(1)LOG DIVISORS FROM A CONFIGURATION SPACE GROUP EQUIPPED WITH ITS COLLECTION OF LOG-FULL SUBGROUPS KAZUMI HIGASHIYAMA Abstract

LOG DIVISORS FROM A CONFIGURATION SPACE GROUP EQUIPPED WITH ITS COLLECTION OF LOG-FULL

SUBGROUPS

KAZUMI HIGASHIYAMA

Abstract. In the present paper, we study configuration space groups. The goal of this paper is to reconstruct group-theoretically the inertia groups asso- ciated to various types of log divisors of a log configuration space of a smooth log curve from the associated configuration space group equipped with its col- lection of log-full subgroups.

0. Introduction

Let l be a prime number; k an algebraically closed field of characteristic ̸= l;

S^def= Spec(k); (g, r) a pair of nonnegative integers such that 2g−2+r >0;X^log→S a smooth log curve of type (g, r) (cf. Notation 1.3, (iv)); n∈Z>1. In the present paper, we study the n-th log configuration spaceX_n^log associated toX^log→S (cf.

Definition 2.1). WriteUX for the interior of the log schemeX^log (cf. Notation 1.2, (vi)). The log scheme X_n^log may be thought of as a certain compactification of the usualn-th configuration spaceUX_n associated to the smooth curveUX. Write Πn

def= π^pro-l₁ (X_n^log) for the pro-lconfiguration space group determined byX_n^log (cf.

[MzTa], Definition 2.3, (i)), i.e., the maximal pro-l quotient of the fundamental group of the log schemeX_n^log (for a suitable choice of basepoint). We shall refer to an irreducible divisor of the underlying scheme ofX_n^logcontained in the complement ofUX_n as alog divisorofX_n^log. Each log divisorV determines, up to Πn-conjugacy, an inertia groupIV(≃Zl)⊆Πn, which plays a central role in the present paper. Let V1, . . . , Vn be distinct log divisors ofX_n^log such thatV1∩· · ·∩Vn̸=∅. Then we shall refer toP ^def= V1∩· · ·∩Vnas alog-full point(cf. Definition 2.2, (ii), and Proposition 2.10). The log-full pointP=V1∩· · ·∩Vndetermines, up to Πn-conjugacy, alog-full subgroupA(≃IV₁×· · ·×IV_n≃Z^⊕_lⁿ)⊆Πn(cf. Definition 2.2, (iii)). It is known that the log-full subgroups of a configuration space group may be characterized group- theoretically whenever the configuration space group is equipped with the action of a profinite group that satisfies certain properties (cf. [HMM], Theorem D). In the present paper, we reconstruct group-theoretically the inertia groups associated to the log divisors from a configuration space group equipped with its collection of log-full subgroups. Moreover, we reconstruct group-theoretically the inertia groups associated to the tripodal divisors (cf. Definition 3.1, (ii)) and the drift diagonals (cf. Definition 3.1, (iv)), as well as the drift collections of Πn (cf. Definition 8.13) and the generalized fiber subgroups of Π_n (cf. Definition 9.1).

Our main result is as follows:

1

Theorem 0.1. For ∈ {◦,•}, let l be a prime number; k an algebraically closed field of characteristic̸=l;S ^def= Spec(k);(g, r)a pair of nonnegative integers such that2g−2 +r>0;

X^log→S

(cf. Notation 1.2, (vi)) a smooth log curve of type(g, r);n∈Z>1;X^log

n the n-th log configuration space associated to X^log → S; Π ^def= π^pro-l₁ (X^log

n ) (for a suitable choice of basepoint);

ϕ: Π^{◦ ∼}→Π^•

an isomorphism of profinite groups. We suppose that r > 0 (cf. the discussion below);ϕinduces a bijection between the set of log-full subgroups ofΠ^◦ and the set of log-full subgroups of Π^•. Then the following hold:

(i) ϕinduces a bijection between the set of inertia groups ofΠ^◦ associated to log divisors ofX_n^log_◦ ^◦ and the set of inertia groups ofΠ^• associated to log divisors ofX_n^log_• ^• (cf. Theorem 5.2).

(ii) ϕ induces a bijection between the set of inertia groups of Π^◦ associated to tripodal divisors of X_n^log◦ ^◦ and the set of inertia groups of Π^• associated to tripodal divisors of X_n^log• ^• (cf. Theorem 6.6).

(iii) ϕ induces a bijection between the set of inertia groups of Π^◦ associated to drift diagonals ofX_n^log_◦ ^◦ and the set of inertia groups ofΠ^• associated to drift diagonals ofX_n^log• ^• (cf. Theorem 7.3).

(iv) ϕinduces a bijection between the set of drift collections of Π^◦ and the set of drift collections ofΠ^• (cf. Theorem 8.14).

(v) ϕinduces a bijection between the set of generalized fiber subgroups of Π^◦ and the set of generalized fiber subgroups ofΠ^• (cf. Theorem 9.3).

Note that, roughly speaking, Theorem 0.1, (i), asserts that we may extract group- theoretically a “geometric direct summandZl” (i.e., an inertia group associated to a log divisor) from “Z^⊕lⁿ” (i.e., a log-full subgroup).

