L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

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L’INSTITUT FOURIER

Manish KUMAR

Embedding problems for open subgroups of the fundamental group Tome 67, n^o6 (2017), p. 2623-2649.

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EMBEDDING PROBLEMS FOR OPEN SUBGROUPS OF THE FUNDAMENTAL GROUP

by Manish KUMAR

Abstract. — LetCbe a smooth irreducible affine curve over an algebraically closed field of positive characteristic and letπ1(C) be its fundamental group. We study various embedding problems forπ1(C) and its subgroups.

Résumé. — SoitCune courbe affine irréductible lisse sur un corps algébrique- ment fermé de caractéristique positive et soitπ1(C) son groupe fondamental. Nous étudions divers problèmes de plongement pourπ1(C) et ses sous-groupes.

1. Introduction

Let C be a smooth irreducible affine curve over an algebraically closed fieldkof characteristicp >0. The existence of wild ramification causes the structure of the étale fundamental groupπ1(C) to be complicated. There have been some attempts to understand this group. One step towards this goal was Raynaud’s [14] and Harbater’s [4] proof of Abhyankar’s conjecture which states that a finite groupGis a quotient ofπ1(C) if and only if the maximal prime-to-pquotientG/p(G) ofGis generated by 2g+r−1 elements whereg is the genus of the smooth completion ofC and ris the number of points in the boundary.

Though this gives a complete description of the finite quotients ofπ1(C), it does not say how these groups fit together in the inverse system for π₁(C). A possible way to understand this is by analyzing which finite embedding problems for π1(C) have a solution (see the “Notation” subsection at the end of this introduction for definition). One crucial result in this direction is by Pop [12] (and independently proved by Harbater

Keywords:ramification, embedding problem, fundamental group, positive characteristic, formal patching.

2010Mathematics Subject Classification:14H30,14G32,12F10.

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as well [7, Theorem 5.3.4]) which says that given an embedding problem E= (α: Π→G, φ: Γ→G) where H= ker(φ) is a quasi-pgroup, there is a proper solutionλ: ΠΓ toE. This clearly strengthens Raynaud’s and Harbater’s result. WhenH is not a quasi-pgroup then it is clear from the Abhyankar’s conjecture that the above embedding problem may not have a solution. But there are finite index open subgroups of π1(C) for which these embedding problems have a solution.

Definition 1.1. — Given an embedding problem E = (α : π1(C) → G, φ : Γ → G) and a finite index subgroup Π of π₁(C), we say the embedding problem restricts to Π if α(Π) = G. Moreover if the restricted embedding problem has a solution then we sayΠ is effectivefor the embedding problemE.

In [8], it was shown that given any finite embedding problem for π₁(C) there exists a finite index effective subgroup for the embedding problem.

In [1, Theorem 1.3] it was shown that it is even possible to find an indexp subgroup ofπ1(C) which is effective for the embedding problem. Our objec- tive is to find some necessary and some sufficient conditions for a subgroup ofπ₁(C) to be an effective subgroup for a given embedding problem.

Suppose Π is a finite index subgroup of π1(C). This corresponds to a coverD →C. Let Z →X be the corresponding morphism between their smooth completions and letnDdenote 2gZ+rD−1. A necessary condition for Π to be effective for the embedding problemE = (α:π1(C)→G, φ : Γ→G) is thatErestricts to Π and the rank of Γ/p(Γ) is at mostnD. Indeed Π is effective implies there is a surjection fromπ1(D) = ΠΓ. Hence there is a surjection from prime-to-p part ofπ₁(D) to the maximal prime-to-p quotient of Γ, denoted Γ/p(Γ). So Γ/p(Γ) is generated by nD elements.

Also, [8, Theorem 5] can be rephrased in terms of sufficient conditions for Π =π1(D) to be an effective subgroup ofπ1(C).

Proposition 1.2. — LetC ⊂X, Π, E, and D ⊂Z be as above. Let ΨX :VX →X be theG-cover corresponding to the embedding problem E and assumeErestricts toΠ. LetΨ :˜ V_X×_XZ →Zbe the pull-back ofΨ_X. ThenΠ is effective if the number of points inZ\D where the morphism Ψ˜ is not branched, is at least the relative rank ofker(φ: Γ→G)inΓ (see

“Notation” subsection for definition of relative rank).

The subset of Z \D for which the morphism Ψ is unramified can be thought of as points available for branching in any cover dominating Ψ.

In the theorem above, having sufficiently many available branch points allows the embedding problem to be solved. A natural question is that can

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the condition on the number of such potential branch points inZ\D be relaxed? For example, could it be replaced by a condition on the genus of Z or n_D? It is also worth noting that in the above proposition the degree of the coverZ→X must be large to ensure that the number of points in Z\D where the morphism ˜Ψ is not branched is sufficiently large. Hence the index of Π inπ1(C) is also large.

