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On Fundamental Groups of Galois Closures of Generic Projections

5.2 The connection with E( − ,n)Before dealing with the general situation we do the cases d ≤ 2 first:For d = 1 we clearly have S n (1) ∼ = S n for all n ≥ 2.Proposition 5.1 For n ≥ 2 there is an isomorphismE(¡ S n (2) ∼ = , n)s 1 ↦→ (1 2)s 2 ↦→ (1, −1, 0, ..., 0) · (1 2)compatible with the respective split surjections onto S n .PROOF. We consider the following elements **of** S n (2)a := (2 n)(s 2 (1 2))(2 n) −1 · (1 n)τ k := (k k + 1) k = 1, ..., n − 1.The affine reflection group Ãn−1 has the following presentation, c.f. Section 5.6W (Ãn−1) := 〈α, τ k | τ 2 k , (τ k τ k+1 ) 2 , (τ k τ j ) 2 for |k − j| ≥ 2,α 2 , (ατ 1 ) 3 , (ατ n−1 ) 3 , (ατ k ) 2 for k ≠ 1, n − 1〉.We define a map ˜ϕ : W (Ãn−1) → S n (2) by sending α to a and τ k to τ k for allk. The relations inside W (Ãn−1) also hold true for the corresponding elementsin the image i.e. ˜ϕ extends to a homomorphism. In a similar fashion we definea homomorphism in the opposite direction being the inverse **of** ˜ϕ. Hence ˜ϕ is anisomorphism.Finally, we identify W (Ãn−1) E(¡ with , n) using the description given inCorollary 3.6 or Example 5.26.□Remark 5.2 There is a general “Coxeter flavour“ in connection with E(−, n).We refer to Section 5.6 for some examples and details.We let F d−1 be the free group **of** rank d − 1 freely generated by elementsf 2 , ..., f d . We denote by θ the action **of** S n on F d−1 n given by permuting thefactors. We recall that we constructed E(−, n) using such a θ in Section 3.1.We want to define a mapφ : S n (d) → F n d−1 ⋊ θ S ns 1 ↦→ (1 2)s a ↦→ (f a , f −1 a , 1, ..., 1) · (1 2) ∀a = 2, ..., dϕ(σ) ↦→ σ ∀σ ∈ S nwhere ϕ is the splitting that comes together with S n (d). Since we have fixed thesplitting ϕ **of** ψ we consider S n as a subgroup **of** S n (d) and do not mention ϕany further. The content **of** the following theorem is that this map φ is not only ahomomorphism but also injective with image E(F d−1 , n):44

Theorem 5.3 For n ≥ 5 there exists an isomorphismφ : S n (d) ∼ = E(Fd−1 , n) ≤ F n d−1 ⋊ θ S ns 1 ↦→ (1 2)s a ↦→ (f a , f −1 a , 1, ..., 1) (1 2) ∀a ≥ 2compatible with the respective split surjections onto S n .PROOF. First we have to check that φ extends to a homomorphism. For thiswe only have to check that all relations **of** S n (d) hold inside the image. Thesecalculations are straight forward and are done in Lemma 5.5.Also, we see from Lemma 5.5 that the image **of** φ is precisely E(F d−1 , n).Fo a = 2, ..., d and i, j = 1, ..., n we define:ij −1f a := (1, ..., 1, f}{{} a , 1, ..., 1, f a , 1, ..., 1) ∈ F}{{}n d .i.th position j.th positionThese elements generate K(F d−1 , n) as can be seen from applying Lemma 3.1using transpositions as generating set for S n .We want to define a homomorphism from K(F d−1 , n) to S n (d) by sendingˆφ : K(F d−1 , n) → S n (d)iif a ↦→ 1ijf a ↦→ (1 i)(2 j) · (s a (1 2)) · (2 j) −1 (1 j) −1 i ≠ jFrom Proposition 5.6 we know all the relations that hold between the f a ij insideK(F d−1 , n). The relations (∗2) and (∗3) hold true in S n (d) by the relations comingfrom cuspidal transpositions. The relations (∗4) hold true because **of** the relationscoming from nodal transpositions. We leave the details to the reader.By definition φ is the identity when restricted to S n . To show that ˆφ extendsto a homomorphism from K(F d−1 , n) ⋊ θ S n to S n (d) we only have to show thatˆφ is S n -equivariant with respect to the S n -action given by conjugation in bothgroups. We leave it to the reader to show that for σ ∈ S nσ · f aij · σ −1= f aσ −1 (i) σ −1 (j)σ · (1 i)(2 j) · (s a (1 2)) · (2 j) −1 (1 j) −1 · σ −1= (1 σ −1 (i))(2 σ −1 (j)) · (s a (1 2)) · (2 σ −1 j) −1 (1 σ −1 (j)) −1holds true proving S n -equivariance.Hence, there is a homomorphism from E(F d−1 , n) to S n (d) prolonging ˆφ andcompatible with the split surjections onto S n . Since φ is surjective and ˆφ◦φ(s a ) =s a for all a it follows that φ is an isomorphism.□45

