Profinite Groups:

On monodromically full points of configuration spaces of hyperbolic curves

Yuichiro Hoshi

AbstractIn the present paper, we introduce and discuss the notion ofmonodromi- cally full pointsof configuration spaces of hyperbolic curves. This notion leads to complements to M. Matsumoto’s result concerning the difference between the ker- nels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to theautomorphismandouter automorphismgroups of the geometric fundamental group of the hyperbolic curve. More concretely, we prove that any hyperbolic curve over a number field hasmany“nonexceptional” closed points, i.e., closed points which donotsatisfy a condition considered by Matsumoto, but that there exist infinitely many hyperbolic curves which admitmany“excep- tional” closed points, i.e., closed points which do satisfy the condition considered by Matsumoto. Moreover, we prove a Galois-theoretic characterization of equiva- lence classes of monodromically full points of configuration spaces, as well as a Galois-theoretic characterization of equivalence classes of quasi-monodromically full points of cores. In a similar vein, we also prove a necessary and sufficient con- dition for quasi-monodromically full Galois sections of hyperbolic curves to be ge- ometric.

Key words: monodromically full point, hyperbolic curve, configuration space, Galois-theoretic characterization, number field

Introduction

In the present§, letlbe a prime number,ka field ofcharacteristic0,kan algebraic closure ofk, andXa hyperbolic curve of type(g,r)overk. Moreover, for an alge-

Yuichiro Hoshi

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN, e-mail:

yuichiro@kurims.kyoto-u.ac.jp

1

braic extensionk⁰⊆kofk, writeGk⁰def

=Gal(k/k⁰)for the absolute Galois group ofk⁰ determined by the given algebraic closurek. In the present paper, we introduce and discuss the notion ofmonodromically full pointsof configuration spaces of hyper- bolic curves. The term “monodromically full” is a term introduced by the author in [9], but the corresponding notion was studied by M. Matsumoto and A. Tamagawa in [12]. If, for a positive integern, we writeXnfor then-th configuration space of the hyperbolic curveX/k, then the natural projectionXn+1→Xnto the firstnfactors may be regarded as afamily of hyperbolic curvesof type(g,r+n). In the present paper, we shall say that a closed pointx∈Xnof then-th configuration spaceXnis l-monodromically fullif thek(x)-rational point, where we writek(x)for the residue field atx, ofXn⊗kk(x)determined byxis anl-monodromically full point with re- spect to the family of hyperbolic curvesXn+1⊗_kk(x)overXn⊗_kk(x)in the sense of [9], Definition 2.1, i.e., roughly speaking, the image of the pro-louter monodromy representation ofπ1(Xn⊗_kk)with respect to the family of hyperbolic curvesXn+1

overXniscontainedin the image of the pro-louter Galois representation ofGk(x)

with respect to the hyperbolic curveXn+1×X_nSpeck(x)overk(x). (See Definition 3 for the precise definition of the notion ofl-monodromically full points — cf. also Remark 5.)

By considering the notion of monodromically full points, one can give some complements to Matsumoto’s result obtained in [13] concerning the difference be- tween the kernels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to theautomorphismandouter automorphism groups of the geometric fundamental group of the hyperbolic curve. To state these complements, let us review the result given in [13]: Write∆_X/k^{l} for the geometric pro-lfundamental group ofX— i.e., the maximal pro-lquotient of the ´etale funda- mental groupπ1(X⊗_kk)ofX⊗_kk— andΠ_X/k^{l} for the geometrically pro-lfunda- mental group ofX— i.e., the quotient of the ´etale fundamental groupπ1(X)ofXby the kernel of the natural surjectionπ1(X⊗kk)∆_X/k^{l}. Then since the closed sub- group∆_X/k^{l} ⊆Π_X/k^{l} isnormalinΠ_X/k^{l}, conjugation by elements ofΠ_X/k^{l} determines a commutative diagram of profinite groups

1 −−−−→ ∆_X/k^{l} −−−−→ Π_X/k^{l} −−−−→ Gk −−−−→ 1



y ^ρeX/k^{l}

 y

 y^ρX/k^{l}

1 −−−−→ Inn(∆_X/k^{l}) −−−−→ Aut(∆_X/k^{l}) −−−−→ Out(∆_X/k^{l}) −−−−→ 1

— where the horizontal sequences are exact, and the left-hand vertical arrow is, in fact, anisomorphism. On the other hand, ifx∈X is a closed point of X, then we have a homomorphismπ1(x):Gk(x)→Π_X/k^{l} induced byx∈X (which is well- defined up toΠ_X/k^{l}-conjugation). In [13], Matsumoto studied the difference between the kernels of the following two homomorphisms:

ρ_X/k^{l}|G_k(x):G_k(x)−→Out(∆_X/k^{l});

G_k(x)^π−→^1(x)Π_X/k^{{l} ρe}

{l}

−→X/kAut(∆_X/k^{l}).

Now we shall say thatE(X,x,l)holdsif the kernels of the above two homomor- phisms coincide and write

X^El⊆X^cl

for the subset of the setX^clof closed points ofX consisting of “exceptional”x∈X^cl such that E(X,x,l)holds (cf. [13], §1,§3, as well as Definition 4 in the present paper). Then the main result of [13] may be stated as follows:

Let g≥3be an integer. Suppose that ldivides2g−2; write l^νfor the highest power of l that divides2g−2. Then there areinfinitely manyisomorphism classes of pairs(K,C) ofnumber fieldsK andproper hyperbolic curvesC of genus g over K which satisfy the following condition: For any closed point x∈C of C with residue field k(x), if l^νdoesnot divide[k(x):k], then E(C,x,l)doesnot hold.

In the present paper, we prove that if a closed pointx∈X of the hyperbolic curveX isl-monodromically full, thenE(X,x,l)doesnot hold(cf. Proposition 11, (ii)). On the other hand, as a consequence of Hilbert’s irreducibility theorem, any hyperbolic curve over a number field hasmany l-monodromically full points (cf.