The two main results of this paper, Theorem 6.1 and Proposition 7.1, use genus and branch points to obtain effective subgroups. LetH be the kernel of the homomorphism φ : Γ → G from the embedding problem E = (α : π₁(C) → G, φ : Γ → G), and let H/p(H) be the maximal prime to p quotient of H. The main tool in this paper is Theorem 5.1 which investigates the relationship between solving embedding problems forπ1(C) and the relative rank ofH/p(H) in Γ/p(H). Roughly speaking, it shows that if theG-Galois cover ΨX :VX →X corresponding toαhas a deformation which is sufficiently degenerate (in terms of having many components that are trivialGcovers), then there exists a proper solution toE. Theorem 6.1 restricts to the case thatC is the affine line and Π has indexpinπ1(A¹k). It uses degenerations and Theorem 5.1 to show that if the curveZin theZ/pZ-coverZ →P¹k corresponding to Π⊂π₁(A¹k) has genus gZ greater than the relative rank of (H/pH) in Γ/p(H) and a technical condition which holds for most values ofgZ(see Corollary 6.2, Corollary 6.3 and Remark 6.4) then Π is effective forE. In particular these results provide sufficient conditions for subgroups of π1(C) to be effective. In Section 7 some of the results proved for affine line case are generalized to general curves though the conclusions obtained are slightly weaker (Proposition 7.1 and Corollary 7.3).

The techniques involved in proving these results include formal patching and deformations within families of covers. Each result requires the construction of a Galois cover of a degenerate curve. Then a formal patching argument is used to obtain a cover of smooth curves over a complete local ring. Finally, a deformation argument similar to [8, Proposition 4] is used to ensure that the original embedding problemE restricts to the subgroup Π ofπ₁(C) that is obtained in the construction. The paper is organized as follows: At the end of the Introduction there is a list of notation. Section 2 looks at some of the group theoretic properties and examples of effective groups. Section 3 defines deformations and degenerations of covers and uses a formal patching result of Harbater to solve the embedding problems in the presence of a sufficiently degenerate deformation of the original

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cover. Section 4 proves a globalization and specialization result using a Lef- schetz type principle and Abhyankar’s lemma. Section 5 applies the formal patching and Lefschetz–Abhyankar result to obtain Theorem 5.1. Finally in Section 6, Theorem 6.1 is proved, and in Section 7, Proposition 7.1 and Corollary 7.3 is proved.

Notation. — If Gis a finite group and pis a prime number, let p(G) denote the subgroup ofG generated by itsp-subgroups. This is a characteristic subgroup ofG, andG/p(G) is the maximal prime-to-pquotient of G. A finite groupGis calledquasi-pifG=p(G).

Let Γ be any finite group and letH be a subgroup of Γ. A subsetS⊂H will be called arelative generating setforH in Γ if for every subsetT ⊂Γ such thatH∪T generates Γ, the subsetS∪T also generates Γ. We define the relative rank ofH in Γ to be the smallest non-negative integer µ :=

rank_Γ(H) such that there is a relative generating set forH in Γ consisting of µelements. Every generating set for H is a relative generating set, so 0 6 rankΓ(H) 6 rank(H). Also, rankΓ(H) = rank(H) if H is trivial or H = Γ, while rank_Γ(H) = 0 if and only ifH is contained in the Frattini subgroup Φ(Γ) of Γ [5, p. 122].

A finite embedding problem E for a group Π is a pair of surjections (α: Π→G, φ: Γ→G), where Γ andGare finite groups. IfH = ker(φ), the embedding problemE can be summarized by

Π

α

1 //H //Γ ^φ //G //

1

Aweak solution toE is a homomorphismγ: Π→Γ such thatφ◦γ =α.

We callγa proper solutiontoE if in addition it is surjective.

Remark 1.3. — Let Π be a profinite group and let E = (α : Π → G, φ : Γ → G) be a finite embedding problem for Π. Suppose that the epimorphismα: Π→Gfactors asrα⁰, where α⁰ : Π→G⁰ and r: G⁰ → G are epimorphisms, for some finite group G⁰. We consider the induced embedding problem E_α⁰ = (α⁰ : Π → G⁰, φ⁰ : Γ⁰ → G⁰) by taking Γ⁰ = Γ×GG⁰ and letting φ⁰ : Γ⁰ →G⁰ be the second projection map. Hereφ⁰ is surjective becauseφis; and soE_α⁰is a finite embedding problem. HereEand Eα⁰ have isomorphic kernels; indeed ker(φ⁰) = ker(φ)×1⊂Γ×GG⁰ = Γ⁰.

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Note that the first projection mapq: Γ⁰ = Γ×GG⁰→Γ is surjective since r:G⁰→Gis surjective; andφq=rφ⁰.

In this situation, every proper solutionλ⁰ : Π→Γ⁰ofE_α⁰ induces a proper solutionλ:=qλ⁰ : Π→Γ ofE; viz. φλ=φqλ⁰ =rφ⁰λ⁰ =rα⁰ =α, andλ is surjective becauseq andλ⁰ are. So we obtain a map PS(Eα⁰)→PS(E), where PS denotes the set of proper solutions to the embedding problem.

In this paper we consider curves over an algebraically closed field k of characteristic p > 0. A cover of k-curves is a morphism Φ : D → C of smooth connected k-curves that is finite and generically separable. If Φ : D → C is a cover, its Galois group Gal(D/C) is the group of k- automorphismsσofD satisfying Φ◦σ= Φ. If Gis a finite group, then a G-Galois cover is a cover Φ :D →C together with an inclusion ρ:G ,→ Gal(D/C) such thatGacts simply transitively on a generic geometric fiber of Φ :D→C. If we fix a geometric point ofCto be a base point, then the pointedG-Galois étale covers ofCcorrespond bijectively to the surjections α:π1(C)→G, whereπ1(C) is the algebraic fundamental group ofC with the chosen geometric point as the base point. The proper solutions to an embedding problemE= (α:π1(C)→G, φ: Γ→G) forπ1(C) then are in bijection to the pointed Γ-Galois coversE→Cthat dominate the pointed G-Galois cover Φ :D →C corresponding toα. In the case thatX is the smooth completion of the affinek-curveC, denote bygX the genus ofX, and definer_C= #(X−C) andn_C= 2g_X+r_C−1.