Note that one may define the notion of a log-full point even ifr= 0 (cf. [HMM], Definition 1.1). On the other hand, since log-full points do not exist when r= 0, we suppose thatr >0 in the present paper.

In the proof of Theorem 0.1, (ii), we use the fact that, in the notation of Theorem 0.1, in fact (g^◦, r^◦, n^◦) = (g^•, r^•, n^•) (cf. Theorem 3.10, (i)), which is proven in [HMM], Theorem A, (i).

This paper is organized as follows: In§1, we explain some notations. In§2, we define log configuration spaces, log-full points, and log divisors. In §3, we define tripodal divisors and drift diagonals and then proceed to study the geometry of various types of log divisors. In§4, we give a group-theoretic reconstruction of the scheme-theoretically non-degenerate elements (cf. Definition 4.6, (i)) of a log-full subgroup. In§5, we reconstruct the inertia groups associated to the log divisors.

In §6, we reconstruct the inertia groups associated to the tripodal divisors. In

§7, we reconstruct the inertia groups associated to the drift diagonals. In §8, we reconstruct the drift collections of a configuration space group. In§9, we reconstruct the generalized fiber subgroups of a configuration space group.

1. Notations

Notation 1.1. (i) Let G be a group. Then we shall write “1_G ∈ G” for the identity element of G.

(ii) LetGbe a group,H ⊆Ga subgroup, andα∈G. Then we shall write ZG(H)^def= {g∈G|gh=hgfor anyh∈H}

for the centralizer of H in G;

ZG(α)^def= ZG(⟨α⟩) ={g∈G|gα=αg} for the centralizer of αin G;

NG(H)^def= {g∈G|gHg⁻¹=H} for the normalizer ofH inG.

Notation 1.2. LetS^log be an fs log scheme (cf. [Naka], Definition 1.7).

(i) WriteS for the underlying scheme ofS^log.

(ii) WriteMS for the sheaf of monoids that defines the log structure ofS^log. (iii) Letsbe a geometric point ofS. Then we shall denote byI(s,MS) the ideal

of OS,s generated by the image of MS,s \ O^×_S,s via the homomorphism of monoids MS,s → OS,s induced by the morphism MS → OS which defines the log structure ofS^log.

(iv) Lets∈Sandsa geometric point ofSwhich lies overs. Write (MS,s/O^×S,s)^gp for the groupification of MS,s/O^×S,s. Then we shall refer to the rank of the finitely generated free abelian group (MS,s/OS,s^× )^gpas thelog rankats. Note that one verifies easily that this rank is independent of the choice of s, i.e., depends only on s.

(v) Letm∈Z. Then we shall write

S^log≤mdef= {s∈S|the log rank atsis≤m}.

Note that sinceS^log≤mis open inS(cf. [MzTa], Proposition 5.2, (i)), we shall also regard (by abuse of notation) S^log≤m as an open subscheme ofS.

(vi) We shall write US

def= S^log≤0 and refer to US as the interior of S^log. When US=S, we shall often use the notationS to denote the log scheme S^log. Notation 1.3. Let (g, r) be a pair of nonnegative integers such that 2g−2 +r >0.

(i) Write Mg,r for the moduli stack (over Z) of pointed stable curves of type (g, r) andMg,r ⊆ Mg,r for the open substack corresponding to the smooth curves. Here, we assume the marked points to be ordered.

(ii) Write

Cg,r → Mg,r

for the tautological curve over Mg,r; Dg,r

def= Mg,r\ Mg,r for the divisor at infinity.

(iii) Write M^logg,r for the log stack obtained by equipping the moduli stack Mg,r

with the log structure determined by the divisors with normal crossingsDg,r.

(iv) The divisor of Cg,r given by the union ofCg,r×_Mg,r Dg,r with the divisor of Cg,r determined by the marked points determines a log structure onCg,r; we denote the resulting log stack by C^logg,r. Thus, we obtain a morphism of log stacks

C^logg,r → M^logg,r

which we refer to as thetautological log curveoverM^logg,r. IfS^logis an arbitrary log scheme, then we shall refer to a morphism

C^log→S^log

whose pull-back to some finite ´etale coveringT →Sis isomorphic to the pull- back of the tautological log curve via some morphism T^{log def}= S^log×S T → M^logg,r as a stable log curve(of type (g, r)). If C → S is smooth, i.e., every geometric fiber of C→S is free of nodes, then we shall refer toC^log →S^log as asmooth log curve(of type (g, r)).

(v) A smooth log curve of type (0,3) will be referred to as a tripod. A vertex of a semi-graph of anabelioids of pro-l PSC-type (cf. [CmbGC], Definition 1.1, (i)) of type (0,3) (cf. [CbTpI], Definition 2.3, (iii)) will also be referred to as a tripod.

Definition 1.4. Let G be a semi-graph of anabelioids of pro-l PSC-type (cf.

