Math. J. Okayama Univ. 51 (2009), 1–26

ON THE FUNDAMENTAL GROUPS OF LOG
CONFIGURATION SCHEMESYuichiro HOSHI
Abstract. In the present paper, we study the cuspidalization problem
for the fundamental group of a curve by means of the log geometry of
the log configuration scheme, which is a natural compactification of theusual configuration space of the curve. The goal of this paper is to showthat the fundamental group of the configuration space is generated by
the images from morphisms from a group extension of the fundamental groups of the configuration spaces of lower dimension, and that thefundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamental groups ofthe configuration spaces of lower dimension.

1. Introduction
In this paper, we study the cuspidalization problem for the fundamentalgroups of curves. This problem may be formulated as follows: Let X be asmooth, proper, geometrically connected curve of genus g ≥ 2 over a fieldK, and r a natural number. Then the r-th configuration space
U(r)
r

z
}|
{
of X is, by definition, the complement in the r-th product X ×K · · · ×K X
rz}|{of the diagonal divisors {(x1 , · · · , xr ) ∈ X ×K · · · ×K X |xi = xj }, where1 ≤ i < j ≤ r; moreover, we have projections
U(r+1) −→ U(r)obtained by forgetting the i-th factor, where 1 ≤ i ≤ r + 1. Roughly
speaking, the cuspidalization problem (due to Mochizuki [cf. [11]]) refers
the problem of “reconstructing” group-theoreticallythe “output data” consisting of the fundamental groups
π1 (U(r) )and morphisms· · · −→ π1 (U(r+1) ) −→ π1 (U(r) ) −→ π1 (U(r−1) ) −→ · · ·· · · −→ π1 (X) = π1 (U(1) )Mathematics Subject Classification. Primary 14H30; Secondary 14H10.
Key words and phrases. cuspidalization, log configuration scheme.
1

2

YUICHIRO HOSHI

induced by the above projections
fromthe “input data” consisting of the morphism
π1 (X) −→ π1 (Spec K) ≃ Gal(K sep /K) ,where K sep is a separable closure of K, induced by the structure morphism of X.
The purpose of this paper is to give an approach to this problem from theviewpoint of log geometry, i.e., we shall make use of the geometry of the log
configuration scheme associated to X
log
X(r)

obtained by equipping a certain canonical compactification X(r) of U(r) withdef

the log structure defined by the divisor with normal crossings D(r) = X(r) \U(r) ⊆ X(r) . Let Mg,r be the moduli stack of r-pointed stable curves of genus
g over K whose r marked points are equipped with an ordering, Mg,r ⊆log
Mg,r the open substack of Mg,r parametrizing smooth curves, and Mg,r thelog stack obtained by equipping Mg,r with the log structure associated tothe divisor with normal crossings Mg,r \ Mg,r ⊆ Mg,r . Then the log scheme
log
log
X(r)
arises as the fiber product of the morphism of stacks Spec K → Mg,0
log

log

determined by the curve X and the morphism of stacks Mg,r → Mg,0obtained by forgetting the marked points. Note that for a set of prime
numbers Σ, it follows from the log purity theorem that if p 6∈ Σ, where p islog
the characteristic of K, then the strict open immersion U(r) ֒→ X(r)induces
an isomorphism of the geometrically pro-Σ fundamental group of U(r) withlog
the geometrically pro-Σ log fundamental group of X(r).
The divisor at infinity D(r) ⊆ X(r) admits a decomposition[
D(r) =
D(r)I
I⊆{1,2,··· ,r};I ♯ ≥2

by considering the configurations of the r marked points. Our first result is
as follows (cf. Theorem 4.1; Remark following Lemma 4.2):
Theorem 1.1. Let r ≥ 3 be an integer, and i, j, k ∈ {1, 2, · · · , r} distinct
elements of {1, 2, · · · , r}. Then the following hold:loglog(i) The images of the log fundamental groups π1 (D(r){i,j}), π1 (D(r){j,k}
),
log
log
log
and π1 (D(r){i,j,k}) in π1 (X(r)) topologically generate π1 (X(r)).

LOG CONFIGURATION SCHEMES

3(ii) If K is of characteristic 0, then there exist exact sequences
log
log
sep
b
1 −→ Z(1)(K) −→ π1 (D(r){i,j}) −→ π1 (X(r−1)) −→ 1 ;log
sepb
1 −→ Z(1)(K) −→ π1 (D(r){i,j,k})log−→ π1 (X(r−2)) ×GK π1 (P1K \ {0, 1, ∞}) −→ 1 .

This result can be regarded as a logarithmic analogue of [8], Remark 1.2.
Note that even if K is of characteristic p > 0, by replacing “log fundamental group” by “geometrically pro-l log fundamental group”, where l is aprime number such that l 6= p, one can obtain a similar result of (ii) in thestatement of the above theorem (cf. Remark following Lemma 4.2).
Our second result (cf. Theorem 5.2) asserts that one can “reconstruct”
partially the log fundamental groups of the log configuration schemes ofhigher dimension from a collection of data concerning the log fundamentalgroups of the log configuration schemes of lower dimension by means of Theorem 1.1; more precisely, one can construct group-theoretically a profinite
loggroup “π1 (X(r+1))G ” and projections
log
log
qi : π1 (X(r+1))G −→ π1 (X(r)),

where 1 ≤ i ≤ r + 1, in such a way thatlog
the morphism qi factors through the projection π1 (X(r+1)
)→
log
log
log
π1 (X(r)) induced by the projection X(r+1)→ X(r)
obtainedby forgetting the i-th factor, and, moreover, the first arrowof the factorization
log
log
log
π1 (X(r+1))G −→ π1 (X(r+1)) −→ π1 (X(r))

of qi is surjectivefrom a collection of data concerning the log fundamental groups of the log
log
configuration schemes X(k), where 0 ≤ k ≤ r. Here, we use the terminology“reconstruct” as a sort of “abbreviation” for the somewhat lengthy but
mathematically precise formulation given in the statement of Theorem 5.2.
By this second result, if one can also reconstruct group-theoretically thelog
log
kernel of the surjection π1 (X(r+1))G → π1 (X(r+1)) which appears as thefirst arrow in the above factorization of qi , then by taking the quotient bylog
this kernel, one can reconstruct the desired profinite group π1 (X(r+1)) (cf.
Proposition 5.1). However, unfortunately, no reconstruction of this kernel
is performed in this paper. Moreover, it seems to the author that if such areconstruction should prove to be possible, it is likely that the method ofreconstruction of this kernel should depend on the arithmetic of K in an

4

YUICHIRO HOSHI

essential way. On the other hand, in a subsequent paper [4], we obtain asolution of a pro-l version of the cuspidalization problem over a finite fieldby means of the results obtained in this paper.
This paper is organized as follows:
In Section 2, we define the log configuration schemes of curves and consider the scheme-theoretic and log scheme-theoretic properties of log configuration schemes. In Section 3, we study the geometry of the divisors at
infinity of log configuration schemes. In Section 4, we study properties ofthe log fundamental groups of log configuration schemes and their divisors
at infinity by means of the results obtained in Sections 2 and 3. In Section 5,
we consider a partial reconstruction of the log fundamental groups of higher
dimensional log configuration schemes as discussed above.