Acknowledgments

The author thanks David Harbater and Kate Stevenson for a number of useful discussions which led to some of the ideas in the paper. There were also many suggestions from them which improved the presentation of this paper. Thanks are also due to the referee for his useful suggestions.

2. Group theory results and examples of effective subgroups

In this section we analyze when a subgroup of an effective subgroup for an embedding problem is effective using some group theory and Galois theory.

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We start with an embedding problem E(2.1) for Π.

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Π

α

1 //H //Γ ^φ //G //

1

1 The following remark is easy to see.

Remark 2.1. — Suppose Π1 is an effective subgroup for the embedding problemE(2.1) and Π2 is a finite index subgroup of Π1. Suppose for some solutionψ of the embedding problemE(2.1) restricted to Π1, the index of ker(ψ)∩Π2 in Π2 is |Γ| then Π2 is also an effective subgroup for E(2.1).

Note that ker(ψ|Π2) = ker(ψ)∩Π₂has index|Γ|in Π₂. Henceψ|Π2 indeed surjects onto Γ.

Corollary 2.2. — SupposeΠ1is an effective subgroup for the embedding problemE(2.1)andΠ2is a finite index subgroup ofΠ1such that the embedding problemE(2.1)restricts toΠ2 and[Π1: Π2]is coprime to|H|.

ThenΠ₂is also effective forE(2.1).

So the index [ker(α|Π₂) : ker(ψ)∩ker(α|Π₂)] =|H|. That the embedding problem restricts to Π₂means that [Π₂: ker(α|Π2)] =|G|. So we obtain that [Π2: ker(ψ)∩ker(α|Π₂)] =|G||H|=|Γ|. Also note that ker(ψ)∩ker(α|Π₂) = ker(ψ)∩ker(α|_Π₁)∩Π₂ = ker(ψ)∩Π₂. Hence the result is obtained from

the previous remark.

In the above corollary the hypothesis guaranteed that if Π1is an effective subgroup for the given embedding problem andψ is a solution of the embedding problem restricted to Π₁thenψ|_Π₂ is a solution to the embedding problem restricted to Π2. Hence Π2 is also an effective subgroup. But this does not hold unconditionally as the following examples show.

Example. — Let Π be the absolute Galois group of the reals, let Γ = H=Z/2Z, and letG= 1. Then the given embedding problem has a proper solution (the complex numbers). So Π₁= Π itself is effective. But if we pull back fromRto C(corresponding to taking the trivial subgroup Π2 of Π) then the embedding problem (which has trivial cokernel) restricts to Π₂. But it no longer has a proper solution. Note here|H|= [Π : Π2] = 2.

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On the contrary, in geometric setting, even if the given solution does not pull back to a proper solution, there might be some other proper solution over the pullback.

Example. — LetC be the affine x-line minus 0 in characteristic 0, let Π =π1(C), Γ =Z/2Z×Z/3Z,H =Z/3Z, andG=Z/2Z. Then there is a G-coverC1 ofC given byy²=x. Over that there’s a proper solutionD to theE(2.1), given by z³ =y. So again Π₁ = Π is an effective subgroup of Π forE(2.1). HereDis the fiber product of C1 with the curveC2 given byw³=x(wherew=z², andz=y/w). Now pull back everything by the degree 3 coverC2ofC. This is linearly disjoint fromC1, soE(2.1) restricts to the subgroup π₁(C₂) of Π. But the degree is not relatively prime to

|H|, the cover C2 is not linearly disjoint from D, and the solution to the givenE(2.1) does not restrict to a proper solution toE(2.1) restricted to π₁(C₂). ButE(2.1) restricted toπ₁(C₂) does have a proper solution, given byv³=z.

3. Formal Patching Results

In this section we develop some formal patching results which are used in later sections to find solutions to various embedding problems. Propo- sition 3.7 is the main result of this section and one of the main technical results of this paper.

Notation. — Given a schemeX, denote byM(X) the category of coherent sheaves of O_X-modules,AM(X) the category of coherent sheaves of OX-algebras and SM(X) the subcategory of AM(X) for which the sheaves of algebras are generically separable and locally free. Given a finite groupGdenote byGM(X) the category of generically separable coherent locally free sheaves of OX-algebras S together with a G-action which is transitive on the geometric generic fibers of Spec_O

X(S)→X. Given cate- gories A,B,C and functors F :A → C and G:B → C, denote by A ×_CB the associated fiber category.

The following result is due to Harbater [7, Theorem 3.2.12].

Theorem 3.1. — Let(A,p)be a complete local ring and T a proper A-scheme. Let{τ₁, . . . , τ_N} be a finite set of closed points of T and U = T\ {τ1, . . . , τN}. Denote by OˆT ,τ_i the completion of the local ring OT ,τ_i

and letTi= Spec( ˆOT ,τ_i). LetU^∗be thep-adic completion ofU andKithe

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p-adic completion ofT_i\ {τi}. Then the base change functor M(T)→ M(U^∗)×

M SN

i=1Ki

M

N

[

i=1

T_i

!

is an equivalence of categories. The same remains true if we replaceMby AM, SMorGMfor a fixed finite groupG.

