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On Fundamental Groups of Galois Closures of Generic Projections

Lemma 3.1 Let S be a subgroup **of** S n , n ≥ 3 that is generated by transpositions.ThenE(G, n) Sdef= 〈⃗g σ ⃗g −1 |⃗g ∈ G n , σ ∈ S〉= 〈⃗g σ ⃗g −1 |⃗g ∈ K(G, n), σ ∈ S〉K(G, n) Sdef= 〈[⃗g, σ] |⃗g ∈ G n , σ ∈ S〉= 〈[⃗g, σ] |⃗g ∈ K(G, n), σ ∈ S〉.Moreover, it is enough that σ runs through a system **of** generating transpositions**of** S in the expressions above.PROOF. We will first assume that S = 〈τ〉 for the transposition τ = (1 2) **of** S n .For (g 1 , ..., g n ) ∈ G n we calculate(g 1 , ..., g n )τ(g 1 , ..., g n ) −1 = (g 1 g 2 −1 , g 2 g 1 −1 , 1, ..., 1)τ.In this case the subgroup K(G, n) S **of** G n is generated by (g, g −1 , 1, ..., 1), g ∈ G.Since we assumed n ≥ 3 we may consider the element (g, 1, g −1 , 1, ...). Byapplying the previous calculation to the transposition (1 3) this is also an element**of** K(G, n). From(g, 1, g −1 , 1, ..., 1)τ(g, 1, g −1 , 1, ..., 1) −1 = (g, g −1 , 1, ..., 1)τwe deduce that 〈⃗gσ⃗g −1 |⃗g ∈ K(G, n), σ ∈ S〉 is generated by the same elementsas E(G, n) S . So both subgroups are equal. A similar calculation yields the resultfor K(G, n) S .We now let S be a subgroup **of** S n generated by transpositions. Then we canwrite σ ∈ S as a product τ 1 · ... · τ d **of** transpositions all lying in S. For ⃗g ∈ G nwe get( d∏ d∏⃗gσ⃗g −1 = ⃗g τ i)⃗g −1 = ⃗gτ i ⃗g −1 .i=1j=iWe have seen above that all ⃗gτ i ⃗g −1 can be written as products **of** **of** conjugates **of**τ i under K(G, n). So ⃗gσ⃗g −1 can be written as a product **of** K(G, n)-conjugates **of**elements **of** S.To prove the remaining assertion we assume that σ ∈ S can be written as aproduct **of** d transpositions **of** S. The case d = 1 was already done above. We canfind a transposition τ and an element ν that can be written as a product **of** strictlyless that d transpositions such that σ = τ · ν. Then also τντ −1 can be written as aproduct **of** strictly less than d transpositions and writing[⃗g, τν] = [⃗g, τ] · τ[⃗g, ν]τ −1 = [⃗g, τ] · [τ⃗gτ −1 , τντ −1 ]we can apply induction.□12

Remark 3.2 The assumption n ≥ 3 is crucial:1. If n = 1 then E(G, 1) = K(G, 1) = 1.2. If n = 2 then K(G, 2) is the subgroup **of** G 2 generated by (g, g −1 ) and〈⃗gσ⃗g −1 |⃗g ∈ G 2 , σ ∈ S 2 〉 = 〈(g, g −1 ), g ∈ G〉 = K(G, 2)〈⃗gσ⃗g −1 |⃗g ∈ K(G, 2), σ ∈ S 2 〉 = 〈(g 2 , g −2 ), g ∈ G〉 ≤ K(G, 2).So in this case it depends on the structure **of** G whether these two subgroupscoincide.3.2 PropertiesThere exists a quite different description **of** K(−, n) given by the followingProposition 3.3 Let n ≥ 2 be a natural number and G be an arbitrary group.ThenK(G, n) = ker ( G n → G ab )(g 1 , ..., g n ) ↦→ g 1 · ... · g nas subgroups **of** G n .PROOF. Lemma 3.1 tells us that K(G, n) is generated by elements **of** the form(1, ..., 1, g, 1, ..., 1, g −1 , 1, ..., 1). Since these elements lie in the kernel **of** the mapG n → G ab it follows that we already have K(G, n) ≤ ker(G n → G ab ).Conversely, suppose that (g 1 , ..., g n ) lies in the kernel **of** G n → G ab . Multiplyingby (1, ..., g n , g −1 n ) we obtain an element **of** the form (g 1 , ..., g n−1, ′ 1). Wemultiply this element by (1, ..., 1, g n−1 ′ , g′ n−1 −1 , 1). Proceeding inductively, we seethat every element **of** ker(G n → G ab ) can be changed by elements from K(G, n)to an element **of** the form (g 1, ′ 1, ..., 1). Then necessarily g 1 ′ ∈ [G, G]. This meansthat g 1 ′ is a product **of** commutators. Since n ≥ 2 we can write a commutator as aproduct **of** elements **of** K(G, n):[(h 1 , 1, ..., 1), (h 2 , 1, ..., 1)]= (h 1 , h 1 −1 , 1, ..., 1) (h 2 , h 2 −1 , 1, ..., 1) ((h 2 h 1 ) −1 , (h 2 h 1 ), 1, ..., 1).For computations later on we remark that for n ≥ 3 such a commutator is even acommutator **of** elements **of** K(G, n):[(h 1 , 1, ..., 1), (h 2 , 1, ..., 1)] = [(h 1 , h 1 −1 , 1, ...), (h 2 , 1, h 2 −1 , 1, ...)].Hence every element **of** ker(G n → G ab ) is a product **of** elements **of** K(G, n). Thisproves the converse inclusion and so we are done.□13

- Page 1: 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: 3 Semidirect products by symmetric
- 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 and 56: 5 A generalised symmetric group5.1
- Page 57 and 58: Theorem 5.3 For n ≥ 5 there exist
- 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

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This can be also formulated as foll

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Severi claimed that a curve with on

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where ϕ denotes the splitting of

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Corollary 6.3 Under the assumptions

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Proposition 6.5 Under the isomorphi

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Remark 6.8 Proposition 6.5 shows us

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We remark that the group on the lef

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E(π top1 (Z), n). By Corollary 3.3

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7.3 Surfaces in 3Let X m be a smoot

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If we denote by Π g the fundamenta

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NotationsVarieties and morphismsf :

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[GR1][GR2][GH][SGA1]H. Grauert, R.