Notations and Terminologies
Groups:Let G be a profinite group, Σ a non-empty set of prime numbers, and n
an integer. We shall say that n is a Σ-integer if the prime divisors of n arein Σ. We shall refer to the quotientlim G/H←−of G, where the projective limit is over all open normal subgroups H ⊆ Gsuch that the index [G : H] of H is a Σ-integer, as the maximal pro-Σ
quotient of G. We shall denote by G(Σ) the maximal pro-Σ quotient of G.
Log schemes:
For a log scheme X log , we shall denote by X (respectively, MX ) theunderlying scheme (respectively, the sheaf of monoids that defines the log
structure) of X log . For a morphism f log of log schemes, we shall denote by
f the underlying morphism of schemes.
Let P be a property of schemes [for example, “quasi-compact”, “connected”, “normal”, “regular”] (respectively, morphisms of schemes [for example, “proper”, “finite”, “étale”, “smooth”]). Then we shall say that a log
scheme (respectively, a morphism of log schemes) satisfies P if the underlying scheme (respectively, the underlying morphism of schemes) satisfies
P.
For fs log schemes X log , Y log , and Z log , we shall denote by X log ×Y log
log
Z
the fiber product of X log and Z log over Y log in the category of fs log
schemes. In general, the underlying scheme of X log ×Y log Z log is not naturally
isomorphic to X ×Y Z. However, since strictness (note that a morphism
f log : X log → Y log is called strict if the induced morphism f ∗ MY → MX
on X is an isomorphism) is stable under base-change in the category ofLOG CONFIGURATION SCHEMES

5

arbitrary log schemes, if X log → Y log is strict, then the underlying schemeof X log ×Y log Z log is naturally isomorphic to X ×Y Z. Note that since thenatural morphism from the saturation of a fine log scheme to the original
fine log scheme is finite, properness and finiteness are stable under fs basechange.
If there exist both schemes and log schemes in a commutative diagram,then we regard each scheme in the diagram as the log scheme obtained byequipping the scheme with the trivial log structure.
We shall refer to the largest open subset (possibly empty) of the underlying scheme of an fs log scheme on which the log structure is trivial as theinterior of the fs log scheme.
Let X log and Y log be log schemes, and f log : X log → Y log a morphismof log schemes. Then we shall refer to the quotient of MX by the image ofthe morphism f ∗ MY → MX induced by f log as the relative characteristic
sheaf of f log . Moreover, we shall refer to the relative characteristic sheaf of∗ ֒→ Mthe morphism X log → X induced by the natural inclusion OX
X aslog
the characteristic sheaf of X .
Fundamental groups:
For a connected scheme X (respectively, log scheme X log ) equipped witha geometric point x → X (respectively, log geometric point x
elog → X log ), weshall denote by π1 (X, x) (respectively, π1 (X log , x
elog )) the fundamental grouplogof X (respectively, log fundamental group of X ). Since one knows that thefundamental group is determined up to inner automorphisms independentlyof the choice of base-point, we shall often omit the base-point, i.e., we shalloften denote by π1 (X) (respectively, π1 (X log )) the fundamental group of X
(respectively, log fundamental group of X log ).
For a set Σ of prime numbers and a scheme X (respectively, log schemelogX ) which is of finite type and geometrically connected over a field K,
we shall refer to the quotient of π1 (X) (respectively, π1 (X log )) by theclosed normal subgroup obtained as the kernel of the natural projection from
π1 (X ⊗K K sep ) (respectively, π1 (X log ⊗K K sep )), where K sep is a separable
closure of K, to its maximal pro-Σ quotient π1 (X ⊗K K sep )(Σ) (respectively,
π1 (X log ⊗K K sep )(Σ) ) as the geometrically pro-Σ fundamental group of X
(respectively, geometrically pro-Σ log fundamental group of X log ). Thus,
the geometrically pro-Σ fundamental group π1 (X)(Σ) of X (respectively,
geometrically pro-Σ log fundamental group π1 (X log )(Σ) of X log ) fits into thefollowing exact sequence:
1 −→ π1 (X ⊗K K sep )(Σ) −→ π1 (X)(Σ) −→ Gal(K sep /K) −→ 1
(respectively,

6

YUICHIRO HOSHI

1 −→ π1 (X log ⊗K K sep )(Σ) −→ π1 (X log )(Σ) −→ Gal(K sep /K) −→ 1).
2. Log configuration schemes
In this Section, we define the log configuration scheme of a curve over a
field and consider the geometry of such log configuration schemes.
Throughout this Section, we shall denote by X a smooth, proper, geometrically connected curve of genus g ≥ 2 over a field K whose (not necessarily
positive) characteristic we denote by p.
Let Mg,r be the moduli stack of r-pointed stable curves of genus g over
K whose r marked points are equipped with an ordering, and Mg,r ⊆ Mg,rthe open substack of Mg,r parametrizing smooth curves (cf. [7]). Let us
def

def

write Mg = Mg,0 and Mg = Mg,0 . For 1 ≤ i ≤ r + 1, we shall denote by pM(r)i : Mg,r+1 → Mg,r the morphism of stacks obtained by forgetting the i-th marked point. Then by considering the morphism of stacks
pM(r)r+1 : Mg,r+1 → Mg,r , we obtain a natural isomorphism of Mg,r+1 withthe universal r-pointed stable curve over Mg,r (cf. [7], Corollary 2.6). Now
the complement Mg,r \ Mg,r is a divisor with normal crossings in Mg,rlog
(cf. [7], Theorem 2.7). Let us denote by Mg,r the fs log stack obtained byequipping Mg,r with the log structure associated to the divisor with normal
crossings Mg,r \ Mg,r . Then since a natural action of the symmetric groupon r letters Sr on Mg,r given by permuting the marked points preserves thelogdivisor Mg,r \ Mg,r , the action of Sr on Mg,r extends to an action on Mg,r .
log
First, we define the log configuration scheme X(r)as follows:
Definition 1. Let r be a natural number. Then we shall define X(r) as thefiber product of the classifying morphism of stacks Spec K → Mg determined by the curve X → Spec K and the morphism of stacks Mg,r → Mg
obtained by forgetting the marked points. Since Mg,r → Mg is reprelog
sentable, X(r) is a scheme. We shall denote by X(r)the fs log scheme
obtained by equipping X(r) with the log structure induced by the log struclog

log, and by DX(r)ture of Mg,r . We shall denote by UX(r) the interior of X(r)the reduced closed subscheme of X(r) obtained as the complement of UX(r)in X(r) . Note that by definition, the scheme UX(r) is naturally isomorphic
to the usual r-th configuration space of X, and the action of Sr on Mg,rlog
log). For(respectively, Mg,r ) determines an action on X(r) (respectively, X(r)simplicity, we shall write U(r) (respectively, D(r) ) instead of UX(r) (respectively, DX(r) ) when there is no danger of confusion.

LOG CONFIGURATION SCHEMES

7

As is well-known, the pull-back of the divisor Mg,r \ Mg,r via pM(r)r+1 :Mg,r+1 → Mg,r is a subdivisor of the divisor Mg,r+1 \ Mg,r+1 (cf. [7], theproof of Theorem 2.7). Thus, there exists a unique morphism of log stacks
log
log
log
M
pM
(r)r+1 : Mg,r+1 → Mg,r whose underlying morphism of stacks is p(r)r+1 .
Moreover, for an integer 1 ≤ i ≤ r, since pM(r)i is obtained as the compositeof the automorphism of Mg,r+1 determined by the action of an element ofM
Sr and pM(r)r+1 , the morphism of stacks p(r)i also extends to a morphism of
log

log

log stacks Mg,r+1 → Mg,r . We shall denote this morphism of log stacks bylogpM(r)i .
Definition 2. Let r be a natural number.
(i) Let 1 ≤ i ≤ r + 1 be an integer. Then pM(r)i : Mg,r+1 → Mg,r
log

log

log
(respectively, pM: Mg,r+1 → Mg,r ) determines a morphism
(r)i
log
log
X(r+1) → X(r) (respectively, X(r+1)→ X(r)). We shall denote thismorphism by pX(r) i (respectively, plog
X(r) i ). Note that by the definitionof stable curves, pX(r) i is proper, flat, geometrically connected, andgeometrically reduced. For simplicity, we shall write p(r)i (respeclog
tively, plog(r)i ) instead of pX(r) i (respectively, pX(r) i ) when there is nodanger of confusion.
(ii) Let I = {i1 , i2 , · · · , iI ♯ }, where i1 < i2 < · · · < iI ♯ , be a non-empty
subset of {1, 2, · · · , r}. Then we shall denote by
log
log
prlog
X(r) I : X(r) −→ X(r−I ♯ )the morphism obtained as the compactification of the projection
UX(r) → UX(r−I ♯ ) given by mapping (x1 , · · · , xr ) to (xi1 , · · · , xiI ♯ ),i.e., if the morphism UX(r) → UX(r−I ♯ ) which induces a morphism
UX(r) (S)−→ UX(r−I ♯ ) (S)(x1 , · · · , xr ) 7→ (xi1 , · · · , xiI ♯ )for any scheme S over K, where xi is an S-valued point of X, isgiven as the compsite
pUX

