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On Fundamental Groups of Galois Closures of Generic Projections

Remark 5.4 The really hard part **of** this pro**of** is Proposition 5.6. It says thatthe relations **of** K(F d , n) are only some “obvious“ ones, i.e. a certain set **of**commutator relations.The author’s original pro**of** used a Reidemeister-Schreier rewriting process toobtain a presentation **of** the subgroup E(F d−1 , n) **of** F d−1 n ⋊ θ S n . However, sincethe subgroup has infinite index in the ambient group he obtained an infinite set **of**relations. The computations were a ten page flow **of** quite messy calculations.Meanwhile, [RTV] appeared and the author decided to copy their pro**of**.Lemma 5.5 Let G be an arbitrary group and ⃗g i , i = 1, 2 two elements **of** K(G, n).We defines i := ⃗g i (1 2)⃗g i −1 , i = 1, 2Then the following relations hold inside E(G, n)2s i i = 1, 2(s i · τ) 2 if τ and (1 2) are nodal transpositions(s i · τ) 3 if τ and (1 2) are cuspidal transpositions(σs i σ −1 · s j ) 2 if σ(1 2)σ −1 and (1 2) are nodal transpositions(σs i σ −1 · s j ) 3 if σ(1 2)σ −1 and (1 2) are cuspidal transpositions.If n ≥ 3 and if the elements g 1 , ..., g s generate G then E(G, n) is generated by[(g i , 1, ..., 1), (1 2)] and an arbitrary generating set **of** S n .PROOF. The first relation is straight forward from Lemma 3.1. Furthermore itallows us to view the remaining relations as commutator relations or triple commutatorrelations, respectively.We do the computations inside G n ⋊ S n as usual. We set τ = (3 4) and⃗g = (g 1 , g 2 , ..., g n ) ∈ G n , and check that ⃗g(1 2)⃗g −1 and τ commute:((⃗g(1 2)⃗g −1 ) · τ) 2= [⃗g(1 2)⃗g −1 , τ]= ⃗g(1 2)⃗g −1 · τ ( (g 1 g −12 , g 2 g −11 , 1, ..., 1) −1 (1 2) −1) τ −1= ⃗g(1 2)⃗g −1 · τ ( (g 1 g −12 , g 2g −11 , 1, ..., 1)−1) τ −1 (1 2) −1= ⃗g(1 2)⃗g −1 · (g 1 g −12 , g 2 g −11 , 1, ..., 1) −1 (1 2) −1= ⃗g(1 2)⃗g −1 · (⃗g(1 2)⃗g −1 ) −1= 1We leave the remaining relations to the reader.We have already seen in Lemma 3.1 that E(G, n) is generated by S n and allelements **of** the form (g, g −1 , 1, ..., 1). Let g 1 , ..., g s be a generating set for G. Wedefine ⃗g i := (g i , 1, ..., 1) and compute for n ≥ 3[⃗g i , (1 3)] · [⃗g j , (1 2)] · [⃗g i , (1 3)] = (g i g j , (g i g j ) −1 , 1, ..., 1)So we get all elements (g, g −1 , 1, ...., 1) from the set [⃗g i , (1 2)] and S n .46□

Proposition 5.6 (Rowen, Teicher, Vishne) We let F d be the free group **of** rank dand assume that it is freely generated by elements f 1 , ..., f d . We set:ij −1nf a := (1, ..., 1, f }{{} a , 1, ..., 1, f a , 1, ..., 1) ∈ F}{{}di.th position j.th positionIf n ≥ 2 then K(F d , n) is generated by f a ij with a = 1, ..., d and i, j = 1, ..., n.And if n ≥ 5 then all relations inside K(F d , n) follow from the following relations:iif a = 1 (∗1)ij jkf a · f a =ikf a (∗2)ik ij ikf[ a · f a = f a (∗3)fa , f ] kl b = 1 if i, j, k, l are all different. (∗4)In other words we have a finite presentation **of** K(F d , n) for n ≥ 5.PROOF. The pro**of** is taken from [RTV, Theorem 5.7]. However, we adapted thenotations to our situation.First **of** all, the f ij a ’s generate K(F d , n). This follows from Lemma 3.1 appliedto the generating set f i **of** F d and taking as generating set for S n the set **of** alltranspositions.We leave it to the reader to show that the relations given in the statement **of**Proposition 5.6 hold true in F n d and hence in K(F d , n).We define K d,n to be the group generated by elements f ij a with a = 1, ..., dand i, j = 1, ..., n subject to the relations given by Proposition 5.6. We haveshown above that there is a surjective homomorphism from K d,n onto K(F d , n).Next, we define Kd,n ∗ to be the group generated by elementsf aijand t a with a = 1, .., d, i, j = 1, ..., nsubject to the relations **of** K d,n and the relations[t a , f ij b ] = [ f nk a , f ] ij b k ≠ i, j (†1)[t a , t b ] = [f ni a , f nj b ] i ≠ j and i, j ≠ n (†2)Then we define the following mapµ : K ∗ d,n → F dnt a ↦→ f anf aij↦→ (f a j ) −1 f aiwhere f a i denotes the element (1, ..., 1, f a , 1, ..., 1) **of** F d n having its non-trivialentry in the i.th position. By Lemma 5.7 this map µ defines an isomorphism **of**groups.47

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