[CmbGC], Definition 1.1, (i)) andGthe underlying semi-graph ofG. Write Cusp(G) (resp. Node(G), Vert(G), Edge(G))

for the set of cusps (resp. nodes, vertices, edges) ofGand Cusp(G)^def= Cusp(G), Node(G)^def= Node(G),

Vert(G)^def= Vert(G), Edge(G)^def= Edge(G).

2. Log configuration spaces and log divisors

Let l be a prime number; k an algebraically closed field of characteristic ̸= l;

S^def= Spec(k); (g, r) a pair of nonnegative integers such that 2g−2 +r >0;

X^log→S

(cf. Notation 1.2, (vi)) a smooth log curve of type (g, r); n ∈ Z>0. We suppose that the marked points ofX^log are equipped with an ordering, and that

r >0

(cf. the discussion at the end of the Introduction). In the present§2, we define log configuration spaces, log-full points, and log divisors.

Definition 2.1. The smooth log curveX^log over S determines, up to a choice of ordering of the marked points (which will in factnotaffect the following construc- tion), a classifying morphism S → M^logg,r. Thus, by pulling back the morphism M^logg,r+n → M^logg,r given by forgetting the lastn marked points via this morphism S→ M^logg,r, we obtain a morphism of log schemes

X_n^log →S.

We shall refer toX_n^log as then-th log configuration space associated to X^log →S.

Note thatX₁^log=X^log. WriteX₀^logdef= S.

Definition 2.2. (i) Write

Π_n^def= π₁^pro-l(X_n^log)

for the maximal pro-l quotient of the fundamental group of the log scheme X_n^log (for a suitable choice of basepoint). We refer to [Hsh], Theorems B.1, B.2, for more details on fundamental groups of log schemes.

(ii) Let P be a closed point of Xn. By abuse of notation, we shall use the notation“P” both for the corresponding point of the scheme Xn and for the reduced closed subscheme ofXn determined by this point. Then we shall say that P is a log-full pointofX_n^log if

dim(OXn,P/I(P,MXn)) = 0 (cf. Notation 1.2, (iii)).

(iii) LetP be a log-full point ofX_n^log andP^logthe log scheme obtained by restrict- ing the log structure ofX_n^logto the reduced closed subscheme ofXndetermined byP. Then we obtain an outer homomorphism π1(P^log)→Πn (for suitable choices of basepoints). We shall refer to the subgroup Im(π1(P^log) → Πn), which is well-defined up to Πn-conjugation, as alog-full subgroupatP. (iv) We shall often refer to a point of the scheme Xn as a point of X_n^log. Let

P be a point of X_n^log. Then P parametrizes a pointed stable curve of type (g, r+n) (cf. Definition 2.1). Thus, any geometric point of X_n^log lying over P determines a semi-graph of anabelioids of pro-l PSC-type, which is in fact easily verified to be independent of the choice of geometric point lying over P. We shall writeGP for this semi-graph of anabelioids of pro-l PSC-type.

(v) Let us fix an ordered set Cr,n

def= {c1, . . . , cr, x1

def= cr+1, . . . xn

def= cr+n}.

Thus, by definition, for each point P of X_n^log, we have a natural bijection Cr,n →∼ Cusp(GP). In the following, let us identify the set Cusp(GP) with Cr,n.

(vi) We shall refer to an irreducible divisor of Xn contained in the complement Xn\UX_n of the interior UX_n of X_n^log as a log divisor of X_n^log. That is to say, a log divisor of X_n^log is an irreducible divisor ofXn whose generic point parametrizes a pointed stable curve with precisely two irreducible components (cf. Definition 2.1).

(vii) LetV be a log divisor ofX_n^log. Then we shall writeGV for “GP” in the case where we take “P” to be the generic point ofV.

(viii) For eachi∈ {1, . . . , n}, writepi:X_n^log →X^logfor the projection morphism of co-profile{i}(cf. [MzTa], Definition 2.1, (ii)). Writeι^def= (p1, . . . , pn) :X_n^log→ X^log×S· · · ×SX^log.

Definition 2.3. Let m ≥ 2 and y1, . . . , ym ∈ Cr,n distinct elements such that

♯({y1, . . . , ym} ∩ {c1, . . . , cr})≤1. Then one verifies immediately — by considering clutching morphisms (cf. [Knu], Definition 3.8) — that there exists a unique log divisor V of X_n^log, which we shall denote by V({y₁, . . . , y_m}), that satisfies the following condition: GV has precisely two vertices v₁, v₂ such that v₁ is of type

(0, m+ 1),v2 is of type (g, n+r−m+ 1), andy1, . . . , ym are cusps ofGV|v₁ (cf.

[CbTpI], Definition 2.1, (iii)).

Remark 2.4. LetV be a log divisor ofX_n^log. Then let usobservethat there exists a unique collection of distinct elementsy₁, . . . , y_m∈C_r,nsuch that♯({y₁, . . . , y_m}∩

{c₁, . . . , c_r}) ≤1 and V =V({y₁, . . . , y_m}). (Note that uniqueness holds even in the case whereg= 0 (in which caser≥3), as a consequence of the condition that

♯({y₁, . . . , y_m} ∩ {c₁, . . . , c_r})≤1.) This observation is essentially a special case of Proposition 2.6, (iii), below.