j

♯
(r−I ♯ ) r−I

◦ pUX

j

♯
(r−I ♯ +1) r−I +1◦ · · · ◦ pUX(r−2) jr−2 ◦ pUX(r−1) jr−1 ,where pUX j is the projection UX(i+1) → UX(i) obtained by forgetting(i)the j-th factor, then we shall write
def

log
prlog
X(r) I = pXj(r−I ♯ ) r−I ♯◦ plogX

j(r−I ♯ +1) r−I ♯ +1log
◦ · · · ◦ plog
X(r−2) jr−2 ◦ pX(r−1) jr−1 .

8

YUICHIRO HOSHI

Note that it is immediate that this composite only depends on I,
i.e., this composite is independent of the choice of the sequence(jr−I ♯ , · · · , jr−1 ). For simplicity, we shall write prlog(r)I instead of
prlog
X(r) I when there is no danger of confusion.
Next, let us consider the scheme-theoretic and log scheme-theoretic proplog
erties of X(r)in more detail.
Proposition 2.1. Let r be a natural number, 1 ≤ i ≤ r + 1 an integer, andI a non-empty subset of {1, 2, · · · , r}.
(i) The scheme X(r) is connected.
log(ii) The morphism plog(r)i is log smooth. In particular, pr(r)I is also log
smooth.
log(iii) The log scheme X(r)is log regular.
log(iv) The scheme X(r) is regular, and the log structure of X(r)is given by
a divisor with normal crossings.

Proof. Assertion (i) follows from the fact that X(0) = Spec K is connected,
together with the fact that the p(r)i ’s are proper and geometrically connected.
Next, we prove assertion (ii). The assertion for plog(r)r+1 follows from the
log

log

log
fact that pM(r)r+1 : Mg,r+1 → Mg,r is log smooth (cf. [5], Section 4). Sincelog
plog(r)i is a composite of an automorphism of X(r) obtained by permuting oflog
the marked points and plog(r)r+1 , the morphism p(r)i is also log smooth.
Assertion (iii) follows from assertion (ii), together with the log regularity
logof the log scheme X(0)= Spec K.
log

log
Finally, we prove assertion (iv). Since the natural morphism X(r)→ Mg,r∗is strict, for any geometric point x → X(r) , the stalk (MX(r) /OX(r) )x of thelogcharacteristic sheaf of X(r)at x → X(r) is isomorphic to N⊕n for somenatural number n. Thus, assertion (iv) follows from assertion (iii).
Proposition 2.2. Let r be a natural number, 1 ≤ i ≤ r + 1 an integer,log
a strict geometric point, i.e., a strict morphism whose
and xlog → X(r)underlying morphism of schemes is a geometric point (cf. [3], Definition
1.1, (i)). Then the following sequence is exact:
via prlog
log) −→1 π1 (X(r+1))
lim π1 (X(r+1)
×X log xlog
λ
←−
(r)

via plog(r)i

log−→ π1 (X(r)) −→ 1 .

LOG CONFIGURATION SCHEMES

9log
Here, the projective limit is over all reduced covering points xlog
λ → x ,
i.e., a morphism obtained as the composite of a connected Kummer finite log′ log whoseétale covering (x′λ )log → xlog and the strict morphism xlog
λ → (xλ )defunderlying morphism of schemes is the closed immersion xλ = (x′λ )red ֒→ x′λ
defined by the ideal generated by the nilpotent elements of Ox′λ (cf. [3],Definition 1.1, (ii)).
Proof. This follows immediately from Proposition 2.1; [3], Theorem 2.3.



3. Divisors at infinity of log configuration schemes
We continue with the notation of the preceding Section. In this Section,
we consider the scheme-theoretic and log scheme-theoretic properties of thedivisors defining the log structures of the log configuration schemes.
Definition 3. Let r ≥ 2 be an integer, and I a subset of {1, 2, · · · , r}of cardinality I # ≥ 2 equipped with the natural ordering. Then we shalldenote by(C(r)I −→ X(r−I # +1) ×K M0,I # +1 )the r-pointed stable curve of genus g obtained by applying the clutchingmorphism of stacks (cf. [7], Definition 3.8)
βg,0,{1,2,··· ,r}\I,I : Mg,r−I # +1 ×K M0,I # +1 → Mg,r ,where {1, 2, · · · , r} \ I is equipped with the natural ordering, to the (r −I # + 1)-pointed stable curve of genus gX(r−I # +2) ×K M0,I # +1 −→ X(r−I # +1) ×K M0,I # +1
obtained by base-changing pX(r−I # +1)

r−I # +2 : X(r−I # +2) → X(r−I # +1) andthe (I # + 1)-pointed stable curve of genus 0X(r−I # +1) ×K M0,I # +2 −→ X(r−I # +1) ×K M0,I # +1
obtained by base-changing the universal curve M0,I # +2 → M0,I # +1 overM0,I # +1 . Note that the clutching locus ofX(r−I # +2) ×K M0,I # +1 −→ X(r−I # +1) ×K M0,I # +1
(respectively, X(r−I # +1) ×K M0,I # +2 −→ X(r−I # +1) ×K M0,I # +1 )is the (r − I # + 1)-st (respectively, (I # + 1)-st) section (cf. [7], Definition
3.8).
Then it is immediate that the classifying morphism of stacks X(r−I # +1) ×K
M0,I # +1 → Mg,r determined by the r-pointed stable curve of genus g(C(r)I −→ X(r−I # +1) ×K M0,I # +1 )

10

YUICHIRO HOSHI

factors through X(r) , and the resulting morphism X(r−I # +1) ×K M0,I # +1 →X(r) is a closed immersion since it is a proper monomorphism. We shalldenote by δX(r) I this closed immersion, by DX(r) I the scheme-theoretic imagelog
of δX(r) I , by DX
the log scheme obtained by equipping DX(r) I with the
(r) I
log
log
log
log structure induced by the log structure of X(r), and by δX
: DX
→
(r) I
(r) I
log
X(r)
the strict closed immersion whose underlying morphism of schemes islog
δX(r) I . For simplicity, we shall write D(r)I (respectively, D(r)I; respectively,log
log
δ(r)I ; respectively, δ(r)I) instead of DX(r) I (respectively, DX; respectively,
(r) I
log
δX(r) I ; respectively, δX) when there is no danger of confusion.
(r) I