Proposition 2, as well as, [12], Theorem 1.2, or [9], Theorem 2.3). By applying these observations, one can prove the following result, which may be regarded as a partial generalizationof the above theorem due to Matsumoto (cf. Theorem 1):

Theorem A (Existence of many nonexceptional closed points).Let l be a prime number, k anumber field(i.e., a finite extension ofQ), and X ahyperbolic curve over k. Then if we naturally regard the set X^clof closed points of X as a subset of X(C), then the complement

X^cl\X^El⊆X(C)

isdensewith respect to thecomplex topologyof X(C). Moreover, the intersection X(k)∩X^El⊆X(k)

isfinite.

On the other hand, in [13],§2, Matsumoto proved that for any prime numberl, the triple

(P¹_Q\ {0,1,∞},01,^→ l)

— where01 is a^→ Q-rational tangential base point — is a triple for which “E(X,x,l)”

holds. As mentioned in [13],§2, the fact that “E(X,x,l)” holds for this triple was observed by P. Deligne and Y. Ihara. However, by definition, in fact, a tangential base point is not a point. In this sense,no example of a triple“(X,x,l)” for which E(X,x,l)holds appears in [13]. The following result is a result concerning theexis- tence of triples“(X,x,l)” for whichE(X,x,l)holds (cf. Theorem 2):

Theorem B (Existence of many exceptional closed points for certain hyperbolic curves). Let l be a prime number, k a field ofcharacteristic 0, X ahyperbolic curvewhich is either oftype(0,3)or oftype(1,1), and Y→X a connected finite

´etale covering over k whicharises froman open subgroup of the geometrically pro- l fundamental groupΠ_X/k^{l} of X and isgeometrically connectedover k.(Thus, Y is ahyperbolic curveover k.)Then the subset Y^El⊆Y^clisinfinite. In particular, the subset X^El⊆X^clisinfinite.

Note that in Remark 15, we also give an example of a triple “(X,x,l)” such that Xis aproperhyperbolic curve, and, moreover,E(X,x,l)holds.

Ifx∈Xn(k)is ak-rational point of then-th configuration spaceXn of the hy- perbolic curveX/k, then it follows from the various definitions involved that the k-rational pointx∈Xn(k)determinesn distinct k-rational points ofX. Write

X[x]⊆X

for the hyperbolic curve of type(g,r+n)overkobtained by taking the complement inXof the images ofndistinctk-rational points ofXdetermined byx, i.e.,X[x]may be regarded as the fiber product of the diagram of schemes

Xn+1

 y Speck −−−−→

x Xn.

Here, for twok-rational pointsxandyofXn, we shall say thatxisequivalenttoyif X[x]'X[y]overk. In [9], the author proved that the isomorphism class of a certain (e.g.,split— cf. [9], Definition 1.5, (i))l-monodromically full hyperbolic curve of genus 0 over a finitely generated extension ofQis completely determined by the kernel of the natural pro-louter Galois representation associated to the hyperbolic curve (cf. [9], Theorem A). By a similar argument to the argument used in the proof of [9], Theorem A, one can prove the following Galois-theoretic characterization of equivalence classes ofl-monodromically full points of configuration spaces (cf.

Theorem 3):

Theorem C (Galois-theoretic characterization of equivalence classes of mon- odromically full points of configuration spaces). Let l be a prime number, n a positive integer, k afinitely generated extension ofQ, k an algebraic closure of k, and X ahyperbolic curveover k. Write Gkdef

=Gal(k/k)for the absolute Galois group of k determined by the given algebraic closure k and Xn for the n-thcon- figuration spaceof X/k. Then for two k-rational points x and y of Xn which are l-monodromically full(cf. Definition 3), the following three conditions are equiv- alent:

(i) x isequivalentto y, i.e., X[x]is isomorphic to X[y]over k.

(ii) Ker(ρ_X[x]/k^{l} ) =Ker(ρ_X[y]/k^{l} ).

(iii) If we writeφx(respectively,φy)for the composite

Gkπ1(x)

−→π1(Xn)^ρe

Xn/k{l}

−→Aut(∆_X^{l}_n/k) (respectively, Gkπ1(y)

−→π1(Xn)^eρ

Xn/k{l}

−→Aut(∆_X^{l}_n/k)) thenKer(φx) =Ker(φy).

In [17], S. Mochizuki introduced and studied the notion of ak-core (cf. [17], Definition 2.1, as well as [17], Remark 2.1.1). It follows from [14], Theorem 5.3, together with [17], Proposition 2.3, that if 2g−2+r>2, thena general hyperbolic curve of type(g,r)over k is a k-core(cf. also [17], Remark 2.5.1). For a hyperbolic curve overkwhich is ak-core, the following stronger Galois-theoretic characteriza- tion can be proven (cf. Theorem 4):

Theorem D (Galois-theoretic characterization of equivalence classes of quasi- monodromically full points of cores). Let l be a prime number, k afinitely gen- erated extension ofQ, k an algebraic closure of k, and X ahyperbolic curveover k which is a k-core (cf. [17], Remark 2.1.1). Write Gkdef

=Gal(k/k)for the abso- lute Galois group of k determined by the given algebraic closure k. Then for two k-rational points x and y of X which arequasi-l-monodromically full(cf. Defini- tion 3), the following four conditions are equivalent:

(i) x=y.

(ii) x isequivalentto y.

(iii) If we write

Uxdef

=X\Im(x); Uydef

=X\Im(y),

then the intersectionKer(ρ_U^{l}_x/k)∩Ker(ρ_U^{l}_y/k)isopeninKer(ρ_U^{l}_x/k)andKer(ρ_U^{l}_y/k).