Definition 2.5. LetGbe a semi-graph of anabelioids of pro-lPSC-type andGthe underlying semi-graph ofG. Suppose thatGis a tree. Lete∈Edge(G),v∈Vert(G) be such thateabuts tov. Writebfor the branch ofethat abuts tov. By replacing eby open edges e1, e2 such thate1 abuts tov, and e2 abuts to the vertex̸=v to whicheabuts (resp. e1 abuts tov, ande2 is an edge which abuts to no vertex) if e∈Node(G) (resp.e∈Cusp(G)), we obtain two connected semi-graphs. WriteG̸∋b

for the semi-graph (among these two connected semi-graphs) that does not contain b. Write G∋b for the semi-graph (among these two connected semi-graphs) that containsb. Observe that

• for arbitrarye∈Edge(G),Gdetermines a natural semi-graph of anabelioids of pro-lPSC-typeG_∋bwhose underlying semi-graph may be identified with G_∋b;

• ife∈Node(G), thenGalso determines a natural semi-graph of anabelioids of pro-lPSC-typeG̸∋bwhose underlying semi-graph may be identified with G̸∋b.

Proposition 2.6. LetP be a point ofX_n^log. WriteGfor the underlying semi-graph of GP (cf. Definition 2.2, (iv)). Then the following hold:

(i) Gis a tree.

(ii) Cusp(GP) ={c1, . . . , cr, x1, . . . , xn}.

(iii) There exists a unique vertex v_g ∈Vert(GP) that satisfies the following prop- erties:

(a) The genus ofGP|vg (cf. [CbTpI], Definitions 2.1, (iii); 2.3, (ii)) isg.

(b) Let e ∈Node(GP) that abuts to v_g and b_g the branch of e that abuts to v_g. Then♯(Cusp((G)_∋bg)∩ {c₁, . . . , c_r})≥r−1.

(c) For eachv∈Vert(GP)\ {vg}, the genus ofGP|v is0.

Proof. Assertion (i) follows immediately from the definition of GP. Assertion (ii) follows from Definition 2.2, (v). Finally, we verify assertion (iii). Existence is immediate. Ifg̸= 0, uniqueness is immediate. Ifg= 0, it follows thatr≥3. Now assume that there exists a vertexv^′_g∈Vert(GP) such thatv^′_g̸=vg, andv_g^′ satisfies conditions (a), (b). It follows immediately from the connectedness ofGthat there exists a node e∈Node(GP) such that eabuts to v_g, andv^′_g ∈Vert(G_̸∋bg), where we write bg for the branch of e that abuts to vg. By condition (b) in the case of vg, bg, it holds that ♯(Cusp(G∋bg)∩ {c1, . . . , cr}) ≥ r−1 ≥ 2. On the other hand, it follows immediately from the connectedness ofGthat there exists a node e^′∈Node(GP) such thate^′ abuts tov^′_g, andv_g∈Vert(G̸∋b^′_g), where we writeb^′_gfor the branch ofe^′ that abuts to v_g^′. Next observe that it follows immediately from the fact thatGis a tree thatG∋b_g is a sub-semi-graph ofG̸∋b^′_g, which implies that

♯(Cusp(G∋b^′_g)∩ {c1, . . . , cr})≤r−2. Thus, by condition (b) in the case ofv_g^′,b^′_g,

we obtain a contradiction.

Definition 2.7. LetP be a point of X_n^log. WriteGfor the underlying semi-graph of GP, v_g ∈Vert(GP) for the vertex of Proposition 2.6, (iii). For e∈ Node(GP), writeb_efor the branch ofesuch thatv_g ∈Vert(G∋be). Then we shall write

I_G^def= {Cusp((G)_̸∋be)∩C_r,n|e∈Node(GP)} ⊆2^Cr,n,

where we write 2⁽⁻⁾for the set of subsets of (−). Note that it follows immediately from Proposition 2.6, (iii), that for eachI∈I_G,♯I≥2.

Proposition 2.8. Let P, P^′ be points of X_n^log. Write G, G^′ for the respective underlying semi-graphs ofGP, GP′;vg, v_g^′ for the respective vertices characterized in Proposition 2.6, (iii). IfI_G=I_G′ ⊆2^Cr,n, then there exists a unique isomorphism of semi-graphsG→^∼ G^′ that mapsvg7→v_g^′ and is compatible with the labels of cusps

∈Cr,n. Moreover,♯Vert(G) =♯I_G+ 1,♯Node(G) =♯Node(GP) =♯I_G.

Proof. Let J ∈ I_G. Write J_⊆ ^def= {I ∈ I_G | I ⊆ J ⊆ C_r,n}. Then one verifies immediately that one may construct a (well-defined)semi-graphGJ satisfying the following properties:

(i) The elements of Vert(GJ) are equipped with labels ∈ J_⊆ that determine a bijection Vert(GJ)→^∼ J_⊆.