Proposition 3.1. Let r ≥ 2; 1 ≤ i ≤ r + 1 be integers, and I a subset of{1, 2, · · · , r} of cardinality I # ≥ 2.
(i) The scheme D(r)I is irreducible.
S(ii) The divisor D(r) of X(r) is J D(r)J , where J ranges over the subsetsof {1, 2, · · · , r} of cardinality ≥ 2.
(iii) The closed subscheme of X(r+1) determined by the composite of thenatural closed immersions defined in Definition 3
X(r−I # +1) ×K M0,I # +2 ֒→ C(r)I ֒→ X(r+1)(respectively, X(r−I # +2) ×K M0,I # +1 ֒→ C(r)I ֒→ X(r+1) )is D(r+1)I∪{r+1} (respectively, D(r+1)I ).
(iv) Let J be a subset of {1, 2, · · · , r} of cardinality ≥ 2. Then D(r)I ∩D(r)J 6= ∅ if and only if I ⊆ J, J ⊆ I, or I ∩ J = ∅.
(v) The inverse image of D(r)I ⊆ X(r) via p(r)i is D(r+1)(I∪{r+1})δi ∪D(r+1)I δi , whereδi = ((1, 2, · · · , r + 1) 7→ (1, 2, · · · , i − 1, i + 1, · · · , r, r + 1, i)) ∈ Sr+1 ,and I δi = {δi (k) | k ∈ I}.
Proof. First, we prove assertion (i). The log smoothness of the morphism
log
log
log
log
plog
(s)s+1 : X(s+1) → X(s) and the morphism M0,t+4 → M0,t+3 obtained byforgetting the (t+4)-th marked point, where s, t are natural numbers, imply
log
log
the log regularity of X(r−I# +1) ×K M0,I # +1 ; therefore, D(r)I is normal (cf.
[6], Theorem 4.1). Moreover, by a similar argument to the argument usedin the proof of Proposition 2.1, (i), D(r)I is connected. Thus, in light of thenormality just observed, D(r) is irreducible.
Assertions (ii), (iii), (iv), and (v) follow from the construction of D(r)I . LOG CONFIGURATION SCHEMES

11

Proposition 3.2. Let r ≥ 2 be an integer, and I a subset of {1, 2, · · · , r}log
of cardinality I ♯ ≥ 2. Then the morphism D(r)I→ D(r)I = X(r−I ♯ +1) ×K∗
֒→ MD(r)I determines a
M0,I ♯ +1 induced by the natural inclusion OD(r)Imorphism
log

log
log
log
νX
: D(r)I−→ X(r−I♯ +1) ×K M0,I ♯ +1(r) I

which is of type N, i.e., the underlying morphism of schemes is an isomorphism, and the relative characteristic sheaf is locally constant with stalk
isomorphic to N (cf. [3], Definition 4.1). Moreover, let I [i] be the uniquesubset of {1, 2, · · · , r − 1} such that for j ∈ {1, 2, · · · , r − 1}, j ∈ I [i] if andonly ifj∈I(if j < i)j + 1 ∈ I (if j ≥ i) .
Then the following hold:(i) If i ∈ I, then (I [i] )♯ = I ♯ − 1, and the diagram
log
log
X(r−I
♯ +1) ×K M0,I ♯ +1



y

log

log
X(r−I
♯ +1) ×K M0,I ♯

log
νX

(r) I

←−−−−

log
D(r)I




via plog(r−1)i y

log
δ(r)I

−−−−→

log
X(r)

plog
y (r−1)iloglog←−−−−−−− D(r−1)I−−−−−→ X(r−1)
[i] −
ν log

X(r−1) I [i]δlog(r−1)I [i]commutes, where if I = {i1 , i2 , · · · , iI ♯ }, i1 < i2 < · · · < iI ♯ , andi = ij , then the left-hand vertical arrow is the morphism induced
log
log
by the morphism M0,I ♯ +1 → M0,I ♯ obtained by forgetting the j-thmarked point.
(ii) If i 6∈ I, then (I [i] )♯ = I ♯ , and the diagram
log
log
X(r−I
♯ +1) ×K M0,I ♯ +1



y

log

log
X(r−I
♯ ) ×K M0,I ♯ +1

log
νX

(r) I

←−−−−

log
D(r)I




via plog(r−1)i y

log
δ(r)I

−−−−→

log
X(r)

plog
y (r−1)iloglog←−−−−−−− D(r−1)I−−−−−→ X(r−1)
[i] −
ν log

X(r−1) I [i]δlog(r−1)I [i]commutes, where if {1, 2, · · · , r} \ I = {i1 , i2 , · · · , ir−I ♯ }, i1 < i2 <· · · < ir−I ♯ , and i = ij , then the left-hand vertical arrow is thelog
logmorphism induced by plog
: X(r−I♯ +1) → X(r−I ♯ ) .
(r−I ♯ )j12

YUICHIRO HOSHI

log
Proof. By the definition of D(r)I, the log scheme obtained by forgetting thelog
portion of the log structure of D(r)I
defined by the divisor D(r)I of X(r) islog

log
isomorphic to X(r−I♯ +1) ×K M0,I ♯ +1 ; moreover, it follows from [3], Lemma
log
4.12, (i), together with the definition of the log structure of D(r)I, thatlog∗
the morphism D(r)I֒→→ D(r)I induced by the natural inclusion OD(r)IMD(r)I factors through the morphism obtained by forgetting the portion oflog
the log structure of D(r)I
defined by the divisor D(r)I of X(r) . Thus, we
log

log
log
obtain a morphism D(r)I→ X(r−I♯ +1) ×K M0,I ♯ +1 . On the other hand, byconstruction, it is a morphism of type N.
The assertion that the diagrams in the statement of Proposition 3.2 comlog
mute follows from Proposition 3.1, (v), together with the definitions of D(r)I
log
and νX.
(r) IDefinition 4. Let r ≥ 2 be an integer, and I a subset of {1, 2, · · · , r} of
cardinality I ♯ ≥ 2. Then we shall denote by νX(r) I the underlying morphism
logof schemes of the morphism νX, and by UX(r) I the open subscheme of(r) IDX(r) I determined by the open immersion
−1
νX

(r) I

∼

UX(r−I ♯ +1) ×K M0,I ♯ +1 ֒→ X(r−I ♯ +1) ×K M0,I ♯ +1 → DX(r) I .
log
For simplicity, we shall write ν(r)I(respectively, ν(r)I ; respectively, U(r)I )
log
instead of νX(respectively, νX(r) I ; respectively, UX(r) I ) when there is no(r) I
danger of confusion.

4. Fundamental groups of the log configuration schemes
We continue with the notation of the preceding Section. In this Section,
we study fundamental facts concerning the log fundamental groups of log
configuration schemes and their divisors at infinity. Let Σ be a non-empty
set of prime numbers. We shall fix a separable closure K sep of K and denoteby GK the absolute Galois group Gal(K sep /K) of K. Moreover, we shallsep ).
bdenote by Λ the maximal pro-Σ quotient of Z(1)(KDefinition 5. Let r be a natural number. Then we shall denote by Πlog
X(r)(respectively, Πlog
Mr+3,K ) the geometrically pro-Σ log fundamental group of
log

def

log
X(r)
(respectively, the log scheme M0,r+3 ) and write ΠX = Πlog
X(1) , andLOG CONFIGURATION SCHEMES

13

def

log
Πlog
PK = ΠM4,K . Moreover, if r ≥ 2, then for a subset I of {1, 2, · · · , r} of

cardinality ≥ 2, we shall denote by Πlog
X(r) I the geometrically pro-Σ log funlog
damental group of DX. For simplicity, we shall write Πlog(r) (respectively,
(r) I
log
log
log
Πlog
Mr+3 ; respectively, ΠP ; respectively, Π(r)I ) instead of ΠX(r) (respectively,
log
log
Πlog
Mr+3,K ; respectively, ΠPK ; respectively, ΠX(r) I ) when there is no danger ofconfusion.