Next we state a lemma useful for putting covers into a situation where the theorem above holds. Its proof is just an application of a generalization of the Noether Normalization Lemma.

Lemma 3.2. — LetC be an affinek-curve andX be its smooth projective completion. Then there exists a finite morphism ΘX : X → P¹x such thatΘ_X is étale atΘ⁻¹_X ({x= 0})and such thatΘ⁻¹_X ({x=∞}) =X\C.

Proof. — By a stronger version of Noether normalization (cf. [2, Corol- lary 16.18]), there exists a finite generically separable k-morphism from C→A¹xwhereA¹xis the affinek-lines with local coordinatex. The branch locus of this morphism is of codimension 1, and hence it is étale away from finitely many points. By translation we may assume that x= 0 is not a branch point ofC → A¹x. This morphism extends to a finite proper morphism ΘX : X → P¹x. Note that ΘX is étale at Θ⁻¹_X ({x = 0}) and that

Θ⁻¹_X ({x=∞}) =X\C.

The patching results will be applied to covers of reducible curves where there are sufficiently many components of the base over which the cover is trivial. First, we need some terminology.

Definition 3.3. — LetΦ :V →X be aG-cover of smooth irreducible projective curves over k. Assume X has r marked points and Φ is étale away from theser points. We say that aG-cover of connected projective curves Φ⁰ : V⁰ → X⁰ with r marked points on X⁰ is a deformation of V →X if there exist a smooth irreduciblek-schemeS, a cover ofS-curves ΦS :VS →XS, rsectionsp1, . . . , pr :S →XS and closed points sand s⁰ inS such that the following three conditions hold.

(1) ΦS inducesΦandΦ⁰ atsands⁰ respectively.

(2) {p1(s), . . . , pr(s)}and{p1(s⁰), . . . , pr(s⁰)}are the marked points of X andX⁰ respectively.

(3) Φ_S is aG-cover étale away from the sectionsp₁, . . . , p_r.

Notation. — Given a Φ : V → X and a deformation Φ⁰ : V⁰ → X⁰, we will call (S; Φ_S :V_S → X_S;p₁, . . . , p_r;s;s⁰) the associated data of the deformation.

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Definition 3.4. — The deformationV⁰→X⁰will be called aSNC de- formationif all irreducible components ofX⁰are smooth and they intersect transversely. Moreover, it will be called adegenerationif it is SNC and for some irreducible componentX1ofX⁰, the restriction of theG-cover toX1

is induced from a trivial cover, i.e. V⁰×X⁰ X₁ ∼= Ind^G_e X₁ as G-covers of X1. In this situationX1 will be called a trivial component ofΦ⁰.

Remark 3.5. — Let Φ :V_X →X be aG-cover. Suppose Φ has a deformation Φ⁰:V_X⁰ →X⁰with associated data (S; ΦS:VS→XS;p1, . . . , pr;s;s⁰).

By taking a smooth irreduciblek-curve in S passing throughsand s⁰ and pulling back ΦS : VS →XS to this curve, we may assumeS is a smooth k-curve. In other words, if Φ⁰ is a deformation of Φ then we may assume Φ can be deformed to Φ⁰ along a smooth curve.

Example. — Let Φ :V_X →X be a G-cover of irreducible smooth pro- jective curves. Let τ be a closed point in X. Let S = A¹t and XS be the blowup of (τ, t = 0) in X×S and X⁰ be the total transform of the zero locus of t = 0 in X ×S. Note that X⁰ has two irreducible components, a copy of X and the exceptional divisor isomorphic to P¹ intersecting at τ. One can obtain aG-cover Φ_S :V_X_S →X_S obtained by pullback along XS →X×S →X. The fiber overt= 0 induces aG-cover Φ⁰:VX⁰ →X⁰. The exceptional divisorP¹is the trivial component of Φ⁰.

Lemma 3.6. — Let Φ : V → S be a flat family of reduced projective irreducible curves in which S is a smooth connected variety and V is a normal variety. Suppose for every point s ∈ S the normalization of the fiberV_shas the same genus. Then for every closed points∈Sthe fiberV_s is smooth.

Proof. — Let η be the generic point of S. Note that Vη being a local- ization of V is normal. Hence there exists a nonempty open subset U of S such that for all closed point s∈U, V_s is a normalk-curve and hence smooth. Letgbe the genus of such a curve. Suppose there exists s⁰ ∈S a closed point such thatV_s⁰ is singular. Then the arithmetic genus ofV_s⁰ is g, since Φ is a flat family. ButVs⁰ is singular so the geometricpg(Vs⁰)< g.

But pg(Vs⁰) is same as the genus of the normalization of Vs⁰. This con- tradicts the hypothesis that the normalization of every fiber has the same

genus.