(ii) Let us call a subset{J1, J2} ⊆J_⊆of cardinality≤2 anadjacent pairofJ_⊆ if J1(J2, and there does not exist an elementI∈I_Gsuch thatJ1(I(J2. For e∈Node(GJ), write Vert(e)⊆Vert(GJ)→^∼ J_⊆ for the subset (of cardinality

≤2) of vertices to whicheabuts. Then the assignment Node(GJ)∋e7→Vert(e)∈2^Vert(GJ)→^∼ 2^J⊆

determines a bijection of Node(GJ) onto the set of adjacent pairs ofJ_⊆. (iii) The cusps ofGJ are equipped with labels∈C_r,nin such a way that, for each

I∈J_⊆, these labels determine a bijection from the set of cusps of the vertex labeled byIonto the subsetI\(∪

I_G∋J^∗(IJ^∗)⊆Cr,n. Moreover, these labels determine a bijection Cusp(GJ)→^∼ J (⊆Cr,n).

Next, one verifies immediately that one may construct a (well-defined)semi-graph GI_G satisfying the following properties:

(I) There exists a unique vertex ofGI_Gequipped with a labelvg. The set of cusps of this vertexvgare equipped with labels∈Cr,n which determine a bijection from the set of cusps of this vertexvg to the subsetCr.n\(∪

I∈I_GI)⊆Cr,n. (II) The semi-graphGI_G is obtained fromv_g(together with its associated cusps) bygluingv_gtoGJ, whereJ ∈I_Granges over the elements ofI_Gthat aremax- imalwith respect to the relation of inclusion, along a node e_J ∈Node(GI_G) that abuts tovg and the vertex ofGJ labeledJ (cf. (i)).

(III) The cusps ofGI_G are equipped with labels∈Cr,n that are compatible with the labels of (I) (in the case of the cusps associated to the vertex labeledvg) and (i) (in the case of the cusps associated to vertices∈Vert(GJ), for J as in (II)). These labels determine a bijection Cusp(GI_G)→^∼ Cr,n.

Then it follows immediately from Definition 2.7 that there exists a unique isomor- phism of semi-graphsG→^∼ GI_G that is compatible with the label “v_g”, as well as

with the labels of cusps∈Cr,n. SinceGis a tree, it follows thatGI_G←∼ Gis also a tree. On the other hand, observe that it follows immediately from the construction ofGI_G (cf. (i), (I), (II)), together with the definition ofI_G(cf. Definition 2.7), that

♯Vert(GI_G) =♯I_G+ 1. SinceGI_Gis a tree, we thus conclude that♯Node(GI_G) =♯I_G. Finally, sinceGI_G iscompletely determinedby the subsetI_G⊆2^Cr,n, the remainder

of Proposition 2.8 follows immediately.

Proposition 2.9. LetP be a point ofX_n^logandI⊆C_r,nsuch that♯(I∩{c₁, . . . , c_r})

≤1. WriteGfor the underlying semi-graph of GP. Then the following conditions are equivalent:

(i) P ∈V(I)(cf. Definition 2.3).

(ii) I∈I_G.

(iii) GV(I) is obtained from GP by generization (with respect to some subset of Node(GP) (cf.[CbTpI], Definition 2.8)).

Proof. The equivalence (i)⇐⇒(iii) follows immediately — by consideringclutching morphisms (cf. [Knu], Definition 3.8) — from the latter portion of Definition 2.2, (vi). The equivalence (ii)⇐⇒(iii) follows immediately from Definition 2.7.

Proposition 2.10. Let m ∈ {1, . . . , n}; V1, . . . , Vm a collection of distinct log divisors ofX_n^log such that V1∩ · · · ∩Vm̸=∅. Then there exist nonnegative integers i0, . . . , im such that

i0+· · ·+im=n−m, and the intersectionV1∩ · · · ∩Vmis isomorphic, over S, to

X_i0×S(M0,i1+3×_Z· · · ×_ZM0,im+3×_ZS).

In particular, the intersection V₁∩ · · · ∩V_m is irreducible of dimensionn−m; if m=n, thenV₁∩ · · · ∩V_n is (the reduced closed subscheme determined by) a log-full point.

Proof. LetP be a generic point ofV1∩· · · ∩Vm. WriteGP for the underlying semi- graph ofGP. Recall from Proposition 2.8 that♯Vert(GP)−1 =♯Node(GP) =♯I_GP. Thus, we conclude from Remark 2.4, together with the equivalence (i)⇐⇒ (ii) of Proposition 2.9, that

♯Node(GP) =♯I_GP =♯{V |V is a log divisor ofX_n^log such thatP ∈V} ≥m.

Since the divisor that determines the log structure ofX_n^log is a divisor with normal crossings, we thus conclude that♯Vert(GP)−1 =♯Node(GP) =m, and hence that

{V |V is a log divisor ofX_n^log such thatP ∈V}={V1, . . . , Vm}. Thus, it follows from Proposition 2.6, (ii), that

♯{branches of edges (i.e., cusps and nodes) ofGP}=n+r+ 2m.

Next, observe that it follows from Proposition 2.6, (iii), that there exists aclutch- ing morphism

ρ_P:X_i0×S(M0,i1+3×Z· · · ×ZM0,im+3×ZS)→X_n

(cf. [Knu], Definition 3.8) such thatP lies in the image of this morphismρP. Since the morphism ρP is a proper monomorphism (cf. Propositions 2.6, (iii); 2.8), it follows that the morphism ρP is a closed immersion. Thus, if we write XP for the scheme-theoretic closure ofP inX_n andX_ρP for the image of ρ_P in X_n, then X_P ⊆X_ρP.