First, we show the following theorem.
Theorem 4.1. Let X be a smooth, proper, geometrically connected curve of
genus ≥ 2 over a field K, r ≥ 3 an integer, and i, j, k ∈ {1, 2, · · · , r} distinct
elements of {1, 2, · · · , r}. Then, for any non-empty set of prime numbers
log
Σ, the geometrically pro-Σ log fundamental group of X(r)is topologicallygenerated by the respective images of the geometrically pro-Σ log fundamental
log
log
log
groups of D(r){i,j}, D(r){j,k}, and D(r){i,j,k}.
Proof. It is immediate that it is enough to show the assertion in the casewhere Σ is the set of all prime numbers; thus, assume that Σ is the set of
all prime numbers. Moreover, it is immediate that it is enough to show theassertion in the case where {i, j, k} = {1, 2, 3}; thus, assume that {i, j, k} ={1, 2, 3}.
log
δ(r){2,3}plog
log
log (r−1)3
log
It follows that the composite D(r){2,3} → X(r) → X(r−1)
coincides
logwith ν(r){2,3}; in particular, this composite is of type N (cf. Proposition

the composite Πlog(r){2,3}log
via δ(r){2,3}

Πlog
(r)

via plog(r−1)33.2). Therefore,
→
→
Πlog
(r−1)
is surjective (cf. [3], Lemma 4.5). Thus, it is enough to show that thelog
log
respective images of Πlog(r){1,2} and Π(r){1,2,3} in Π(r) topologically generlog
log
ate the kernel of the morphism Πlog(r) → Π(r−1) induced by p(r−1)3 . Let
log
log
xlog → X(r−1)
be a strict geometric point of X(r−1)
whose image lies inU(r−1){1,2} . Then it follows from Proposition 2.2 that the kernel of the morlog
log
phism Πlog(r) → Π(r−1) induced by p(r−1)3 is included in the image of the
log
log
log
natural morphism π1 (X(r)xlog ) → Π(r) , where X(r)xlog is the fs log schemelog
log
log → X log .
obtained as the fiber product of plog(r−1)(r−1)3 : X(r) → X(r−1) and x
On the other hand, it follows from the definition of the open subscheme
U(r−1){1,2} ⊆ D(r−1){1,2} , together with Proposition 3.1, (v), that the log
schemes obtained by equipping the irreducible components of X(r)xlog withlog
the log structures induced by the log structure of X(r)x
log are the log schemes

14

YUICHIRO HOSHI

obtained as the respective fiber products of the diagramslog
D(r){1,2}via plogy(r−1)3 ;log
xlog −−−−→ X(r−1)log
D(r){1,2,3}
via plogy(r−1)3log
xlog −−−−→ X(r−1).

Therefore, the assertion follows from the logarithmic analogue of [13], Example 5.5 (cf. also [13], §5.5).
Remark. Theorem 4.1 can be regarded as a logarithmic analogue of [8],Remark 1.2.
Definition 6. We shall say that a set of prime numbers Σ is innocuous inK if
the set of all prime numbers or {l} if p = 0Σ={l}if p ≥ 2 ,where l is a prime number which is invertible in K.
In the rest of this paper, we assume that
Σ is innocuous in K.
Lemma 4.2. Let r, r ′ be natural numbers, and I a subset of {1, 2, · · · r} of
cardinality I ♯ ≥ 2.
log

log(i) The natural morphism U(r) ×K M0,r′ +3 → X(r)×K Mr′ +3 induces∼log
an isomorphism π1 (U(r) ×K M0,r′ +3 )(Σ) → Πlog(r) ×GK ΠM ′ , wherer +3“π1 (−)(Σ) ” is the geometrically pro-Σ fundamental group of “(−)”.
log
In particular, the natural morphism U(r) → X(r)(respectively,log
log
U(r)I → X(r−I♯ +1) ×K M0,I ♯ +1 ) induces an isomorphism∼

π1 (U(r) )(Σ) → Πlog
(r)
∼

×
Πlog
(respectively, π1 (U(r)I )(Σ) → Πlog
M ♯ ).
(r−I ♯ +1) GKI +1(ii) Let 1 ≤ i ≤ r + 1 (respectively, 1 ≤ i ≤ r + 4) be an integer,
and x → U(r) (respectively, x → M0,r+3 ) a geometric point of U(r)(respectively, M0,r+3 ). Then the sequence
via prlog
log
1 −→ π1 (X(r+1)
×X log x)(Σ) −→1 Πlog(r+1) −→ Π(r) −→ 1(r)(respectively,log

1 −→ π1 (M0,r+4 ×Mlog

0,r+3

via prlogx)(Σ) −→1 Πlog
Mr+4 −→ ΠMr+3 −→ 1 )LOG CONFIGURATION SCHEMES

15

is exact, where the third arrow is the morphism induced by the morlog
log
phism plog(r)i (respectively, M0,r+4 → M0,r+3 obtained by forgetting
the i-th marked point).
(iii) For a profinite group Γ (respectively, a scheme S), we shall denote byS(Γ) (respectively, Sét ) the classifying site of Γ, i.e., the site definedby considering the category of finite sets equipped with a continuous
action of Γ (respectively, the étale site of S). Then we have naturalmorphisms of siteslog(U(r) ×K M0,r′ +3 )ét −→ S(π1 (U(r) ×K M0,r′ +3 )(Σ) ) ←− S(Πlog(r) ×GK ΠMr‘+3 ) .
log
Let A be a finite Πlog(r) ×GK ΠMr ′ +3 -module whose order is a Σ-integer,
and n an integer. Then the morphismsnHét(U(r) ×K M0,r′ +3 , FA ) ←− H n (π1 (U(r) ×K M0,r′ +3 )(Σ) , A)log−→ H n (Πlog(r) ×GK ΠM ′ , A)r +3

induced by the above morphisms of sites are isomorphisms, where
FA is the locally constant sheaf on U(r) ×K M0,r′ +3 determined byA.
Proof. Assertion (i) follows immediately from the log purity theorem (cf.
[9], Theorem 3.3, also [3], Remark 1.10), together with [3], Proposition 2.4,(ii).
Next, we prove assertion (ii). To prove assertion (ii), by base-changing,
we may assume that K is a separably closed field. Moreover, if Σ is theset of all prime numbers, then this follows from [8], Lemma 2.4, together
with assertion (i). Thus, we may assume that Σ = {l} for a prime number
l which is invertible in K. Then, to prove assertion (ii), it follows from [8],Lemma 3.1, (i), the assertion in the case where Σ is the set of all prime
numbers, together with [1], Proposition 3, that it is enough to show thatthe representationlog
π1 (U(r) ) −→ Aut((π1 (X(r+1)
×X log x)(Σ) )ab )
(r)

log

log

(respectively, π1 (M0,r+3 ) −→ Aut((π1 (M0,r+4 ×Mlog

0,r+3x)(Σ) )ab ))determined by the exact sequence appearing in the statement of assertion(ii) in the case where Σ is the set of all prime numbers factors through alog
pro-Σ quotient of π1 (U(r) ) (respectively, π1 (M0,r+3 )). On the other hand,
it is immediate that this representation is trivial. This completes the proofof assertion (ii).
Finally, we prove assertion (iii). The assertion that the second morphism
in question is an isomorphism follows immediately from assertion (i); thus,

16

YUICHIRO HOSHI

we prove the assertion that the first morphism in question is an isomorphism. We prove this assertion by induction on r + r ′ . If r + r ′ = 0,then the assertion is well-known. If r 6= 0, then by considering the par(pr ,pr )1
3tial compactification U(r−1) ×K X ×K M0,r′ +3 →U(r−1) ×K M0,r′ +3 of(p(r−1)r , id)|U(r) ×K M0,r′ +3 : U(r) ×K M0,r′ +3 → U(r−1) ×K M0,r′ +3 , it follows
from [2], Corollary 10.3, that the sheaf

def

F ′ = Rq ((p(r−1)r , id)|U(r) ×K M0,r′ +3 )∗ FA
is locally constant and constructible; moreover, it follows from [2], Theolog′
rem 7.3, that the Πlog(r−1) ×GK ΠMr ′ +3 -module Fx is naturally isomorphic
to H q (U, FA |U ), where x → U(r−1) ×K M0,r′ +3 is a geometric point ofdef

U(r−1) ×K M0,r′ +3 , and U = U(r) ×U(r−1) x. On the other hand, the natdefn (U, F | ) is an isomorphism, where π =ural morphism H n (π, A) → HétA U
(Σ)π1 (U ) . Indeed, one then verifies immediately that it is enough to verify
that every étale cohomology class of U with coefficients in FA |U vanishes
upon pull-back to some connected finite étale Σ-covering V → U . Moreover, by passing to an appropriate V , we may assume that FA |U is trivial.
Then the vanishing assertion in question is immediate (respectively, a tautology) for n = 0 (respectively, n = 1). Moreover, the vanishing assertionin question is immediate for n ≥ 3 by [2], Theorem 9.1. If U is affine, thenn (U, F | ) vanishes for n = 2 (cf. [2], Theorem 9.1), the assertion
since Hét
A U
is immediate. If U is proper, then it is enough to take V → U so that thedegree of V → U annihilates A (cf. e.g., the discussion at the bottom of [2],p. 136).
Therefore, by considering the Hochschild-Serre spectral sequence (cf. [12],Theorem 2.1.5) associated to the exact sequence