Proposition 3.7. — Let Γ be a finite group. Let G be a subgroup of Γ and let H₁, . . . , H_m be subgroups of Γ of order prime-to-p. Assume that G, H1, . . . , Hm generate Γ. Let C be an affine k curve with smooth

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completionX. LetΦ :V_X→X be aG-Galois cover ofX étale overC. Let BX=X−C,r= #(BX), and consider the setBX to be rmarked points on X. Suppose Φ has a degeneration Φ⁰ : V_X⁰ → X⁰ with an associated data(S; ΦS:VS →XS;p1, . . . , pr;s;s⁰)such thatS is a smooth curve and X₁, . . . , Xm are the trivial components of X⁰. LetB be the union of the images of the sectionsp1, . . . , pr : S → XS and let ri be the number of smooth marked points ofX⁰ lying onXi fori= 1, . . . , m. Further assume that for each1 6i 6m, there exists an H_i-coverΦXi :W_X_i →X_i étale away from theseri points. LetT =XS×SSˆs⁰ whereSs⁰ is the completion ofS ats⁰ andΦ_T :V_T →T be the pullback ofΦ_S toT. Then there exists aΓ-coverΨ :W →T such thatΨis étale away from B×X_S T. Moreover, ifΓ =HoGandH₁, . . . , Hm6H thenΨdominates the G-coverΦT.

Proof. — By Remark 3.5 we may assume that there exists an associated data (S; ΦS :VS →XS;p1, . . . , pr;s;s⁰) for the degeneration of Φ to Φ⁰such that S is a smooth curve. Let t be the local coordinate of S at s⁰. Then Sˆs⁰ = Spec(k[[t]]). By construction, the closed fiber ofTisX⁰. LetX0be the closure ofX⁰\∪^m_i=1X_iinX⁰. SoX₀is made up of the nontrivial components of Φ⁰. Since Φ⁰ is a degeneration of Φ it is a SNC deformation, so all irreducible components ofX⁰ are smooth and they intersect transversely.

Let τ1, . . . , τN be the closed points of T where Xi and Xj intersect for some 06i 6= j 6 m. LetX_i^o = Xi\({τ1, . . . , τN} ∩Xi) for 0 6i 6m.

Note thatτ₁, . . . , τ_N will be used to denote the points ofT,X⁰ and various components ofX⁰ as well, but this should not lead to any confusion.

LetTô=T\ {τ₁, . . . , τ_N} and ˜Tô be the formal scheme obtained by the completion ofTôalong the closed fiber (t= 0) (i.e. the (t)-adic completion).

LetT_iô =X_iô×kSpec(k[[t]]) and ˜T_iô be the (t)-adic completion of T_iô (i.e.

alongX_iô) fori= 0, . . . , m. Since the closed fiber ofTôis the disjoint union ofX₀ô, X₁ô, . . . , X_mô,

T˜ô= ˜T₀ô∪T˜₁ô∪ · · · ∪T_mô

By base change of ΦT to ˜Tô we obtain a G-cover ΦT˜ô : ˜VTô →T˜ô and hence aG-cover of the component ˜T₀ôwhich we will denote by Φ_T₀ : ˜V₀→ T˜₀ô. Note that ΦT restricted to the closed fiber is the G-cover Φ⁰ :V_X⁰ → X⁰. Since Xi, for i = 1, . . . , m, are the trivial components of the cover Φ⁰ : V_X⁰ →X⁰, the G-covers of the components ˜T_iô are induced from the trivial cover. Let ΦT_i : ˜Vi →T˜_iô be theHi-cover obtained by pulling back Φ_X_i :W_X_i →X_i along the composition of morphisms ˜T_iô→T_iô→X_iô → Xi, fori= 1, . . . , m. Let Φô : Ind^Γ_GV˜0∪Ind^Γ_H₁V˜1∪ · · · ∪Ind^Γ_H_mV˜m→T˜ô be the Γ-cover of ˜Tô obtained from ΦT_i fori= 0, . . . , m.

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Let ˆT_τ_j = Spec( ˆOT ,τj) be the formal neighbourhood of τ_j in T and ˆKj

be the (t)-adic completion of ˆT_τ_j \ {τ_j} for 1 6j 6N, i.e. ˆK_j is the (t)- adic completion of the punctured formal neighbourhood of τj in T. We have a natural morphism ˆKj →T˜^o. Note that Φ⁰ : V_X⁰ → X⁰ is étale at τ1, . . . , τN ∈ X⁰, since τ1, . . . , τN lie in Xi for some 1 6 i 6m and over X_i, Φ⁰ is induced from a trivial cover. Since Φ⁰ is the restriction of Φ_T to the closed fiber, ΦT is étale over the pointsτ1, . . . , τN inT. Also note that Φ_T_i is the pullback of the H-cover Φ_X_i : W_X_i → X_i which is étale over τ1, . . . , τN. Hence the pullback of Φ^o to ˆKj is the Γ-cover of ˆKj induced from the trivial cover for 16j6N.

Apply Theorem 3.1 to obtain a Γ-cover Ψ :W →T which induces the Γ- cover Φ^oof ˜T^oand trivial Γ-cover over∪^N_j=1Tˆτ_j. The coverW is connected because Γ is generated by G, H₁, . . . , H_m and Γ acts transitively on the fibers of Ind^Γ_GV˜0∪Ind^Γ_H₁V˜1∪ · · · ∪Ind^Γ_H_mV˜m.