(iv) If we writeφx(respectively,φy)for the composite

Gkπ₁(x)

−→π1(X)^ρe

{l}

−→X/kAut(∆_X/k^{l}) (respectively, Gkπ₁(y)

−→π1(X)^ρe

{l}

X/k

−→Aut(∆_X/k^{l})), then the intersectionKer(φx)∩Ker(φy)isopeninKer(φx)andKer(φy).

Finally, in a similar vein, we prove a necessary and sufficient condition for a quasi-l-monodromically fullGalois section (cf. Definition 5) of a hyperbolic curve to begeometric(cf. Theorem 5):

Theorem E (A necessary and sufficient condition for a quasi-monodromically full Galois section of a hyperbolic curve to be geometric). Let l be a prime number, k a finitely generated extension of Q, k an algebraic closure of k, Gkdef

=Gal(k/k)the absolute Galois group of k determined by the given algebraic closure k, X ahyperbolic curveover k, and s:Gk→Π_X/k^{l} a pro-lGalois section

of X (i.e., a continuous section of the natural surjectionΠ_X/k^{l} Gk — cf. [10], Definition 1.1,(i))which isquasi-l-monodromically full(cf. Definition 5). Write φsfor the composite

Gk s

−→Π_X/k^{{l} eρ}

X/k{l}

−→Aut(∆_X/k^{l}).

Then the following four conditions are equivalent:

(i) The pro-l Galois section s isgeometric(cf. [10], Definition 1.1,(iii)).

(ii) The pro-l Galois section sarises froma k-rational point of X(cf. [10], Definition 1.1,(ii)).

(iii) There exists aquasi-l-monodromically fullk-rational point(cf. Definition 3) x∈X(k)of X such that if we writeφxfor the composite

Gkπ1(x)

−→Π_X/k^{{l} ρe}

X{l}/k

−→Aut(∆_X/k^{l}),

then the intersectionKer(φs)∩Ker(φx)isopeninKer(φs)andKer(φx).

(iv) There exists aquasi-l-monodromically fullk-rational point(cf. Definition 3) x∈X(k)of X such that if we write

U^def=X\Im(x),

then the intersectionKer(φs)∩Ker(ρ_U/k^{l})isopeninKer(φs)andKer(ρ_U/k^{l}).

The present paper is organized as follows: In§1, we introduce and discuss the notion of monodromically full points of configuration spaces of hyperbolic curves.

In§2, we consider the fundamental groups of configuration spaces of hyperbolic curves. In §3, we consider the kernels of the outer representations associated to configuration spaces of hyperbolic curves. In§4, we prove Theorems A and B. In

§5, we prove Theorems C, D, and E.

0 Notations and Conventions Numbers:

The notationPrimeswill be used to denote the set of all prime numbers. The nota- tionZwill be used to denote the set, group, or ring of rational integers. The notation Qwill be used to denote the set, group, or field of rational numbers. The notationC will be used to denote the set, group, or field of complex numbers.

Profinite Groups:

IfGis a profinite group, andH⊆Gis a closed subgroup ofG, then we shall write NG(H)for thenormalizerofHinG, i.e.,

NG(H)^def={g∈G|gHg⁻¹=H} ⊆G, ZG(H)for thecentralizerofHinG, i.e.,

ZG(H)^def={g∈G|ghg⁻¹=hfor anyh∈H} ⊆G, Z_G^loc(H)for thelocal centralizerofHinG, i.e.,

Z_G^loc(H)^def= lim−→

H⁰⊆H

ZG(H⁰)⊆G

— whereH⁰⊆H ranges over the open subgroups ofH—Z(G)^def=ZG(G)for the centerofG, andZ^loc(G)^def=Z_G^loc(G)for thelocal centerofG. It is immediate from the various definitions involved thatH⊆NG(H)⊇ZG(H)⊆Z^loc_G (H)and that ifH1, H2⊆Gare closed subgroups ofGsuch thatH1⊆H2(respectively,H1⊆H2;H1∩H2

isopeninH1andH2), thenZG(H2)⊆ZG(H1)(respectively,Z_G^loc(H2)⊆Z_G^loc(H1);

Z_G^loc(H1) =Z_G^loc(H2)).

We shall say that a profinite groupGiscenter-free(respectively,slim) ifZ(G) = {1}(respectively,Z^loc(G) ={1}). Note that it follows from [16], Remark 0.1.3, that a profinite groupGisslimif and only if every open subgroup ofGiscenter-free.

IfGis a profinite group, then we shall denote by Aut(G)the group of [continu- ous] automorphisms ofG, by Inn(G)the group of inner automorphisms ofG, and by Out(G)the quotient of Aut(G)by the normal subgroup Inn(G)⊆Aut(G). If, more- over,Gistopologically finitely generated, then one verifies easily that the topology ofGadmits a basis ofcharacteristic open subgroups, which thus induces aprofinite topologyon the group Aut(G), hence also aprofinite topologyon the group Out(G).

Curves:

LetSbe a scheme andC a scheme overS. Then for a pair(g,r)of nonnegative integers, we shall say thatCis asmooth curve of type (g,r)overSif there exist a schemeC^cpt which issmooth,proper,geometrically connected, and ofrelative dimension1 overSand a closed subschemeD⊆C^cpt ofC^cpt which isfiniteand

´etaleoverSsuch that the complement ofDinC^cpt is isomorphic toCoverS, any geometric fiber ofC^cpt→Sis (a necessarily smooth, proper, and connected curve) of genusg, and, moreover, the degree of the finite ´etale coveringD,→C^cpt→Sis r. Moreover, we shall say thatCis ahyperbolic curve(respectively,tripod) overS

if there exists a pair(g,r)of nonnegative integers such thatCis a smooth curve of type(g,r)overS, and, moreover, 2g−2+r>0 (respectively,(g,r) = (0,3)).