1 −→ π −→ Πlog(r) ×GK ΠMr‘+3,id)
via (plog(r−1)r−→Πlog(r−1) ×GK ΠMr‘+3 −→ 1

obtained by assertion (ii) and the Leray spectral sequence associated tothe morphism (p(r−1)r , id)|U(r) ×K M0,r′ +3 , it follows that it is enough to show
the assertion in the case where the pair of natural numbers “(r, r ′ )” in thestatement of assertion (iii) is (r − 1, r ′ ). Moreover, if r ′ 6= 0, then by asimilar argument to the above argument, it is verified that it is enough toshow the assertion in the case where the pair of natural numbers “(r, r ′ )”in the statement of assertion (iii) is (r, r ′ − 1). This completes the proof ofassertion (iii).
Remark. Let r ≥ 2 be an integer, and I a subset of {1, 2, · · · , r} of cardinality I ♯ ≥ 2. Then by Lemma 4.2, (i), (iii), it follows from Proposition

LOG CONFIGURATION SCHEMES

17

3.2, together with similar arguments to the arguments used in the proves
log
of [10], Lemmas 4.3; 4.4, and [3], Proposition 4.22, that the morphism ν(r)I
determines an exact sequence
1 −→ Λ −→ Πlog
(r)I

log
via ν(r)I−→ Πlog×Πlog(r−I ♯ +1) GK M ♯I +1−→ 1 .

Lemma 4.3. Let r ≥ 2; 1 ≤ i ≤ r be integers, I a subset of {1, 2, · · · , r}of cardinality I ♯ ≥ 2, and I [i] the subset of {1, 2, · · · , r − 1} defined in thestatement of Proposition 3.2. Then the following hold:(i) If i ∈ I, then the diagram
Πlog
(r)I



via plog(r−1)i ylog
via ν(r)I−−−−−→

Πlog
×Πlog
MI ♯ +1(r−I ♯ +1) GKy−−−−−−−−→ Πlog
Πlog
×
Πlog
M ♯
(r−I ♯ +1) GK(r−1)I [i]via ν logI(r−1)I [i]is cartesian, where if I = {i1 , i2 , · · · , iI ♯ }, i1 < i2 < · · · < iI ♯ , andi = ij , then the right-hand vertical arrow is the morphism induced
log
log
by the morphism M0,I ♯ +1 → M0,I ♯ obtained by forgetting the j-thmarked point.
(ii) If i 6∈ I, then the diagram
Πlog
(r)I



via plog(r−1)i ylog
via ν(r)I−−−−−→−−−−−−−−→
Πlog(r−1)I [i]
via ν

log

Πlog
×
Πlog
MI ♯ +1(r−I ♯ +1) GK


y
Πlog
×
Πlog
M ♯
(r−I ♯ ) GK

I +1

(r−1)I [i]is cartesian, where if {1, 2, · · · , r} \ I = {i1 , i2 , · · · , ir−I ♯ }, i1 < i2 <· · · < ir−I ♯ , and i = ij , then the right-hand vertical arrow is thelog
logmorphism induced by plog
: X(r−I♯ +1) → X(r−I ♯ ) .
(r−I ♯ )j
Proof. This follows from Proposition 3.2; Remark following Lemma 4.2, tolog
log
gether with the fact that the restriction of D(r)I→ D(r−1)I[i] to the genericlog
point of D(r)I
is strict.



18

YUICHIRO HOSHI

Lemma 4.4.
(i) Let r ≥ 2 be an integer, and I = {i, i + 1}, where i = 1, 2. Then thefollowing diagram is cartesian:via prlog
(r)I

Πlog
(r)I

−−−−−−→



logvia ν(r)IyΠlog(r−1) −−−−−−−−→
via prlog(r−1){i}Πlog(2){1,2}via ν logy(2){1,2}
ΠX .

(ii) Let r ≥ 3 be an integer. Then the following diagram is cartesian:via prlog(r){1,2}

Πlog(r){1,2,3}−−−−−−−−→



logvia ν(r){1,2,3}
y

log
Πlog
(r−2) ×GK ΠPΠlog−−−−−−−−→(r−2) −−−−−→
pr1

via prlog(r−2){1}Πlog(2){1,2}via ν logy(2){1,2}
ΠX .

Proof. This follows from Lemma 4.3 by induction on r.

Definition 7.
(i) Let r ≥ 2 be an integer, and I = {i, i + 1}, where i = 1, 2. Then weshall writedeflog
ΠGX(r) I = Πlog
X(r−1) ×ΠX ΠX {1,2} ,(2)where the morphism implicit in the fiber product Πlog
X(r−1) → ΠX (relog
spectively, Πlog
X {1,2} → ΠX ) is the morphism induced by pr(r−1){i}(2)log(respectively, νX{1,2} ). On the other hand, by Lemma 4.4, (i), the
(2)

log
log
morphism Πlog
X(r) I → ΠX(r−1) induced by νX(r) I and the morphism
log
log
Πlog
X(r) I → ΠX(2) {1,2} induced by prX(r) I induce an isomorphism∼

ΠGX(r) I −→ Πlog
X(r) I .
We shall denote this isomorphism by αlog
X(r) I .
(ii) Let r ≥ 3 be an integer. Then we shall write
def loglog= ΠPK ×GK Πlog
X(r−2) ×ΠX ΠX(2) {1,2} ,{1,2,3}(r)

ΠGX

where the morphism implicit in the fiber product Πlog
X(r−2) → ΠX (respectively, Πlog
→ ΠX ) is the morphism induced by prlog
X {1,2}(r−2){1}(2)LOG CONFIGURATION SCHEMES

19

log(respectively, νX{1,2} ). On the other hand, by Lemma 4.4, (ii), the
(2)

log
log
log
morphism Πlog
X(r) {1,2,3} → ΠX(r−2) ×GK ΠPK induced by νX(r) {1,2,3}log
log
and the morphism Πlog
X(r) {1,2,3} → ΠX(2) {1,2} induced by prX(r) {1,2}induce an isomorphismΠGX∼(r) {1,2,3}−→ Πlog
X {1,2,3} .
(r)

We shall denote this isomorphism by αlog
X {1,2,3} .
(r)

5. Partial reconstruction of the fundamental groups ofhigher dimensional log configuration schemes
We continue with the notation of the preceding Section; in particular, we
assume that the set of prime numbers Σ is innocuous in K. In this Section,
we show that the fundamental group of the configuration space can be partially reconstructed from a collection of data concerning the fundamentalgroups of the configuration spaces of lower dimension.
Definition 8. Let r ≥ 3 be an integer. Then we shall denote by ΠGX(r) the
profinite group obtained as the free profinite product of
log
log
Πlog
X {1,2} ; ΠX {2,3} ; ΠX {1,2,3} .
(r)

(r)

(r)

We shall denote byfX(r) : ΠGX(r) −→ Πlog
X(r)log
the morphism determined by the morphisms DX
(r) I

log
δX

(r) I

log
→ X(r)
where I ={1, 2}, {2, 3}, and {1, 2, 3}. Note that it follows from Theorem 4.1 that fX(r)is surjective. Let I = {1, 2}, {2, 3}, or {1, 2, 3}. Then by the definition ofΠGX(r) , we have a natural morphismGΠlog
X(r) I −→ ΠX(r) .