Recall that B is the union of the images of the sectionsp₁, . . . , p_r:S→ XS. The branch locus of Ψ is clearly contained inTôbecause in the formal neighbourhood ofτi’s, Ψ restricts to the Γ-cover induced from the trivial cover. Since the pullback of Ψ to ˜Tô is Φô, the branch locus of Φô maps to the branch locus of Ψ under the morphism ˜Tô → Tô. Note that for i = 1, . . . , m, Φ_T_i is étale away from B_i ×_Xô

i

T˜_iô in ˜T_iô where B_i is the set ofri smooth marked points ofX⁰ lying onXi. Moreover, the image of B_i×_Xô

i

T˜_i^o under the morphism ˜T_i^o→T is contained in B×_X_S T. Note that ΦT₀ : ˜V0→T˜0 is the pullback of theG-cover ΦS :VS →XS

under the morphism ˜T₀→T˜^o→T^o→T →X_S. Also the branch locus of ΦS is contained inB. So combining all these we see that Ψ is étale away fromB×X_ST.

Finally if Γ =HoGthenHis a normal subgroup of Γ and by quotienting one obtains aG-cover W/H →T. Since H1, . . . Hm 6H the pullback of W/Hon ˜T^ois ˜V_To. Also the pullback over ˆT_τ_j ofW/H is a trivial cover for 16j 6N. Hence by Theorem 3.1 theG-coversW/H →T and VT →T

are isomorphic. So Ψ dominates Φ_T.

4. Lefschetz–Abhyankar Result

Now we can use a Lefschetz type principle and Abhyankar’s lemma to obtain a Γ-cover ofX étale overCand dominating the givenG-cover. This is a deformation argument similar to [8, Proposition 4].

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Proposition 4.1. — In the context of Proposition 3.7, there exists a Γ-cover Ws → X dominating the G-cover Φ : VX → X which is étale overC.

Proof. — From the conclusion of Proposition 3.7, there is a Γ-cover Ψ : W → T of ˆSs⁰-curves dominating ΦT : VT → T where VT = VS ×S Sˆs⁰. As in the proof of Proposition 3.7, let t be local coordinate ofS at s⁰ so that ˆSs⁰ = Spec(k[[t]]). By hypothesis T =XS×S Sˆs⁰, VS →XS is a G- cover and its fiber ats ∈ S is theG-cover Φ : V_X →X. Also Ψ is étale away from B×X_S T. Since Ψ is a finite morphism, there exists a finitely generated O_S-algebra R ⊂k[[t]] such that the morphisms W →V_T → T descend to the morphisms ofS0= Spec(R)-curvesWS₀ →VS₀ →TS₀. Note that the morphism ˆSs⁰ →S is the composition of the structure morphism π:S0→S and the morphism ˆSs⁰ →S0induced by the inclusionR⊂k[[t]].

SoTS₀ =XS×SS0 andVS₀=VS×SS0, sinceT andVT were base change ofX_S andV_S respectively to ˆS_s⁰. By shrinkingS₀we may assume thatS₀ is smooth and for every point τ ∈ S0 the fiber of the cover WS₀ → TS₀

is a smooth irreducible Γ-cover W_τ → X_π(τ) which is étale away from {p1(π(τ)), . . . , pr(π(τ))}and dominates theG-coverVτ →Xπ(τ).

SinceS0is an affineS-scheme, we choose an embeddingS0,→AⁿS, define S¯0 to be the closure of S0 in PⁿS and let ¯π : ¯S0 → S be the structure morphism. Since ¯π is dominating and projective, it is surjective onto S.

Let TS¯₀ = X_S ×S S¯₀ and VS¯₀ = V_S ×S S¯₀. Note that TS¯₀ → S¯₀ and VS¯0→S¯₀extendT_S₀ →S₀ andV_S₀→S₀respectively.

LetWS¯₀be the normalization ofVS¯₀ink(WS₀). Note thatWS₀ →VS₀ → T_S₀are finite morphism of normal varieties, so it is the restriction ofWS¯₀ → VS¯0→TS¯0. We summarize the setup in the following diagram.

W //

WS₀ //

WS¯0

V_S

V_T

oo //

V_S₀ //

VS¯₀

XS

oo T //

TS₀ //

TS¯₀

S ooii Sˆs⁰ //S0 //S¯0

Let τS ∈S¯0 be such that ¯π(τS) = s. Let ¯A0 be a curve in ¯S0 passing throughτ_S such thatA₀= ¯A₀∩S₀is non empty. Note that ifτ_S∈S₀then the result follows from the assumptions onS0.

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Replacing ¯A₀by an open neighbourhood ofτ_S, we may assume that the fiber at all points of ¯A0of the morphismVA¯₀ →A¯0 are smooth irreducible curves. LetA and ¯Abe the normalization of A0 and ¯A0 respectively. Let τ_A∈A¯be a point lying aboveτ_S ∈A¯₀.

Let B1 be the finitely many points in ¯A\A. Let bi : ¯A → VA¯ be the section obtained by the pullback ofp_i: S→V_S along the composition of the morphisms ¯A→A¯0→S¯0→S. Recall that the sectionspi’s correspond to the deformation of marked points onX (see Proposition 3.7). LetB₂=

∪^r_i=1im(bi) and BV be the union of fibers ∪ζ∈B₁Vζ. Note thatWA¯→VA¯

is anH-cover étale away fromB2∪BV. SinceH is a prime-to-pgroup, the least common multiplemof the ramification indices at the generic points of BV is coprime top. LetA⁰→A¯be a cyclic branched cover totally ramified at the points inB₁with ramification indicesm. LetW_A⁰→V_A⁰ be the pull back ofWA¯ → VA¯. Then applying Abhyankar’s lemma we conclude that WA⁰ →VA⁰ is étale away fromB₂∪BV and it is unramified at the generic points ofBV. SinceWA⁰ →VA⁰ is a finite morphism of normal varieties, the purity of Branch locus implies that the morphism is étale away fromB2.