For a pair(g,r) of nonnegative integers such that 2g−2+r>0, writeM_g,r for the moduli stack ofr-pointed smooth curves of genusgoverZwhose marked points are equipped with orderings (cf. [4], [11]) andM_g,[r]for the moduli stack of hyperbolic curves of type(g,r)overZ. Then we have a natural finite ´etale Galois Sr-coveringM_g,r→M_g,[r] — where we writeSr for the symmetric group onr letters.

List of major notation:

Primes §0, Numbers NG(H) §0, Profinite Groups ZG(H) §0, Profinite Groups Z^loc_G (H) §0, Profinite Groups Aut(G) §0, Profinite Groups Inn(G) §0, Profinite Groups Out(G) §0, Profinite Groups M_g,r §0, Curves

M_g,[r] §0, Curves

∆_X/S^Σ §1, Definition 1, (ii) Π_X/S^Σ §1, Definition 1, (ii) ρe_X/S^Σ §1, Definition 1, (iii) ρ_X/S^Σ §1, Definition 1, (iii) Φ_X/S^Σ §1, Definition 1, (iv) Γ_X/S^Σ §1, Definition 1, (iv) Xn §1, Definition 2, (i) X[x] §1, Definition 2, (ii) E(X,x,l) §4, Definition 4 X^El §4, Definition 4

1 Monodromically full points

In the present§, we introduce and discuss the notion ofmonodromically full points of configuration spaces of hyperbolic curves. LetΣ⊆Primesbe a nonempty sub- set ofPrimes (cf. the discussion entitled “Numbers” in§0)andSa regular and connected scheme.

Definition 1.LetXbe a regular and connected scheme overS.

(i) Let 1→∆→Π →G→1 be an exact sequence of profinite groups. Suppose that ∆ istopologically finitely generated. Then conjugation by elements ofΠ determines a commutative diagram of profinite groups

1 −−−−→ ∆ −−−−→ Π −−−−→ G −−−−→ 1



y y y

1 −−−−→ Inn(∆) −−−−→ Aut(∆) −−−−→ Out(∆) −−−−→ 1

— where the horizontal sequences areexact, and we refer to the discussion en- titled “Profinite Groups” in§0 concerning the topology of Aut(∆)(respectively, Out(∆)). We shall refer to the continuous homomorphism

Π−→Aut(∆) (respectively,G−→Out(∆))

obtained as the middle (respectively, right-hand) vertical arrow in the above di- agram as therepresentation associated to1→∆→Π→G→1 (respectively, outer representation associated to1→∆→Π→G→1).

(ii) We shall write

∆_X/S^Σ

for the maximal pro-Σ quotient of the kernel of the natural homomorphism π1(X)→π1(S)and

Π_X/S^Σ

for the quotient ofπ1(X)by the kernel of the natural surjection from the kernel ofπ1(X)→π1(S)to∆_X/S^Σ . (Note that since Ker(π1(X)→π1(S))is anormal closed subgroup ofπ1(X), and the kernel of the natural surjection Ker(π1(X)→ π1(S))∆_X/S^Σ is acharacteristicclosed subgroup of Ker(π1(X)→π1(S)), it holds that the kernel of Ker(π1(X)→π1(S))∆_X/S^Σ is anormalclosed subgroup ofπ1(X).) Thus, we have a commutative diagram of profinite groups

1 −−−−→ Ker(π1(X)→π1(S)) −−−−→ π1(X) −−−−→ π1(S)



y y

1 −−−−→ ∆_X/S^Σ −−−−→ Π_X/S^Σ −−−−→ π1(S)

— where the horizontal sequences areexact, and the vertical arrows are sur- jective. IfSis the spectrum of a ringR, then we shall write∆_X/R^{Σ def}=∆_X/S^Σ and Π_X/R^{Σ def}=Π_X/S^Σ .

(iii) Suppose that the natural homomorphism π1(X)→π1(S) is surjective — or, equivalently,Π_X/S^Σ →π1(S)is surjective— and that the profinite group∆_X/S^Σ istopologically finitely generated. Then we have an exact sequence of profinite groups

1−→∆_X/S^Σ −→Π_X/S^Σ −→π1(S)−→1.

We shall write

ρe_X/S^Σ :Π_X/S^Σ −→Aut(∆_X/S^Σ )

for the representation associated to the above exact sequence (cf. (i)) and refer to ρe_X/S^Σ as thepro-Σrepresentation associated to X/S. Moreover, we shall write

ρ_X/S^Σ :π1(S)−→Out(∆_X/S^Σ )

for the outer representation associated to the above exact sequence (cf. (i)) and refer toρ_X/S^Σ as thepro-Σouter representation associated to X/S. IfSis the spec- trum of a ringR, then we shall writeρe_X/R^{Σ def}=ρe_X/S^Σ andρ_X/R^{Σ def}=ρ_X/S^Σ . Moreover, if lis a prime number, then for simplicity, we write “pro-lrepresentation associated toX/S” (respectively, “pro-louter representation associated toX/S”) instead of

“pro-{l}representation associated toX/S” (respectively, “pro-{l}outer repre- sentation associated toX/S”).

(iv) Suppose that the natural homomorphism π1(X)→π1(S) is surjective — or, equivalently,Π_X/S^Σ →π1(S)is surjective— and that the profinite group∆_X/S^Σ istopologically finitely generated. Then we shall write

Π_X/S^Σ Φ_X/S^{Σ def}=Im(eρ_X/S^Σ )

for the quotient ofΠ_X/S^Σ determined by the pro-Σrepresentationeρ_X/S^Σ associated toX/S. Moreover, we shall write

π1(S)Γ_X/S^{Σ def}=Im(ρ_X/S^Σ )

for the quotient ofπ1(S)determined by the pro-Σouter representationρ_X/S^Σ asso- ciated toX/S. IfSis the spectrum of a ringR, then we shall writeΦ_X/R^{Σ def}=Φ_X/S^Σ andΓ_X/R^{Σ def}=Γ_X/S^Σ .