We shall denote this morphism by δXG(r) I .
Next, we define the collection of data used in the partial reconstruction of
the log fundamental groups of higher dimensional log configuration schemes
performed in Theorem 5.2 below.
Definition 9. Let r ≥ 2 be an integer.
(i) We shall denote by DX (Σ), or DX(1) (Σ), the collection of data consisting of

20

YUICHIRO HOSHI(i-1) the profinite groups
log
log
Πlog
X(2) , ΠX {1,2} , ΠX , GK , ΠPK ;(2)(i-2) the morphisms
via plog
X

(1) i

Πlog
X(2)

−→

ΠX (i = 1, 2) ,and the morphisms induced by the respective structure morphisms
ΠX −→ GK ; Πlog
PK −→ GK ;(i-3) the morphismΠlog
X(2) {1,2}log
via δX(2) {1,2}−→

Πlog
X(2) .

(ii) We shall denote by DX(r) (Σ) the collection of data consisting of(ii-1) the profinite groups
log
log
Πlog
X(k) (1 ≤ k ≤ r + 1), ΠX {1,2} , GK , ΠPK ;(2)(ii-2) the morphisms
Πlog
X(k)

via plogX(k−1) i−→Πlog
X(k−1) (2 ≤ k ≤ r + 1, 1 ≤ i ≤ k),Πlog
X(2) {1,2}log
via νX(2) {1,2}

−→

ΠX ,

and the morphisms induced by the respective structure morphisms
ΠX −→ GK ; Πlog
PK −→ GK ;(ii-3) the composites
αlogX(r+1) IΠGX(r+1) I

∼

−→

Πlog
X(r+1) I

log
via δX(r+1) I−→

Πlog
X(r+1) ,where I = {1, 2}, {2, 3}, and {1, 2, 3}.
G(iii) We shall denote by DX(Σ) the collection of data consisting of(r)G(iii-1) the data obtained by replacing Πlog
X(r+1) in (ii-1) by ΠX(r+1) ;LOG CONFIGURATION SCHEMES(iii-2) the data obtained by replacing Πlog
X(r+1)
r + 1) in (ii-2) by ΠGX(r+1)
r + 1);(iii-3) the dataobtainedfX(r+1)→by21

via plogXΠlog
X(r+1)→(r) i

via plogX→Πlog
X(r) (1 ≤ i ≤

(r) i

Πlog
X(r) (1 ≤ i ≤Πlog
X(r+1)

replacing(respectively,

log“via δX”) in (ii-3) by ΠGX(r+1) (respectively, “via δXG(r+1) I ”).
(r+1) I

In the following, let Y be a smooth, proper, geometrically connected curveof genus ≥ 2 over a field L, and ΣY a non-empty set of prime numbers whichis innocuous in L. We shall fix a separable closure Lsep of L and denote by
GL the absolute Galois group Gal(Lsep /L) of L.
Definition 10. Let r ≥ 2 be an integer.
(i) We shall refer to isomorphisms
∼

∼

log
log
log
Πlog
X(k) −→ ΠY(k) (k = 1, 2) ; ΠX {1,2} −→ ΠY {1,2} ;
(2)

∼

(2)

∼

log
GK −→ GL ; Πlog
PK −→ ΠPL

which are compatible with the morphisms given in the definitions ofDX (Σ) and DY (ΣY ) as an isomorphism of DX (Σ) with DY (ΣY ).
(ii) We shall refer to isomorphisms
∼

∼

log
log
log
Πlog
X(k) −→ ΠY(k) (1 ≤ k ≤ r + 1) ; ΠX {1,2} −→ ΠY {1,2} ;
(2)

∼

(2)

∼

log
GK −→ GL ; Πlog
PK −→ ΠPL

which are compatible with the morphisms given in the definitions ofDX(r) (Σ) and DY(r) (ΣY ) as an isomorphism of DX(r) (Σ) with
DY(r) (ΣY ).
(iii) We shall refer to isomorphisms
∼

∼

∼

∼

log
ΠGX(r+1) −→ ΠGY(r+1) ; Πlog
X(k) −→ ΠY(k) (1 ≤ k ≤ r) ;
∼

log
log
log
Πlog
X {1,2} −→ ΠY {1,2} ; GK −→ GL ; ΠPK −→ ΠPL
(2)

(2)

which are compatible with the morphisms given in the definitions ofGG(Σ) with(Σ) and DYG(r) (ΣY ) as an isomorphism of DX
DX
(r)
(r)
DYG(r) (ΣY ).

∼GG(Σ) with(Σ) → DYG(r) (ΣY ) be an isomorphism of DX(iv) Let φG(r) : DX
(r)
(r)
∼

DYG(r) (ΣY ). Then by forgetting the isomorphism ΠGX(r+1) → ΠGY(r+1)
in φG(r) , we obtain an isomorphism of DX(r−1) (Σ) with DY(r−1) (ΣY ).

22

YUICHIRO HOSHI

We shall denote this isomorphism by F−1 (φG(r) ). Note that it is immediate that the correspondenceφG(r) 7→ F−1 (φG(r) )is functorial.
Remark. If there exists an isomorphism of DX (Σ) (respectively, DX(r) (Σ);
G
respectively, DX(Σ)) with DY (ΣY ) (respectively, DY(r) (ΣY ); respectively,(r)
DYG(r) (ΣY )), then Σ = ΣY . Indeed, this follows from the fact that the abelianization of the kernel of ΠX → GK (respectively, ΠY → GL ) is a non-trivial
module which is free over Z(Σ) (respectively, Z(ΣY ) ).
Proposition 5.1. Let r ≥ 2 be an integer, Σ a set of prime numbers that∼G(Σ) → DYG(r) (Σ) an isomorphism.
is innocuous in K and L, and φG(r) : DX(r)∼

Then if the isomorphism ΠGX(r+1) → ΠGY(r+1) in φG(r) induces an isomorphismof the kernel of the morphism fX(r+1) : ΠGX(r+1) → Πlog
X(r+1) with the kernel ofthe morphism fY(r+1) : ΠGY(r+1) → Πlog
Y(r+1) , then there exists an isomorphism∼

F (φG(r) ) : DX(r) (Σ) → DY(r) (Σ). Moreover, the correspondenceφG(r) 7→ F (φG(r) )is functorial.
Proof. Since the morphism fX(r+1) : ΠGX(r+1) → Πlog
X(r+1) (respectively, fY(r+1) :ΠGY(r+1) → Πlog
Y(r+1) ) is surjective (cf. Theorem 4.1), by the assumption,∼log
we obtain an isomorphism φ : Πlog
X(r+1) → ΠY(r+1) induced by the isomor∼

phism ΠGX(r+1) → ΠGY(r+1) in φG(r) . Therefore, by replacing the isomorphism∼

ΠGX(r+1) → ΠGY(r+1) in φG(r) by φ, we obtain an isomorphism F (φG(r) ) of thedesired type.

Theorem 5.2. Let r ≥ 2 be an integer, Σ a set of prime numbers which∼is innocuous in K and L, and φ(r−1) : DX(r−1) (Σ) → DY(r−1) (Σ) an iso∼Gmorphism. Then there exists an isomorphism F+1 (φ(r−1) ) : DX
(Σ)
→
(r)
DYG(r) (Σ) such that
F−1 (F+1 (φ(r−1) )) = φ(r−1) .
Moreover, the correspondenceφ(r−1) 7→ F+1 (φ(r−1) )is functorial.

