We set (n, m) := \frac{1}{2}(n − 1)nm. The following identities hold in the group G/G^{(4)}:

x^{i_{1}}y^{i_{2}}z^{i_{3}}x^{j_{1}}y^{j_{2}}z^{j_{3}}
= x^{i_{1}+j_{1}}x^{−j_{1}}y^{i_{2}}z^{i_{3}}x^{j_{1}}z^{−i_{3}}y^{−i_{2}}y^{i_{2}}z^{i_{3}}y^{j_{2}}z^{j_{3}}
= x^{i_{1}+j_{1}}[x^{j_{1}}, z^{−i_{3}}y^{−i_{2}}]y^{i_{2}+j_{2}}y^{−j_{2}}z^{i_{3}}y^{j_{2}}z^{−i_{3}}z^{i_{3}+j_{3}}
= x^{i_{1}+j_{1}}[x^{j_{1}}, y^{−i_{2}}][x^{j_{1}}, z^{−i_{3}}][[x^{j_{1}}, z^{−i_{3}}], y^{−i_{2}}]y^{i_{2}+j_{2}}[y^{j_{2}}, z^{−i_{3}}]z^{i_{3}+j_{3}}
= x^{i_{1}+j_{1}}r^{−i_{2}j_{1}}[r, x]^{(j_{1},−i_{2})}[r, y]^{(−i_{2},j_{1})}t^{−i_{3}j_{1}}[t, x]^{(j_{1},−i_{3})}[t, z]^{(−i_{3},j_{1})}
y^{i_{2}+j_{2}}s^{−i_{3}j_{2}}[s, y]^{(j_{2},−i_{3})}[s, z]^{(−i_{3},j_{2})}z^{i_{3}+j_{3}}[t, y]^{i_{2}i_{3}j_{1}}
= x^{i_{1}+j_{1}}y^{i_{2}+j_{2}}z^{i_{3}+j_{3}}r^{−i_{2}j_{1}}s^{−i_{3}j_{2}}t^{−i_{3}j_{1}}[r, y]^{−i_{2}j_{1}(i_{2}+j_{2})}[t, y]^{−i_{3}j_{1}(i_{2}+j_{2})}[r, z]^{−i_{2}j_{1}(i_{3}+j_{3})}
[t, z]^{−i_{3}j_{1}(i_{3}+j_{3})}[s, z]^{−i_{3}j_{2}(i_{3}+j_{3})}[r, x]^{(j_{1},−i_{2})}[r, y]^{(−i_{2},j_{1})}[t, x]^{(j_{1},−i_{3})}[t, z]^{(−i_{3},j_{1})}
[s, y]^{(j_{2},−i_{3})}[s, z]^{(−i_{3},j_{2})}[r, z]^{j_{1}i_{2}i_{3}}[s, x]^{j_{1}i_{2}i_{3}}
= x^{i_{1}+j_{1}}y^{i_{2}+j_{2}}z^{i_{3}+j_{3}}r^{−i_{2}j_{1}}s^{−i_{3}j_{2}}t^{−i_{3}j_{1}}[r, x]^{(j_{1},−i_{2})}[r, y]^{−(i_{2},j_{1})−i_{2}j_{1}j_{2}}[r, z]^{−i_{2}j_{1}j_{3}−i_{3}j_{1}j_{2}−i_{2}i_{3}j_{1}}
[s, x]^{−i_{3}j_{1}j_{2}}[s, y]^{(j_{2},−i_{3})}[s, z]^{−(i_{3},j_{2})−i_{3}j_{2}j_{3}}[t, x]^{(j_{1},−i_{3})}[t, z]^{−(i_{3},j_{1})−i_{3}j_{1}j_{3}}.

Also

r^{i_{4}}s^{i_{5}}t^{i_{6}}x^{j_{1}}y^{j_{2}}z^{j_{3}}r^{j_{4}}s^{j_{5}}t^{j_{6}} = x^{j_{1}}y^{j_{2}}z^{j_{3}}r^{i_{4}}s^{i_{5}}t^{i_{6}}[r^{i_{4}}s^{i_{5}}t^{i_{6}}, x^{j_{1}}y^{j_{2}}z^{j_{3}}]r^{j_{4}}s^{j_{5}}t^{j_{6}}
= x^{j_{1}}y^{j_{2}}z^{j_{3}}r^{i_{4}+j_{4}}s^{i_{5}+j_{5}}t^{i_{6}+j_{6}}[r, x]^{i_{4}j_{1}}[r, y]^{i_{4}j_{2}}[r, z]^{i_{4}j_{3}+i_{6}j_{2}}
[s, x]^{i_{5}j_{1}+i_{6}j_{2}}[s, y]^{i_{5}j_{2}}[s, z]^{i_{5}j_{3}}[t, x]^{i_{6}j_{1}}[t, z]^{i_{6}j_{3}}.