(v) Let π1(X)Q be a quotient ofπ1(X). Then we shall say that a finite ´etale coveringY→Xis afinite ´etale Q-coveringifY is connected, and the finite ´etale coveringY →X arises from an open subgroup ofQ, i.e., the open subgroup of π1(X)corresponding to the connected finite ´etale coveringY →X contains the kernel of the surjectionπ1(X)Q.

Remark 1.In the notation of Definition 1, ifSis the spectrum of a fieldk, then it follows from [5], Expos´e V, Proposition 6.9, thatπ1(X)→π1(S)issurjectiveif and only ifX isgeometrically connectedover k, i.e.,X⊗_kk, wherekis an algebraic closure ofk, isconnected— or, equivalently,X⊗_kk^sep, wherek^sepis a separable closure ofk, isconnected. Suppose, moreover, that X isgeometrically connected and offinite typeoverS. Then it follows from [5], Expos´e IX, Th´eor`eme 6.1, that the natural sequence of profinite groups

1−→π1(X⊗kk^sep)−→π1(X)−→π1(S)−→1

isexact. Thus, it follows from the various definitions involved that∆_X/k^Σ isnaturally isomorphicto themaximal pro-Σ quotientof the ´etale fundamental groupπ1(X⊗_k k^sep)ofX⊗_kk^sep. In particular, if, moreover,kis ofcharacteristic0, then it follows from [6], Expos´e II, Th´eor`eme 2.3.1, that∆_X/k^Σ istopologically finitely generated.

Remark 2.In the notation of Definition 1, suppose, moreover, thatX is ahyper- bolic curve over S (cf. the discussion entitled “Curves” in§0). Then since Sis regular, it follows immediately from [5], Expos´e X, Th´eor`eme 3.1, that the nat- ural homomorphism π1(ηS)→π1(S), where we write ηS for the generic point ofS, issurjective. Thus, in light of thesurjectivityof the natural homomorphism π1(X×SηS)→π1(ηS)(cf. Remark 1), we conclude that the natural homomorphism π1(X)→π1(S)issurjective. In particular, we have an exact sequence of profinite groups

1−→∆_X/S^Σ −→Π_X/S^Σ −→π1(S)−→1.

If, moreover, every element ofΣ isinvertibleonS, then it follows from a similar argument to the argument used in the proof of [7], Lemma 1.1, that∆_X/S^Σ is nat- urally isomorphic to the maximal pro-Σ quotient of the ´etale fundamental group π1(X×_Ss)ofX×_Ss, wheres→Sis a geometric point ofS. In particular, it follows immediately from the well-known structure of the maximal pro-Σ quotient of the fundamental group of a smooth curve over an algebraically closed field of charac- teristic6∈Σthat∆_X/S^Σ istopologically finitely generatedandslim, where we refer to the discussion entitled “Profinite Groups” in§0 concerning the term “slim”. Thus, we have continuous homomorphisms

ρe_X/S^Σ :Π_X/S^Σ −→Aut(∆_X/S^Σ ); ρ_X/S^Σ : π1(S)−→Out(∆_X/S^Σ ).

Moreover, there exists anatural bijectionbetween the set of the cusps ofX/Sand the set of the conjugacy classes of the cuspidal inertia subgroups of∆_X/S^Σ .

Lemma 1 (Outer representations arising from certain extensions).Let 1−→∆−→Π−→G−→1

be an exact sequence of profinite groups. Suppose that∆ istopologically finitely generatedandcenter-free. Write

ρe:Π−→Aut(∆); ρ:G−→Out(∆)

for the continuous homomorphisms arising from the above exact sequence of profi- nite groups(cf. Definition 1,(i)). Then the following hold:

(i)Ker(ρe) =Z_Π(∆). Moreover, the natural surjectionΠG induces an isomor- phism

Ker(ρe) (=Z_Π(∆))−→^∼ Ker(ρ).

In particular,∆∩Ker(eρ) ={1}.

(ii)The normal closed subgroup Ker(eρ)⊆Π is themaximalnormal closed sub- group N ofΠsuch that N∩∆={1}.

(iii)Write

Aut(∆⊆Π)⊆Aut(Π)

for the subgroup of Aut(Π) consisting of automorphisms whichpreserve the closed subgroup∆ ⊆Π. Suppose that ZΠ(∆) ={1}. Then the natural homo- morphism Aut(∆ ⊆Π)→Aut(∆) is injective, and its image coincides with N_Aut(∆)(Im(ρe))⊆Aut(∆), i.e.,

Aut(∆⊆Π)−→^∼ N_Aut(∆)(Im(ρe))⊆Aut(∆).

Proof. Assertion (i) follows immediately from the various definitions involved.

Next, we verify assertion (ii). First, let us observe that ifN⊆Πis a normal closed subgroup ofΠsuch that∆∩N={1}, then since∆ andNarenormalinΠ, for any x∈∆,y∈N, it holds thatxyx⁻¹y⁻¹∈∆∩N={1}; in particular, we obtain that N⊆Z_Π(∆) =Ker(eρ)(cf. assertion (i)). LetN⊆Πbe a normal closed subgroup of Πsuch that Ker(eρ)⊆N, and, moreover,N∩∆ ={1}. WriteN⊆Gfor the image ofNvia the natural surjectionΠG. Then since the image of Ker(eρ)⊆Π via the natural surjectionΠGis Ker(ρ)(cf. assertion (i)), we obtain a commutative diagram of profinite groups

1 −−−−→ ∆ −−−−→ Π/Ker(ρe) −−−−→ G/Ker(ρ) −−−−→ 1

 y

 y

1 −−−−→ ∆ −−−−→ Π/N −−−−→ G/N −−−−→ 1

— where the horizontal sequences areexact, and the vertical arrows are surjec- tive. Thus, it follows immediately from theexactnessof the lower horizontal se- quence of the above diagram that the homomorphism ρ factors through G/N.

