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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Proposition 4.2 We can

Proposition 4.2 We can split the short exact sequence (∗) using inertia groups.With respect to this splitting there are the following isomorphisms(Gal(M/L)/C) / (Gal(M/L)/C) Sn∼ = Gal(M ∩ k nr /k)(Gal(M/L)/C) / (Gal(M/L)/C) (1) S∼= Gal(M ∩ K nr /K)n−1where the notations are the ones introduced in Section 3.1If Question 2.14 has an affirmative answer for the finite étale cover Y → X galthen the group C is trivial.PROOF. For every transposition (1 k) we choose a prime ideal Q (1 k),i and denoteby r k the non-trivial element of its inertia group. We denote by ¯r k the image ofr k inside Gal(M/k)/C. The elements ¯r k fulfill ¯r 2 k = 1 and map to (1 k) underthe induced surjection onto Gal(L/k). Since we took the quotient by C also thefollowing relations hold true:(¯r i ¯r i+1 ) 3 = 1 and (¯r i ¯r j ) 2 = 1 for |i − j| ≥ 2.These are precisely the Coxeter relations for S n (cf. Section 5.6) and hence the¯r k define a group isomorphic to a quotient of S n . Since there is a surjectivemap from this group onto S n it must be equal to S n . This defines a splittings : Gal(L/k) → Gal(M/k)/C.From Lemma 4.1 we know that C is a subgroup of N. So we see that the mapfrom Gal(M/k) onto Gal(M ∩ k nr /k) factors over Gal(M/k)/C. The kernelof the map from Gal(M/k)/C onto Gal(M ∩ k nr /k) clearly is the image ¯N ofN inside Gal(M/k)/C. The group ¯N is generated by the images of the inertiagroups.From Lemma 3.1 we know that (Gal(M/L)/C) Sn is generated by the commutators[g, s(τ)]’s where g runs through Gal(M/L) and τ runs through the transpositionsof S n . The element gs(τ)g −1 is the non-trivial element of the inertiagroup of some prime ideal lying above P τ . With this said it is easy to concludethe equalitiesN = (Gal(M/L)/C) Sn · s(S n )and N ∩ (Gal(M/L)/C) Sn = (Gal(M/L)/C) SnApplying the second isomorphism theorem of group theory we obtainGal(M/L)/C(Gal(M/L)/C) Sn==Gal(M/L)/CN ∩ Gal(M/L)/C = N · Gal(M/L)/CNGal(M/k)/CN∼ =Gal(M ∩ k nr /k).28

Hence the induced homomorphism from Gal(L/K)/C to Gal(M ∩ k nr /k) is surjectivewith kernel (Gal(L/K)/C) Sn .The assertion about the quotient of Gal(M/L)/C by (Gal(M/L)/C) (1) Sisn−1proved similarly and left to the reader.Now suppose that the curves corresponding to the Q τ,i ’s fulfill the connectivityproperties of Question 2.14.For two disjoint transpositions τ 1 and τ 2 we choose two prime ideals Q τ1 ,i andQ τ2 ,j and let r 1 and r 2 be the non-trivial elements of their inertia groups. Sincethe curves corresponding to the two prime ideals intersect there is a maximal idealcontaining both of them. The inertia group of this maximal ideal is isomorphic to¡ 2 × ¡ 2 and is generated by r 1 and r 2 . Hence these two elements commute andc(r 1 , r 2 ) = 1.If τ 1 and τ 2 have exactly one index in common then there is a maximal idealwith inertia group S 3 that is generated by r 1 and r 2 . So there is a triple commutatorrelation between r 1 and r 2 and so also c(r 1 , r 2 ) = 1 holds true in thiscase.So if Question 2.14 has an affirmative answer for Yc(r 1 , r 2 )’s are equal to 1 and so C is trivial.→ X gal then all the□We now pass to the limit of all finite étale covers of X gal and keep track of theinduced homomorphisms between the corresponding field extensions and theirGalois groups. We will denote the limit of the subgroups C by C proj . UsingProposition 4.2 we arrive at surjective homomorphismsGal(L nr /L)/C proj ↠ Gal(L nr ∩ k nr /k)Gal(L nr /L)/C proj ↠ Gal(L nr ∩ K nr /K).Taking the compositum of L with K nr we get a subfield of Ω that corresponds toa limit of étale extensions X gal . Hence this compositum must be contained in L nrand hence already K nr was contained in L nr . So K nr ∩ L nr is equal to K nr andthe second surjective homomorphism above takes the formGal(L nr /L)/C proj ↠ Gal(K nr /K).Its kernel is (Gal(L nr /L)/C proj ) (1) S.n−1Up to now have actually never needed that k is the function field of the projectiveplane over the complex numbers. This means that everything done in thissection works equally well in the affine situation. We denote by L nr,aff the compositumof all fields corresponding to finite étale extensions of Xgal aff inside Ω. Wethen define C aff to be the subgroup of Gal(L nr,aff /k) normally generated by thec(r 1 , r 2 )’s where the r i ’s run through inertia groups in this extension.29

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