Combining these proves the lemma. □ Lemma 6.2 (i_{1}, ⋯, i_{14})^{n} =

( \begin{matrix}ni_{1} \\ ni_{2} \\ ni_{3} \\ ni_{4} − \frac{1}{2}(n − 1)ni_{2}i_{1} \\ ni_{5} − \frac{1}{2}(n − 1)ni_{3}i_{2} \\ ni_{6} − \frac{1}{2}(n − 1)ni_{3}i_{1} \\ ni_{7} + \frac{1}{2}(n − 1)ni_{4}i_{1} − \frac{1}{12}(n − 1)n(2n − 1)i_{1}^{2}i_{2} + \frac{1}{4}(n − 1)ni_{1}i_{2} \\ ni_{8} + \frac{1}{2}(n − 1)ni_{4}i_{2} − \frac{1}{6}(n − 1)n(2n − 1)i_{1}i_{2}^{2} + \frac{1}{4}(n − 1)ni_{1}i_{2} − \frac{1}{4}(n − 1)ni_{1}i_{2}^{2} \\ ni_{9} + \frac{1}{2}(n − 1)ni_{4}i_{3} + \frac{1}{2}(n − 1)ni_{6}i_{2} − \frac{1}{3}(n − 1)n(2n − 1)i_{3}i_{1}i_{2} − \frac{1}{2}(n − 1)ni_{1}i_{2}i_{3} \\ ni_{10} + \frac{1}{2}(n − 1)ni_{5}i_{1} + \frac{1}{2}(n − 1)ni_{6}i_{2} − \frac{1}{6}(n − 1)n(2n − 1)i_{3}i_{1}i_{2} \\ ni_{11} + \frac{1}{2}(n − 1)ni_{5}i_{2} − \frac{1}{12}(n − 1)n(2n − 1)i_{2}^{2}i_{3} + \frac{1}{4}(n − 1)ni_{2}i_{3} \\ ni_{12} + \frac{1}{2}(n − 1)ni_{5}i_{3} − \frac{1}{6}(n − 1)n(2n − 1)i_{2}i_{3}^{2} + \frac{1}{4}(n − 1)ni_{3}i_{2} − \frac{1}{4}(n − 1)ni_{2}i_{3}^{2} \\ ni_{13} + \frac{1}{2}(n − 1)ni_{6}i_{1} − \frac{1}{12}(n − 1)n(2n − 1)i_{1}^{2}i_{3} + \frac{1}{4}(n − 1)ni_{1}i_{3} \\ ni_{14} + \frac{1}{2}(n − 1)ni_{6}i_{3} − \frac{1}{6}(n − 1)n(2n − 1)i_{1}i_{3}^{2} + \frac{1}{4}(n − 1)ni_{3}i_{1} − \frac{1}{4}(n − 1)ni_{1}i_{3}^{2} \end{matrix} )

Proof. By induction on n. □ Lemma 6.3 Notations as above.

\begin{aligned}G^{(4),n}/G^{(4)} & ≅ & \{(i_{1}, …, i_{14})|i_{1}, i_{2}, i_{3} ∈ n , i_{4}, i_{5}, i_{6} ∈ n_{2} , \\ & & i_{7}, …, i_{14} ∈ n_{6} , i_{9} + i_{10} ∈ n_{2} \}. \end{aligned}

(Recall that n_{i} := n/gcd(n, i).) Proof. The inclusion “⊂” follows at once from Lemma 4.2 and the following two observations:

−\frac{1}{12}(n − 1)n(2n − 1)i_{1}^{2}i_{2} + \frac{1}{4}(n − 1)ni_{1}i_{2} =

\frac{1}{12}(n − 1)ni_{1}i_{2}[−(2n − 1)i_{1} + 3]

is an element of n_{6} (either i_{1} is even or the two terms between brackets have the same parity). The same holds for

−\frac{1}{6}(n − 1)n(2n − 1)i_{1}i_{2}^{2} + \frac{1}{4}(n − 1)ni_{1}i_{2} − \frac{1}{4}(n − 1)ni_{1}i_{2}^{2} =

\frac{1}{12}(n − 1)ni_{1}i_{2}[−(4n + 1)i_{2} + 3].