Next, observe that since the sum of the cardinalities of the sets of cusps of the pointed stable curves parametrized by the moduli stack factors of the domain of ρP is equal to

(i0+r) +

∑m

j=1

(ij+ 3), it holds that

(i0+r) +

∑m

j=1

(ij+ 3) =♯{branches of edges ofGP}=n+r+ 2m, and hence that

i₀+

∑m

j=1

i_j =n−m.

Since

dim(Xρ_P) = dim(Xi₀×S(M0,i₁+3×_Z· · · ×_ZM0,i_m+3×_ZS)) =i0+

∑m

j=1

ij, andV1∪ · · · ∪Vmis a divisor with normal crossings (which implies that dim(XP) = dim(V1∩ · · · ∩Vm) =n−m), we thus conclude that

dim(X_ρP) = dim(X_P), and hence thatXρ_P =XP. Moreover, since

{V |V is a log divisor ofX_n^log such thatP ∈V}={V1, . . . , Vm},

we thus conclude from Remark 2.4, together with the equivalence (i) ⇐⇒ (ii) of Proposition 2.9, that{V1, . . . , Vm}determinesI_GP, hence, by Proposition 2.8, that {V1, . . . , Vm}determinesGP. But this implies that every generic point ofV1∩· · ·∩Vm

lies in XρP, for some fixedP, and hence that V1∩ · · · ∩Vm is irreducible. This

completes the proof of Proposition 2.10.

3. Various types of log divisors

We continue with the notation introduced at the beginning of §2. In addition, we suppose that n∈ Z>1. In the present §3, we define various types of log divisors and study their geometry.

Definition 3.1. (i) For positive integers i∈ {1, . . . , n−1}, j ∈ {i+ 1, . . . , n}, write

πi,j:X×S· · · ×SX→X×SX

for the projection of the fiber product ofncopies ofX→S to thei-th andj- th factors. Writeδ^′_i,jfor the inverse image viaπ_i,jof the image of the diagonal embeddingX ,→X×SX. Writeδi,j for the uniquely determined log divisor of X_n^log whose generic point maps to the generic point ofδ_i,j^′ via the natural morphismX_n →X×S· · · ×SX (cf. Definition 2.2, (viii)). We shall refer to the log divisorδi,j as anaive diagonalofX_n^log.

(ii) LetV be a log divisor ofX_n^log. We shall say thatV is atripodal divisorif one of the vertices of GV (cf. Definition 2.2, (vii)) is a tripod (cf. Notation 1.3, (v)).

(iii) Let V be a log divisor ofX_n^log. We shall say thatV is a (g, r)-divisorif one of the vertices ofGV is of type (g, r) (cf. [CbTpI], Definition 2.3, (iii)).

(iv) LetV be a log divisor ofX_n^log. We shall say thatV is adrift diagonalif there exist a naive diagonal δ and an automorphism α of X_n^log over S such that V =α(δ).

Remark 3.2. Recall (cf. [NaTa], Theorem D) that:

• when (g, r) = (0,3) or (1,1), any automorphism ofX_n^log overS necessarily arises as the composite of an automorphism (of X_n^log that arises from an automorphism) ofX^log over S with an automorphism of X_n^log that arises from a permutation of the r+n marked points of the stable log curve X_n+1^log →X_n^log;

• when (g, r) ̸= (0,3),(1,1), any automorphism of X_n^log over S necessarily arises as the composite of an automorphism (of X_n^log that arises from an automorphism) ofX^log over S with an automorphism of X_n^log that arises from a permutation of thenfactors ofX_n^log.

Proposition 3.3. The following hold:

(i) It holds that

{naive diagonals}={V({x_i, x_j})|i∈ {1, . . . , n−1}, j∈ {i+ 1, . . . , n}}

(cf. Definition 2.3).

(ii) If(g, r)̸= (0,3), then

{tripodal divisors}

={V({y1, y2})|y1, y2∈Cr,n are distinct elements,{y1, y2} ̸⊆ {c1, . . . , cr}}

(cf. Definition 2.3).

(iii) If(g, r) = (0,3), then

{tripodal divisors}={V({y1, y2})|Cr,n⊇ {y1, y2} ̸⊆ {c1, c2, c3}}

∪ {V({y1, y2})^def= V(Cr,n\ {y1, y2})| {y1, y2} ⊆ {c1, c2, c3}}

(cf. Definition 2.3).

(iv) Let V be a tripodal divisor and α an automorphism of X_n^log over S. Then α(V)is a tripodal divisor.

Proof. First, assertion (i) follows immediately from the various definitions involved.

Next, assertions (ii), (iii) follow immediately from Remark 2.4, together with the definition of tripodal divisors. Finally, we consider assertion (iv). It follows from Re- mark 3.2 thatαlifts to an automorphism ofX_n+1^log relative to the natural morphism X_n+1^log →X_n^log, hence induces an isomorphism of GV with Gα(V). This completes

the proof of assertion (iv).