Therefore, it holds that N=Ker(ρ). In particular, the right-hand vertical arrow, hence also the middle vertical arrow, is anisomorphism. This completes the proof of assertion (ii). Finally, we verify assertion (iii). It follows from the various def- initions involved that the natural homomorphism Aut(∆ ⊆Π)→Aut(∆)factors through N_Aut(∆)(Im(eρ))⊆Aut(∆). On the other hand, since the natural surjec- tion Π Im(eρ) is an isomorphism (cf. assertion (i)), conjugation by elements ofN_Aut(∆)(Im(eρ))determines a homomorphismN_Aut(∆)(Im(eρ))→Aut(Im(eρ))←^∼ Aut(Π), which factors through Aut(∆ ⊆Π)→Aut(Π). Now it may be easily verified that this homomorphism is theinverse of the homomorphism in question Aut(∆ ⊆Π)→N_Aut(∆)(Im(eρ)). This completes the proof of assertion (iii).

Lemma 2 (Certain automorphisms of slim profinite groups).Let G be atopolog- ically finitely generatedandslim(cf. the discussion entitled “Profinite Groups” in

§0)profinite group andα an automorphism of G. Ifα induces theidentity auto- morphismon an open subgroup, thenαis theidentity automorphismof G.

Proof. LetH⊆Gbe an open subgroup ofGsuch thatα induces theidentity au- tomorphismofH. To verify Lemma 2, by replacingH by the intersection of all G-conjugates ofH, we may assume without loss of generality thatHisnormalin G. Then sinceZG(H) ={1}, it follows immediately from Lemma 1, (iii), thatα is theidentity automorphismofG. This completes the proof of Lemma 2.

Proposition 1 (Fundamental exact sequences associated to certain schemes).

Let X be a regular and connected scheme over S. Suppose that the natural ho- momorphismπ1(X)→π1(S)issurjective, and that the profinite group∆_X/S^Σ (cf.

Definition 1,(ii))istopologically finitely generatedandcenter-free. Then the fol- lowing hold:

(i)We have a commutative diagram of profinite groups

1 −−−−→ ∆_X/S^Σ −−−−→ Π_X/S^Σ −−−−→ π1(S) −−−−→ 1

^eρX/SΣ

 y

 y^ρX/SΣ

1 −−−−→ ∆_X/S^Σ −−−−→ Φ_X/S^Σ −−−−→ Γ_X/S^Σ −−−−→ 1

(cf. Definition 1,(ii),(iii),(iv))— where the horizontal sequences areexact, and the vertical arrows aresurjective.

(ii)The quotientΠ_X/S^Σ Φ_X/S^Σ determined byρe_X/S^Σ is theminimalquotientΠ_X/S^Σ Q ofΠ_X/S^Σ such thatKer(Π_X/S^Σ Q)∩∆_X/S^Σ ={1}.

Proof. Assertion (i) (respectively, (ii)) follows immediately from Lemma 1, (i) (re- spectively, (ii)).

Definition 2.Letnbe a nonnegative integer, (g,r)a pair of nonnegative integers such that 2g−2+r>0, Sa regular and connected scheme, andX ahyperbolic curveof type(g,r)overS(cf. the discussion entitled “Curves” in§0).

(i) We shall write

X0def

=S and

Xn

for the n-th configuration space of X/S, i.e., the open subscheme of the fiber product ofncopies ofX overSwhich represents the functor from the category of schemes overSto the category of sets given by

T {(x1,· · ·,xn)∈X(T)^×n|xi6=xjifi6=j}.

For a nonnegative integerm≤n, we always regardXnas a scheme overXm by the natural projectionXn→Xmto the firstmfactors. Then it follows immediately from the various definitions involved that Xn+1 is a hyperbolic curveof type (g,r+n)overXn. In particular, if every element ofΣ isinvertibleonS, then we have continuous homomorphisms

ρe_X^Σ_n+1/Xn:Π_X^Σ_n+1/Xn−→Aut(∆_X^Σ_n+1/Xn); ρ_X^Σ_n+1/Xn: π1(Xn)−→Out(∆_X^Σ_n+1/Xn)

(cf. Remark 2). Moreover, it follows immediately from the various definitions involved thatXnisnaturally isomorphicto the(n−m)-th configuration space of the hyperbolic curveXm+1/Xm.

(ii) Letm≤nbe a nonnegative integer,T a regular and connected scheme overS, andx∈Xm(T)aT-valued point ofXm. Then we shall write

X[x]⊆X×ST

for the open subscheme ofX×_STobtained by taking the complement inX×_ST of the images of them distinct T-valued points ofX×_ST determined by theT- valued pointx. Then it follows immediately from the various definitions involved thatX[x]is equipped with anatural structure of hyperbolic curve of type(g,r+ m)over T and that the base-change ofXn→Xmviaxisnaturally isomorphic to the(n−m)-th configuration spaceX[x]n−mof the hyperbolic curveX[x]/T, i.e., we have acartesiandiagram of schemes

X[x]n−m −−−−→ Xn

 y

 y T −−−−→

x Xm.

(iii) LetT be a regular and connected scheme overSandx,y∈Xn(T)twoT-valued points ofXn. Then we shall say thatx is equivalent to yif there exists an isomor- phismX[x]→^∼ X[y]overT.