Letτ_A⁰ ∈A⁰ be a point lying aboveτ_A. The fiber overτ_A⁰ of the covering VA⁰ →TA⁰ isVs→Ts. But this is same asVX →X. SinceA⁰→A¯is proper and all the fibers ofVA¯₀ →A¯₀ are smooth irreducible curves, same is true for the fibers ofVA⁰ →A⁰.

Let b⁰_i : A⁰ →VA⁰ be the sections obtained by the pullback of bi along A⁰→A. After shrinkingA⁰ to an open neighborhood ofτA⁰ if necessary, we may assume that im(bi) and im(bj) are disjoint fori6=j. In particular, the branch locus of the coverW_A⁰ →V_A⁰ is smooth. Moreover, being a prime- to-pcover, it is étale locally a Kummer cover. Hence the fiber over every point τ ∈A⁰ of the cover W_A⁰ →V_A⁰ is a cover of smooth curves. Since the fibers ofWA⁰ →A⁰ are connected for all but finitely many points ofA⁰, Zariski’s connectedness theorem tells us that every fiber ofWA⁰ →A⁰ must be connected. In particular,Wτ_A0 →Tτ_A0 is a Γ cover of smooth connected curves. This cover dominatesVτ_A0 →Tτ_A0 which is same asVX →X.

5. Degenerations of covers and solving embedding problems

In this section we use Proposition 3.7 and 4.1 to solve certain embedding problems. This is used to obtain more examples of effective subgroups of the fundamental group of a curve for a given embedding problem. The

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method below combines the technique of “adding branch points” as in [8]

and “increasing the genus” as in [9, 10, 1].

Theorem 5.1. — LetC be a smooth affine curve overkandX be the smooth completion ofC. Let g be the genus ofX andr= #(X\C). Let E(5.1)denote the embedding problem

(5.1)

π₁(C)

α

||1 //H //Γ ^φ //G //

1

LetΦ :V_X→X be theG-cover ofX étale overC corresponding toα. Let BX = X −C and r = #(BX) and consider the set BX to be r marked points onX. SupposeΦhas a degeneration Φ⁰:V_X⁰ →X⁰ with associated data(S; ΦS :VS →XS;p1, . . . , pr;s;s⁰)and let X1, . . . , Xm be the trivial components of X⁰. Let ri be the number of smooth marked points ofX⁰ lying onX_i fori= 1, . . . , m. Suppose there exists a group homomorphism

θ:π1(X1\ {r1 points})× · · · ×π1(Xm\ {rmpoints})→H/p(H) such that the image ofθis a relative generating set forH/p(H)inΓ/p(H).

Then there exists a Γ-coverW_X → X which corresponds to a proper solution to E(5.1)(i.e. WX → X dominates Φ :VX → X and is étale over C).

Proof. — Let ¯Γ = Γ/p(H), ¯H =H/p(H), andH_i =θ(π₁(X_i\r_i points)) fori= 1, . . . m. Consider the embedding problemE(5.2)

(5.2)

π1(C)

α

}}

1 //H¯ //Γ¯ ^φ //G //

1

This has prime topkernel ¯H. Since the image ofθis a relative generating set for ¯H in ¯Γ, the subgroupsH₁, . . . H_mandGtogether generate ¯Γ. Moreover, by definition ofHi, there exists anHi-cover ofXiétale away fromripoints for 16 i6 m. So the hypotheses of Proposition 3.7 and 4.1 for the em- bedding problemE(5.2) are satisfied. By the conclusion of Proposition 4.1

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there exists a ¯Γ-coverW_s→Xdominating theG-cover Φ :V_X →X which is étale overC. This is a solution toE(5.2).

Since p(H) is a quasi-pgroup, by [6, Corollary 4.6] or [12, Theorem B], the following embedding problem has a solution.

π1(C)

}}

1 //p(H) //Γ ^φ //Γ¯ //

1

This provides the required Γ-cover.

As a consequence of Theorem 5.1, we obtain the following:

Corollary 5.2. — Let C be a smooth affine curve over k and X be the smooth completion ofC. Let g be the genus ofX and r= #(X\C).

LetE(5.1)denote the embedding problem above forπ₁(C)and letµbe the relative rank ofH/p(H)inΓ/p(H). LetΦ :VX →X be the G-cover ofX étale overC corresponding toα. SupposeΦhas a degenerationΦ⁰:V_X⁰ → X⁰ with a trivial componentX₁. Letr₁be the number of marked points of X⁰ lying onX1, g(X1) be the genus ofX1 andnX₁ = 2g(X1) +r1−1. If one of the following holds:

(1) r1>1andnX₁ >µ (2) r₁= 0andg(X₁)>µ

then there exists aΓ-cover ofX dominatingΦwhich is étale overC.

Proof. — Letu1, . . . , uµbe the relative generators ofH/pHin Γ/pH and H⁰ be the subgroup of H/pH generated by u1, . . . , uµ. Note that H⁰ is a prime-to-pgroup generated byµelements, so in both scenario (1) and (2) by [3] there exists an epimorphism fromπ1(X1\ {marked points}) toH⁰.

Hence the corollary follows from Theorem 5.1.

The above result restated in the terminology of effective subgroups be- comes the following statement.