Proposition 3.4. The following hold:

(i) It holds that

{naive diagonals} ⊆ {drift diagonals} ⊆ {tripodal divisors} ⊆ {log divisors}. (ii) If(g, r)̸= (0,3),(1,1), then

{naive diagonals}={drift diagonals}.

(iii) If(g, r) = (0,3) or(1,1), then

{drift diagonals}={tripodal divisors}.

Proof. First, we verify assertion (i). The first and third inclusions follow imme- diately from the various definitions involved. The second inclusion follows from Proposition 3.3, (i), (iv). This completes the proof of assertion (i). Assertion (ii) follows immediately from Remark 3.2.

Finally, we consider assertion (iii). Let V be a tripodal divisor. Let us first suppose that (g, r) = (0,3). Then X_n^log is naturally isomorphic to the mod- uli stack (M^log0,n+3)k

def= M^log0,n+3×_ZS over S, on which the symmetric group on n+ 3 letters acts naturally. Moreover, it follows from Proposition 3.3, (iii), that V = V({y₁, y₂}), where y₁, y₂ ∈ C_r,n are distinct elements. Thus, there exists a permutationα∈ S_n+3 such that α(V({x₁, x₂})) =V({y₁, y₂}). Assertion (iii) in the case where (g, r) = (0,3) now follows immediately.

Next, let us suppose that (g, r) = (1,1). ThenX_n^log is naturally isomorphic to the fiber productM^log1,n+1×_M^log

1,1

SoverS, where the arrowS→ M^log1,1is taken to be the classifying morphismS→ M^log1,1determined byX^log(cf. Definition 2.1). Thus, one verifies easily, by considering the automorphisms of an elliptic curve given by translation by a rational point, that the action of the symmetric group on n+ 1 letters on M^log1,n+1 induces an action of the symmetric group on n+ 1 letters on X_n^log. Moreover, it follows from Proposition 3.3, (ii), thatV =V({y1, y2}), where y1, y2∈Cr,n arearbitrary distinct elements (cf. Definition 2.3). Thus, there exists a permutationα∈Sn+1 such thatα(V({x1, x2})) =V({y1, y2}). Assertion (iii) in the case where (g, r) = (1,1) now follows immediately.

Definition 3.5. LetG be a semi-graph of anabelioids of pro-lPSC-type.

(i) We shall say that a vertex ofGis aterminal vertexif precisely one node abuts to it.

(ii) We shall say that a node of G is a terminal node if it abuts to a terminal vertex.

(iii) Write

TerNode(G)⊆Node(G) for the set of terminal nodes ofG.

Proposition 3.6. Let P be a closed point ofX_n^log. Then it holds that P is a log-full point⇐⇒Node(Gp) =n.

Proof. This equivalence follows immediately from Definitions 2.1, 2.2, (ii), together with the well-known modular interpretation of the log moduli stacks that appear in the definition of X_n^log (where we recall that the log structure of this log stack

arises from a divisor with normal crossings).

Proposition 3.7. Let P be a log-full point of X_n^log andAa log-full subgroup atP (cf. Definition 2.2, (iii)). Then the following hold:

(i) It holds that ♯Node(GP) = n. The underlying semi-graph of GP is a tree that has preciselyn+ 1 vertices, one of which is of type (g, r) (cf. [CbTpI], Definition 2.3, (iii)); the other vertices are tripods (cf. Notation 1.3, (v)).

(ii) Write Node(GP) ={e1, . . . , en} (cf. (i)). Then for each i∈ {1, . . . , n}, there exists a unique log divisorVi such that there exists an isomorphism ofGV_i with (GP) Node(GP)\{e_i}(cf.[CbTpI], Definition 2.8) which preserves the respective orderings of cusps. In this situation, we shall say that V_i is the log divisor associated to e_i∈Node(GP).

(iii) In the situation of (ii),

P =V₁∩ · · · ∩V_n andA=I_V1× · · · ×I_Vn,

where I_Vi ⊆ Π_n is a suitable inertia group associated to V_i contained in A.

Moreover, for each i∈ {1, . . . , n}, it holds thatI_Vi≃Zl andA≃Z^⊕lⁿ. (iv) Letm be a positive number; W₁, . . . , W_m distinct log divisors such that P =

W₁∩ · · · ∩W_m. Thenm=n, and{W₁, . . . , W_m}={V₁, . . . , V_n} (cf. (iii)).

Proof. Assertion (i) follows immediately from Propositions 2.6, 2.8, and 3.6, to- gether with the observation that a log-full point (cf. Definition 2.2, (ii)) corre- sponds to an intersection of the sort considered in Proposition 2.10, in the case where n =m, and i_j = 0, forj = 0,1, . . . , m. Assertion (ii) follows immediately from Proposition 2.9. Assertion (iii) follows from Propositions 2.9 and 2.10, and [CbTpI], Lemma 5.4, (ii). Assertion (iv) follows immediately from Propositions 2.8,

2.9, 2.10, together with assertion (iii).