Definition 3.Letnbe a nonnegative integer,Sa regular and connected scheme,T a regular and connected scheme overS, andX ahyperbolic curveoverS. Suppose that every element ofΣ isinvertibleonS. Then we shall say that aT-valued point x∈Xn(T) of the n-th configuration space Xn ofX/S(cf. Definition 2, (i)) is Σ- monodromically full (respectively,quasi-Σ-monodromically full) if the following condition is satisfied: For anyl∈Σ, if we writeΓT⊆Γ_X^{l}_n+1/Xn(respectively,Γgeom⊆ Γ_X^{l}_n+1/Xn) for the image of the composite

π1(T)^π→^1(x)π1(Xn)

ρ_Xn+1^{l} _/Xn

Γ_X^{l}_n+1/Xn

(respectively,Ker

π1(Xn)→π1(S)

,→π1(Xn)

ρ_Xn+1^{l} _/Xn

Γ_X^{l}_n+1/Xn)

(cf. Definitions 1, (iii), (iv); 2, (i)), thenΓT containsΓgeom(respectively,ΓT∩Γgeom

is an open subgroup ofΓgeom). Note that since the closed subgroupΓgeom⊆Γ_X^{l}_n+1/Xn

isnormalinΓ_X^{l}_n+1/Xn, one may verify easily that whether or notΓT containsΓgeom

(respectively,ΓT∩Γgeomis an open subgroup ofΓgeom) doesnot dependon the choice of the homomorphism “π1(T)^π→^1(x)π1(Xn)” induced byx∈Xn(T)among the various π1(Xn)-conjugates.

Moreover, we shall say that a pointx∈XnofXnisΣ-monodromically full(re- spectively,quasi-Σ-monodromically full) if for anyl∈Σ, thek(x)-valued point of Xn, where we writek(x)for the residue field atx, naturally determined byxisΣ- monodromically full (respectively, quasi-Σ-monodromically full).

If l is a prime number, then for simplicity, we write “l-monodromically full”

(respectively, “quasi-l-monodromically full”) instead of “{l}-monodromically full”

(respectively, “quasi-{l}-monodromically full”).

Remark 3.The notion of (quasi-)monodromic fullnessdefined in Definition 3, as well as the notion of (quasi-)monodromic fullness defined in [9], Definitions 2.1;

2.2, is motivated by the study by Matsumoto and Tamagawa of the difference be- tween theprofiniteandpro-louter Galois representaions associated to a hyperbolic curve (cf. [12]). A consequence obtained from the main result of [12] is the follow- ing: In the notation of Definition 3, suppose thatSis the spectrum of anumber field k (i.e., a finite extension ofQ). Letkbe an algebraic closure ofk. Then foranyclosed pointxofXn, the image of theprofiniteouter Galois representation associated to the hyperbolic curveX[x]hastrivial intersectionwith the image of theprofiniteouter monodromy representation associated toXn+1⊗kk/Xn⊗kk. On the other hand, for any prime numberl, there existmany l-monodromically fullclosed points ofXn, i.e., a closed pointxsuch that the image of thepro-louter Galois representation associ- ated to the hyperbolic curveX[x]containsthe image of thepro-louter monodromy representation associated toXn+1⊗kk/Xn⊗kk. (cf. Remark 7 below).

Remark 4.In the notation of Definition 3, as the terminologies suggest, it follows immediately from the various definitions involved that theΣ-monodromic fullness ofx∈Xn(T)implies thequasi-Σ-monodromic fullnessofx∈Xn(T).

Remark 5.In the notation of Definition 3, ifSis the spectrum of a fieldkofcharac- teristic0, then it follows immediately from the various definitions involved that for a closed pointx∈XnofXnwith residue fieldk(x), the following two conditions are equivalent:

• The closed point x∈Xn is a Σ-monodromically full (respectively, quasi-Σ- monodromically full) point in the sense of Definition 3.

• Thek(x)-rational point ofXn⊗_kk(x)determined byxis aΣ-monodromically full (respectively,quasi-Σ-monodromically full) point with respect to the hyperbolic curvesXn+1⊗_kk(x)/Xn⊗_kk(x)in the sense of [9], Definition 2.1.

If, moreover,Xis the complementP¹_k\ {0,1,∞}of{0,1,∞}in the projective lineP¹_k overk, then since then-th configuration spaceXnofX/kisnaturally isomorphicto the moduli stackM_0,n+3⊗Zkof(n+3)-pointed smooth curves of genus 0 overk- schemes whose marked points are equipped with orderings, for a closed pointx∈Xn

with residue fieldk(x), the following two conditions are equivalent:

• The closed point x∈Xn is a Σ-monodromically full (respectively, quasi-Σ- monodromically full) point in the sense of Definition 3.

• The hyperbolic curveX[x]overk(x)(cf. Definition 2, (ii)) is aΣ-monodromically full(respectively,quasi-Σ-monodromically full) hyperbolic curve overk(x)in the sense of [9], Definition 2.2.

Remark 6.In the notation of Definition 3, suppose thatS=T. Then it follows from the various definitions involved that the following two conditions are equivalent:

(i) TheS-valued pointx∈Xn(S)is aΣ-monodromically full(respectively,quasi-Σ- monodromically full) point.

(ii) For anyl∈Σ, the composite

π1(S)^π→^1(x)π1(Xn)

ρ_Xn+1^{l} _/Xn

Γ_X^{l}_n+1/Xn issurjective(respectively, hasopen image).

Proposition 2 (Existence of many monodromically full points).LetΣbe a nonempty finiteset of prime numbers, k afinitely generated extension ofQ, X ahyperbolic curveover k(cf. the discussion entitled “Curves” in§0), n a positive integer, Xnthe n-th configuration space of X/k(cf. Definition 2,(i)), X_n^clthe set of closed points of Xn, and X_n^Σ-MF⊆X_n^clthe subset of X_n^clconsisting of closed points of Xnwhich are Σ-monodromically full(cf. Definition 3). If we naturally regard X_n^clas a subset of Xn(C), then the subset

X_n^Σ-MF⊆Xn(C)

isdensewith respect to thecomplex topologyof Xn(C). If, moreover, X is ofgenus 0, then the complement

Xn(k)\(Xn(k)∩X_n^Σ-MF)⊆Xn(k)

forms athin setin Xn(k)in the sense of Hilbert’s irreducibility theorem.

Proof. This follows from [9], Theorem 2.3, together with Remark 5.