LOG CONFIGURATION SCHEMES

23

Proof. First, we define an isomorphism φG{1,2} (respectively, φG{2,3} ; reG
spectively, φG{1,2,3} ) of ΠGX{1,2} , (respectively, ΠX{2,3} ; respectively,
(r+1)

ΠGX

(r+1)

ΠGY

G
{1,2,3} ) with ΠY(r+1)(r+1) {1,2,3}

(r+1)

G
{1,2} , (respectively, ΠY(r+1) {2,3}; respectively,) as follows:∼(i) We define an isomorphism φG{1,2} : ΠGX(r+1) {1,2}→ ΠGY(r+1) {1,2}∼isomorphism induced by the isomorphisms ΠlogX(r)∼as theΠlog
Y(r) ,

→

∼

log
Πlog
X {1,2} → ΠY {1,2} , and ΠX → ΠY in φ(r−1) .
(2)

(2)

∼

(ii) We define an isomorphism φG{2,3} : ΠGX(r+1) {2,3}→ ΠGY(r+1) {2,3}∼isomorphism induced by the isomorphisms ΠlogX(r)∼as theΠlog
Y(r) ,

→

∼

log
Πlog
X {1,2} → ΠY {1,2} , and ΠX → ΠY in φ(r−1) .
(2)

(2)

∼

(iii) We define an isomorphism φG{1,2,3} : ΠGX(r+1) {1,2,3}→ ΠGY

as the isomorphism induced by the isomorphisms∼∼∼(r+1) {1,2,3}

Πlog
PK

∼

→ Πlog
PL ,
∼

log
log
log
Πlog
X(r−1) → ΠY(r−1) , ΠX(2) {1,2} → ΠY(2) {1,2} , GK → GL , and ΠX →
ΠY in φ(r−1) .

Then these isomorphisms φG{1,2} , φG{2,3} , and φG{1,2,3} induce an isomorphismof the profinite group ΠGX(r+1) obtained as the free profinite product ofΠGX(r+1) {1,2}

; ΠGX

; ΠGX

(r+1) {2,3}(r+1) {1,2,3}(cf. Definitions 7; 8) with the profinite group ΠGY(r+1) obtained as the freeprofinite product ofΠGY(r+1) {1,2}; ΠGY(r+1) {2,3}; ΠGY(r+1) {1,2,3}.

Now we denote this isomorphism by φG .
On the other hand, for an integer 1 ≤ i ≤ r + 1, we define a projection
qX(r) i : ΠGX(r+1) → Πlog
X(r) as follows:{1,2}(i) If i = 1 or 2, then we define a morphism qX(r) i : ΠGX(r+1) {1,2}

=

log
log
Πlog
X(r) ×ΠX ΠX {1,2} → ΠX(r) as the first projection. If i ≥ 3, then
(2)

{1,2}

we define a morphism qX(r) i : ΠGX(r+1) {1,2}via plogX(r−1) i−2

ΠGX(r+1) {1,2}

×id

−→

D

→ ΠlogX(r) as the compositelog
X(2) {1,2}

ΠGX(r) {1,2}

24

YUICHIRO HOSHI
αlog
X(r) {1,2}−→

Πlog
X(r) {1,2}log
via δX(r) {1,2}−→

Πlog
X(r) .

{2,3}(ii) We define a morphism qX(r) 1 : ΠGX→ ΠlogX(r) as the mor-phism obtained by replacing ΠGX(respectively, plog
X(r−1) i−2 )(r+1) {2,3}(r+1) {1,2}{1,2}by ΠGX(r+1) {2,3}(respectively, plog
X(r−1) 1 ) in the definition of “qX(r) i for{2,3}

i ≥ 3”. If i = 2 or 3, then we define a morphism qX(r) i : ΠGX=(r+1) {2,3}

log
log
Πlog
X(r) ×ΠX ΠX {1,2} → ΠX(r) as the first projection. If i ≥ 4, then
(2)

{2,3}

(r+1) {2,3}→ ΠlogX(r) as the morphismby ΠGXin the definition of

we define a morphism qX(r) i : ΠGXobtained by replacing ΠGX(r+1) {1,2}(r+1) {2,3}{1,2}“qX(r) i for i ≥ 3”.
{1,2,3}(iii) If i = 1, 2, or 3, then we define a morphism qX(r) i : ΠGX(r+1) {1,2,3}→

Πlog
X(r) as the composite
αlog
X

pr2,3

ΠGX

(r+1)

(r){1,2}

G{1,2,3} −→ ΠX(r) {1,2}−→

Πlog
X {1,2}log
via δX(r) {1,2}

−→

(r)

Πlog
X(r) .

{1,2,3}→ Πlog
If i ≥ 4, then we define a morphism qX(r) i : ΠGX
X(r)(r+1) {1,2,3}
as the composite
via id log ×plog
X
P
K

ΠGX
(r+1) {1,2,3}αlog
X(r) {1,2,3}−→(r−2) i−3

×id

D

−→

Πlog
X(r) {1,2,3}

log
X(2) {1,2}

ΠGX(r) {1,2,3}

log
via δX(r) {1,2,3}

−→

Πlog
X(r) .

Moreover, by a similar procedure to the procedure that we applied to define
{1,2}
{2,3}
{1,2,3}the morphism qX(r) i (respectively, qX(r) i ; respectively, qX(r) i ), we define a{1,2,3}

{2,3}

{1,2}

morphism qY(r) i (respectively, qY(r) i ; respectively, qY(r) i ).
{1,2}

{2,3}

{1,2,3}

{1,2}

{2,3}

Then these morphisms qX(r) i , qX(r) i , and qX(r) i (respectively, qY(r) i , qY(r) i ,{1,2,3}G
and qY(r) i ) induce a morphism ΠGX(r+1) → Πlog
X(r) (respectively, ΠY(r+1) →Πlog
Y(r) ). We denote this morphism by qX(r) i (respectively, qY(r) i ).
Then by Lemma 4.4, together with constructions, for any 1 ≤ i ≤ r+1, themorphism qX(r) i (respectively, qY(r) i ) factors as the composite ΠGX(r+1)Πlog
X(r+1)

via plog
X

→

(r) i

Πlog
X(r)

(respectively, ΠGY(r+1)

fY(r+1)→

Πlog
Y(r+1)

via plog
Y
i
(r)

→

fX(r+1)→Πlog
Y(r) );

LOG CONFIGURATION SCHEMES

25

moreover, the diagram
ΠGX(r+1)yqX(r) i 

Πlog
X(r)

φG

∼

−−−−→ ΠGX(r+1)
q Y i
y (r)
∼

−−−−→

Πlog
Y(r)

commutes, where the bottom horizontal arrow is the isomorphism in φ(r−1) .
Therefore, by equipping φ(r−1) with the isomorphism φG , we obtain an isoGmorphism F+1 (φ(r−1) ) of DX(Σ) with DYG(r) (Σ) of the desired type.
(r)
Acknowledgement
I would like to thank Professor Shinichi Mochizuki for suggesting the
topics, helpful discussions, warm encouragements, and valuable advices. I
would also like to thank the referee for useful suggestions and comments.

References[1] M. P. Anderson, Exactness properties of profinite completion functors, Topology 13(1974), 229-239.
[2] E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture, Springer-Verlag(1988).
[3] Y. Hoshi, The exactness of the log homotopy sequence, RIMS Preprint 1558 (2006).
[4] Y. Hoshi, Absolute anabelian cuspidalizations of configuration spaces over finite fields,
RIMS Preprint 1592 (2007).
[5] F. Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math.
11 (2000), 215-232.
[6] K. Kato, Toric singularities, Amer. J. Math. 116 (1994), 1073-1099.
[7] F. F. Knudsen, The projectivity of the moduli space of stable curves II, Math. Scand.
52 (1983), 161-199.
[8] M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine.
Angew. Math. 474 (1996), 169-219.
[9] S. Mochizuki, Extending families of curves over log regular schemes, J. Reine. Angew.
Math. 511 (1999), 43-71.
[10] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois
Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ. 41 (2003), 119-165.
[11] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J.
Math. Kyoto Univ. 47 (2007), no. 3, 451-539.
[12] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields,
Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag (2000).
[13] J. Stix, A general Seifert-Van Kampen theorem for algebraic fundamental groups,
Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, 763-786.

26

YUICHIRO HOSHI

Yuichiro Hoshi
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
JAPAN
e-mail address: yuichiro@kurims.kyoto-u.ac.jp(Received June 21, 2007 )(Revised April 25, 2008 )