Definition 3.8. Let P be a log-full point of X_n^log and V₁, . . . , V_n the log divisors such that P =V1∩ · · · ∩Vn (cf. Proposition 3.7, (iv)). We shall say thatVi is a terminal divisorofP if there exists a terminal nodee∈TerNode(GP) such thatVi

is the log divisor associated toe∈Node(GP) (cf. Proposition 3.7, (ii)).

Lemma 3.9. LetP be a log-full point ofX_n^log andV₁, . . . , V_n the log divisors such thatP =V₁∩ · · · ∩V_n (cf. Proposition 3.7, (iv)). Then the following conditions are equivalent:

(i) Vi is a terminal divisor of P.

(ii) Vi is a tripodal divisor or a(g, r)-divisor.

Proof. The implication (i) =⇒ (ii) follows from Proposition 3.7, (i), (ii). The implication (ii) =⇒ (i) follows immediately from the various definitions involved.

Theorem 3.10. For ∈ {◦,•}, let l be a prime number; k an algebraically closed field of characteristic̸=l;S ^def= Spec(k);(g, r)a pair of nonnegative integers such that2g−2 +r>0;

X^log→S a smooth log curve of type (g, r);n ∈Z>1;X^log

n the n-th log configuration space associated to X^log → S; Π ^def= π^pro-l₁ (X_n^log) (for a suitable choice of basepoint);

ϕ: Π^{◦ ∼}→Π^•

an isomorphism of profinite groups. Then the following hold:

(i) (g^◦, r^◦, n^◦) = (g^•, r^•, n^•).

(ii) If (g, r) ̸= (0,3),(1,1), then ϕ induces a bijection between the set of fiber subgroups of a given co-length (cf.[MzTa], Definition 2.3, (iii)) ofΠ^◦and the set of fiber subgroups of the same co-length ofΠ^•.

(iii) Suppose that (g, r)̸= (0,3),(1,1). Writeι_Π: Π→Π₁ × · · · ×Π₁ for the outer homomorphism induced byι:X^log

n →X^log×S· · · ×SX^log (cf.

Definition 2.2, (viii)), where Π₁ ^def= π₁^pro-l(X^log) (for a suitable choice of basepoint). Thenϕ induces a commutative diagram

Π^{◦ ϕ} //

ι^◦_Π

Π^•

ι^•_Π

Π^◦₁× · · · ×Π^◦₁ ^∼ //Π^•₁× · · · ×Π^•₁,

where the lower horizontal isomorphism preserves the respective direct product decompositions (but possibly permutes the factors).

Proof. Assertion (i) follows from [HMM], Theorem A, (i). Assertion (ii) follows from [MzTa], Corollary 6.3. Assertion (iii) follows from assertion (ii).

4. Reconstruction of non-degenerate elements of log-full subgroups We continue with the notation of§3. In the present §4, we reconstruct the subset of scheme-theoretically non-degenerate elements (cf. Definition 4.6, (i), below) of a log-full subgroup (cf. Theorem 4.15 below).

Proposition 4.1. Let m < n be an integer, q:X_n^log → X_m^log a projection, V a log divisor of X_n^log. Write q: Πn → Πm for the outer homomorphism induced by q:X_n^log→X_m^log. Suppose thatq(V)(Xm. Then the following hold:

(i) q(V)is a log divisor of X_m^log.

(ii) Let IV ⊆Πn be an inertia group associated to V. Then q(IV)(≃IV) is an inertia group associated toq(V).

Proof. Assertion (i) follows immediately from the latter portion of Definition 2.2, (vi), together with the well-known modular interpretation of the log moduli stacks that appear in the definition ofX_n^logandX_m^log. Assertion (ii) follows from [NodNon], Remark 2.4.2, together with thesurjectivityportion of [NodNon], Lemma 2.7, (ii).

Proposition 4.2. LetP be a log-full point ofX_n^log;V₁, . . . , V_n the log divisors such thatP =V₁∩· · ·∩V_n;A=I_V1×· · ·×I_Vn the log-full subgroup atP (cf. Proposition 3.7, (iii), (iv)). Then the following hold:

(i) There exists a tripodal divisor in {V1, . . . , Vn}. Suppose that V1 is a tripodal divisor. Thus, GV1 has precisely two vertices v1, v^′₁, one of which is a tripod.

Suppose thatv₁ is a tripod.

(ii) If r = 1, then there exists a unique (g, r)-divisor in {V₁, . . . , V_n}. Suppose thatV_n is this unique (g, r)-divisor.

(iii) In the situation of (i), if(g, r)̸= (0,3), then there exists ani₀∈ {1+r, . . . , n+

r}such thatc_i0 is a cusp ofGV1|v1 (cf.[CbTpI], Definition 2.1, (iii)). In this case, write p: X_n^log → X_n^log₋₁ for the projection morphism of profile {i0−r} (cf.[MzTa], Definition 2.1, (ii)).

(iv) In the situation of (i), if(g, r) = (0,3), then there exists ani0∈ {1, . . . ,3 +n} such that c_i0 is a cusp ofGV1|v1. In this case, writep:X_n^log→X_n^log₋₁ for the