Remark 7.Letnbe a positive integer,Σ⊆Primesa nonemtpy subset ofPrimes,k anumber field(i.e., a finite extension ofQ), andXahyperbolic curveoverk. For a closed pointx∈Xnwith residue fieldk(x), as in Definition 3, writeΓ_x^Σ ⊆Γ_X^Σ_n+1/Xn (respectively,Γ_geom^Σ ⊆Γ_X^Σ_n+1/Xn) for the image of the composite

π1(Speck(x))^π→^1(x)π1(Xn)

ρ^Σ_Xn+1/Xn

Γ_X^Σ_n+1/Xn

(respectively,Ker

π1(Xn)→π1(Speck)

,→π1(Xn)

ρ_Xn+1^Σ _/Xn

Γ_X^Σ_n+1/Xn).

Let us recall that for a prime numberl, the closed pointx∈Xnisl-monodromically fullif and only ifΓgeom^{l} ⊆Γx^{l}. Thus, Proposition 2 asserts thatmanyclosed points

ofXnsatisfy this condition, i.e.,Γgeom^{l} ⊆Γx^{l}. On the other hand, it follows imme- diately from [12], Theorem 1.1; [8], Corollary 6.4, that foranyclosed pointx∈Xn, it holds thatΓx^Primes∩Γgeom^Primes={1}; in particular, since one verifies easily that Γgeom^Primes6={1}, the inclusionΓgeom^Primes⊆Γx^Primesnever holds.

2 Fundamental groups of configuration spaces

In the present§, we consider the fundamental groups of configuration spaces of hy- perbolic curves. LetΣ⊆Primesbe a nonempty subset ofPrimes(cf. the discussion entitled “Numbers” in§0),Sa regular and connected scheme, andX ahyperbolic curveoverS(cf. the discussion entitled “Curves” in§0). Suppose that every element ofΣ isinvertibleonS.

Lemma 3 (Fundamental groups of configuration spaces).Let m<n be nonnega- tive integers. Suppose thatΣis eitherPrimesor ofcardinality1. Then the follow- ing hold:

(i)The natural homomorphismπ1(Xn)→π1(Xm)issurjective. Thus, we have an exact sequence of profinite groups

1−→∆_X^Σ_n/Xm−→Π_X^Σ_n/Xm −→π1(Xm)−→1 (cf. Definition 1,(ii)).

(ii)If x→Xmis a geometric point of Xm, then∆_X^Σ_n/Xm isnaturally isomorphicto the maximal pro-Σ quotient of the ´etale fundamental groupπ1(Xn×X_mx)of Xn×X_mx.

(iii)Let T be a regular and connected scheme over S and x∈Xm(T)a T -valued point of Xm. Then the homomorphism

∆_X[x]^Σ _n−m/T −→∆_X^Σ_n/Xm determined by thecartesiansquare of schemes

X[x]n−m −−−−→ Xn



y y T −−−−→

x Xm

(cf. Definition 2,(ii))is anisomorphism. In particular, the right-hand square of the commutative diagram of profinite groups

1 −−−−→ ∆_X[x]^Σ _n−m/T −−−−→ Π_X[x]^Σ _n−m/T −−−−→ π1(T) −−−−→ 1

o

 y

 y

 y^π1(x)

1 −−−−→ ∆_X^Σ_n/Xm −−−−→ Π_X^Σ_n/Xm −−−−→ π1(Xm) −−−−→ 1

— where the horizontal sequences areexact(cf. assertion(i))— iscartesian.

(iv)The natural sequence of profinite groups

1−→∆_X^Σ_n/Xm−→∆_X^Σ_n/S−→∆_X^Σ_m/S−→1 isexact.

(v)The profinite group∆_X^Σ_n/Xm istopologically finitely generatedandslim(cf. the discussion entitled “Profinite Groups” in§0). Thus, we have

ρe_X^Σ_n/Xm:Π_X^Σ_n/Xm−→Aut(∆_X^Σ_n/Xm);

ρ_X^Σ_n/Xm:π1(Xm)−→Out(∆_X^Σ_n/Xm) (cf. assertion(i)).

(vi)Let T be a regular and connected scheme over S and x∈Xm(T)a T -valued point of Xm. Then the diagram of profinite groups

π1(T) ^ρ

ΣX[x]n−m/T

−−−−−−→ Out(∆_X[x]^Σ _n−m/T)

π₁(x)y

 y^o π1(Xm) −−−−→

ρ_Xn/Xm^Σ Out(∆_X^Σ_n/Xm)

(cf. assertion(v))— where the right-hand vertical arrow is theisomorphism determined by the isomorphism obtained in assertion(iii)—commutes.

(vii)The centralizer Z_∆Σ

Xn/S(∆_X^Σ_n/Xm)of∆_X^Σ_n/Xm in∆_X^Σ_n/S(cf. assertion(iv))istrivial.

(viii)The pro-Σouter representation associated to Xn/Xm

ρ_X^Σ_n/Xm:π1(Xm)−→Out(∆_X^Σ_n/Xm)

(cf. assertion(v))factors throughthe natural surjectionπ1(Xm)Π_X^Σ_m/S, and, moreover, the composite of the natural inclusion∆_X^Σ_m/S,→Π_X^Σ_m/Sand the result- ing homomorphismΠ_X^Σ_m/S→Out(∆_X^Σ_n/Xm)isinjective.

Proof. First, we verify assertion (i). By induction onn−m, we may assume without loss of generality thatn=m+1. On the other hand, ifn=m+1, thenXn→Xmis a hyperbolic curveoverXm(cf. Definition 2, (i)). Thus, the desired surjectivity follows from Remark 2. This completes the proof of assertion (i). Next, we verify assertion (ii). It is immediate that there exists a connected finite ´etale coveringY→XmofXm

which satisfies the condition (c) in the statement of [18], Proposition 2.2, hence also