Corollary 5.3. — Let C, X, g, r and µ be as in Corollary 5.2. Let ΠEπ1(C) be of finite index such that the embedding problem E(5.1)restricts to Π. Let Z be the cover of X étale over C with π₁(Z) = Π and Φ :VX×XZ →Z be the inducedG-cover. SupposeΦhas a degeneration with a trivial componentX1 such that one of the following holds:

(1) r1>1andnX₁ >µ

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(2) r₁= 0andg(X₁)>µ

thenΠis an effective subgroup for the embedding problemE(5.1).

Proof. — Note that Φ is a G-cover of Z étale over the preimage of C and it has degeneration with the same properties as in the hypothesis of the above corollary. So using that corollary, we obtain a Γ-cover ofZ dominating Φ which is étale over the preimage of C. This Γ-cover provides a solution to the embedding problemE(5.1) restricted to Π. Hence Π is an

effective subgroup.

6. The case of affine line

Let C be the affine line A¹x = Spec(k[x]) andE = (α:π1(C)→ G, φ : Γ→G) be an embedding problem forπ₁(C) with H = kerφ. In [1, The- orem 1.3], it was shown that there are infinitely many index p effective subgroups ofπ₁(A¹x) for the embedding problemE.

In [8, Theorem 5] (see Proposition 1.2), it was shown that if the étale cover D → A¹x is such that the number of points above x = ∞ in the smooth completion ofDis large enough thenπ1(D) is an effective subgroup ofπ1(C) as long as the embedding problem E restricts toπ1(D).

In this section we will demonstrate some sufficient conditions on a subgroup ofπ1(A¹x) to be effective depending only on rank ofH and the cover corresponding toα. In fact, we will show that in the collection of allp-cyclic étale covers of high enough genus ofA¹x, every member D →A¹x leads to an effective subgroupπ₁(D)Eπ₁(A¹x).

Assume Π is an index pnormal subgroup ofπ1(C). Let D →C be the cover corresponding to Π, i.e.,Dbe the normalization ofCin (K^un)^Πwhere K^un is the compositum of the function fields of all étale covers ofC (in a fixed separable closure of k(C)). Since D is an étale p-cyclic cover of the affine line, by Artin–Schreier theory it is given by the equationz^p−z−f(x) for some non-constant polynomialf(x)∈k[x]. Letrbe the degree off(x).

By changingf(x) if necessary we may assumeris prime top. LetZ→X be the corresponding morphism between their smooth completions. The genus ofZ isgZ = (p−1)(r−1)/2 andZ is totally ramified at infinity. So n_D= 2g_Z+r_D−1 = 2g_Z= (p−1)(r−1).

LetE = (α:π1(C)→G, φ: Γ→G) be an embedding problem forπ1(C) withH = kerφ. As observed in the introduction ifn_Dis less than the rank of Γ/p(Γ) then Π can not be effective. ButnD>rank(Γ/p(Γ)) is certainly not a sufficient condition for Π =π₁(D) to be effective, as indicated by the following example.

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Example. — Let C = A¹, Γ a quasi-p group, G = Z/pZ and H a nontrivial prime-to-p group. Note that rank(Γ/p(Γ)) = 0. Suppose the mapα:π₁(C)→Gbe induced by ap-cyclic étale coverV_C→CwhereV_C is also isomorphic toA¹. LetD →C be any p-cyclic étale cover linearly disjoint fromVC →C and U =D×CVC, then the embedding problemE restricts toπ1(D)Eπ1(C) but ifnU <rank(H/p(H)) then the embedding problemErestricted toπ1(D) has no solution. This is because the existence of a solution to the embedding problem implies thatH/p(H) is a quotient ofπ1(U). But this is impossible if nU <rank(H/p(H)).

Though we shall see that if C = A¹, D → C is p-cyclic étale and nD>2 rank(H/p(H)) then π1(D) is indeed an effective subgroup for the embedding problemE in many cases.

Proposition 6.1. — LetE(6.1)be the embedding problem

(6.1)

π1(A¹)

α

||

1 //H //Γ ^φ //G //

1

LetV →A¹x be theG-Galois cover corresponding toαandV_X→P¹x=X be the morphism corresponding to the smooth completion. Letg be such that there is a homomorphismθfrom the surface groupΠgtoH/p(H)with the property thatim(θ)is a relative generating set forH/p(H)inΓ/p(H).

LetD→A¹xbe an étale p-cyclic cover such that the genus of the smooth completionZ ofD is at leastg. LetV_Z be the normalization ofV_X×XZ. Suppose that the genus of the normalization ofZ⁰×XVX is same asg(VZ) for all but finitelyp-cyclic covers Z⁰ →X branched only at x=∞ with the genus ofZ⁰ same asg(Z). Thenπ1(D)is an effective subgroup for the embedding problemE(6.1).

Proof. — For convenience we first sketch out the idea behind the proof before proceeding with the formal proof. In view of Theorem 5.1, the idea is to construct a degeneration of the cover V_Z →Z to V_s⁰ → T_s⁰ (in the notation below) such thatTs⁰ is the union of the line X and a curveY of genus at leastg. The preimage ofX isV_s⁰ isV_X. And the copy ofY in T_s⁰ is the trivial component (i.e. preimage ofY inVs⁰ is|G|copies of Y). For this one starts with a projective line degenerating to two lines intersecting at a point. This along with a family which parametrizes allp-cyclic covers