The other inclusion follows from a direct computation:

\begin{matrix}a := & (1, 0, …, 0)^{n}(−1, 1, 0, …, 0)^{n}(0, −1, 0, …, 0)^{n} \\ = & (0, 0, 0, \frac{1}{2}(n − 1)n, 0, 0, −\frac{1}{6}(n − 1)n(n + 1), −\frac{1}{6}(n − 1)n(n + 1), 0, …, 0), \\ b := & (−1, 0, …, 0)^{n}(1, 1, 0 …, 0)^{n}(0, −1, …, 0)^{n} \\ = & (0, 0, 0, −\frac{1}{2}(n − 1)n, 0, 0, −\frac{1}{6}(n − 2)(n − 1)n, \frac{1}{6}(n − 1)n(n + 1), 0, …, 0), \\ c := & (1, 0 …, 0)^{n}(−1, −1, 0, …, 0)^{n}(0, 1, 0, …., 0)^{n} \\ = & (0, 0, 0, −\frac{1}{2}(n − 1)n, 0, 0, \frac{1}{6}(n − 1)n(n + 1), −\frac{1}{6}(n − 2)(n − 1)n, 0, …, 0). \end{matrix}

So we have:

ab = (0, 0, 0, 0, 0, 0, −\frac{1}{6}(n − 1)n(2n − 1), 0, …, 0),

ac = (0, 0, 0, 0, 0, 0, 0, −\frac{1}{6}(n − 1)n(2n − 1), 0, …, 0).

The same computation holds for the eleventh till the fourteenth coefficient, so it is possible to get n_{6} as one of the coefficients i_{7}, i_{8}, i_{11}, .., i_{14} and all the other coefficients zero. Thus, looking again at a, we see that it is possible to get n_{2} as the coefficient i_{4} and all the other coefficients zero. By symmetry the same holds for the coefficients i_{5}, i_{6}. Furthermore, if we put

d := (0, 0, −1, 0, …, 0)^{n}(0, 0, 1, 1, 0, …, 0)^{n}(0, 0, 0, −1, 0, ….,)^{n},
e := (−1, 0, …, 0)^{n}(1, 0, 0, 0, 1, 0, …, 0)^{n}(0, 0, 0, 0, −1, 0, ….,)^{n},

then we have:

\begin{matrix}d = (0, 0, 0, 0, 0, 0, 0, 0, \frac{1}{2}(n − 1)n, 0, …, 0), \\ e = (0, 0, 0, 0, 0, 0, 0, 0, 0, \frac{1}{2}(n − 1)n, 0, …, 0). \end{matrix}

Thus, we can get n_{2} in the ninth and tenth coefficient, independently. The rest now follows from what we have proven already and the computation of

(−1, 0, …, 0)^{n}(0, −1, 0, …, 0)^{n}(0, 0, −1, 0, …, 0)^{n}(1, 1, 1, 0, …, 0)^{n}.
□

Corollary 6.4 Notations as above.

1. (G^{(2)} ∩ G^{(3),n})/G^{(3)} ≅ n_{2}G^{(2)}/G^{(3)}.

2. (i_{7}, …, i_{14}) ∈ (G^{(3)}∩G^{(4),n})/G^{(4)} if and only if i_{7}, …, i_{14} ∈ n_{6} , i_{9}+i_{10} ∈ n_{2} .

References

[SGA 1] A. Grothendieck et al., Séminaire de géometrie algébrique 1 - Revêtements étales et groupe fondamentale, Lecture notes in mathematics 224, Springer-Verlag, Heidelberg (1971).

[AMO] M. Asada-M. Matsumoto-T. Oda, Local monodromy on the fundamental groups of algebraic curves along a degenerate stable curve, to appear in J. of Pure and Appl. Alg.

[B] J.-L. Brylinski, Propiétés de ramification à l’infini du groupe modulaire de Teichmüller, Ann. scient. Ec. Norm. Sup. 12 (1979), 295-333.

[Deh] M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-210.

[Del] P. Deligne, Le lemme de Gabber, in Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, ed. L. Szpiro, Astérisque 127 (1985), 131-150.

[DM] P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES 36 (1969), 75-109.

[GO] B. van Geemen, F. Oort, A compactification of a fine moduli space of curves, preprint Utrecht (1983).

[Lab] J. Labute, On the descending central series of groups with a single defining relation, Journal of algebra 14 (1970), 16-23.

[Lam] K. Lamotke, Die Homologie isolierter Singularitäten, Math. Z. 143 (1975), 27-44.

[L] E. Looijenga, Smooth Deligne-Mumford compactifications by means of Prym level structures, J. Algebraic Geometry 3 (1994), 283-293.

[MKS] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, Pure and applied math., Interscience publishers, New York, London (1966).

[Mo] S.M. Mostafa, Die Singularitäten der Modulmannigfaltigkeit \overline{M}_{g}(n) der stabilen Kurven vom Geschlecht g ≥ 2 mit n-Teilungspunktstruktur, Crelle 343 (1983), 81-98.

[Mu] J. Murre, Lectures on an introduction of Grothendieck’s theory of the fundamental group. Lecture notes, Tata institute of fundamental research, Bombay, 1967.

[P] H. Popp, Moduli theory and classification theory of algebraic varieties, Lecture notes in mathematics 620, Springer-Verlag, Berlin-Heidelberg-New York 1977.

M. Pikaart and A.J. de Jong Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. E-mail: pikaart@math.ruu.nl & dejong@math.ruu.nl