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On Fundamental GroupsofGalois Closu

- Page 5 and 6: ContentsIntroductioniii1 A short re
- Page 7 and 8: IntroductionSchon winkt der Wein im
- Page 9 and 10: is defined by a line bundle L on X.
- Page 11 and 12: depends only on G and n and not on
- Page 13 and 14: 1 A short reminder on fundamental g
- Page 15 and 16: Miyaoka [Mi] gave a construction of
- Page 17 and 18: 2 Generic projections and their Gal
- Page 19 and 20: 2.2 Galois closures of generic proj
- Page 21 and 22: Proposition 2.12 Let L be a suffici
- Page 23 and 24: 3 Semidirect products by symmetric
- Page 25 and 26: Remark 3.2 The assumption n ≥ 3 i
- Page 27 and 28: Corollary 3.5 Let n ≥ 2.1. If G i
- Page 29 and 30: The assertions about the torsion an
- Page 31 and 32: For j ≠ i, j ≥ 2 the group X τ
- Page 33 and 34: since the product over all componen
- Page 35 and 36: 4 A first quotient of π 1 (X gal )
- Page 37 and 38: 4.2 The quotient for the étale fun
- Page 39 and 40: of K. The normalisation of X inside
- Page 41 and 42: Hence the induced homomorphism from
- Page 43 and 44: zero. Hence the proof also works al
- Page 45 and 46: an isomorphism between these two gr
- Page 47 and 48: In particular, for H = 1 we obtain
- Page 49 and 50: For an element g of the inertia gro
- Page 51 and 52: eThis automorphism has order e i an
- Page 53 and 54: Hence the homomorphisms from π top
- Page 55: 5 A generalised symmetric group5.1
- Page 59 and 60: Proposition 5.6 (Rowen, Teicher, Vi
- Page 61 and 62: First, assume that i = n. ThenNow a
- Page 63 and 64: We note that for affine subgroups n
- Page 65 and 66: Every element of H 2 (G) maps to an
- Page 67 and 68: PROOF. We let F d /N ∼ = G and F
- Page 69 and 70: 5.4 ExamplesWe now compute ˜K(−,
- Page 71 and 72: Theorem 5.22 For a K(G, 1)-complex
- Page 73 and 74: Example 5.24 The homomorphism ψ ma
- Page 75 and 76: 6 ConclusionJetzt nehmt den Wein! J
- Page 77 and 78: This can be also formulated as foll
- Page 79 and 80: Severi claimed that a curve with on
- Page 81 and 82: where ϕ denotes the splitting of
- Page 83 and 84: Corollary 6.3 Under the assumptions
- Page 85 and 86: Proposition 6.5 Under the isomorphi
- Page 87 and 88: Remark 6.8 Proposition 6.5 shows us
- Page 89 and 90: We remark that the group on the lef
- Page 91: E(π top1 (Z), n). By Corollary 3.3
- Page 94 and 95: 7.3 Surfaces in 3Let X m be a smoot
- Page 96 and 97: If we denote by Π g the fundamenta
- Page 98 and 99: NotationsVarieties and morphismsf :
- Page 100 and 101: [GR1][GR2][GH][SGA1]H. Grauert, R.