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- Generic,
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- Homomorphism,
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- Galois,
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On Fundamental Groups of Galois Closures of Generic Projections

Under the isomorphism given in the example the “extra“ reflection maps to theelement (1, 0, ..., 0, −1)(1 n) **of** E(¡ , n)The upper chain forms a subgraph **of** type A n−1 inside ˜D n . This defines asubgroup isomorphic to S n inside W (˜D n ). We define a split surjectionψ : W (˜D n ) ↠ S nbeing the identity when restricted to the subgroup S n and sending a remainingreflection to the image **of** the respective reflection “lying above“ it in the graph˜D n . We leave it to the reader to show that we get theExample 5.27 The homomorphism ψ makes ker ψ into K(D ∞ , n) and induces anisomorphismW (˜D n ) ∼ = E(D∞ , n)where D ∞ denotes the infinite dihedral group.62

6 ConclusionJetzt nehmt den Wein! Jetzt ist es Zeit, Genossen!Leert eure gold’nen Becher zu Grund!Dunkel ist das Leben, ist der Tod!6.1 The algorithm **of** Zariski and van KampenLet C be a reduced but not necessarily smooth or irreducible projective curve **of**degree d in the complex projective plane. We choose a generic line ˜l ⊂2 , i.e.a line that intersects C in d distinct points. We ¢ set 2 :=2 − ˜l and denote theintersection C ¢ ∩ 2 again by C. We are interested in computing the fundamentalgroupsπ top1 ( 2 − C) and π top (¢ 1 2 − C).An algorithm that yields presentations **of** these groups is given in van Kampen’sarticle [vK]. The result was known to Zariski before and also Enriques, Lefschetzand Picard should be mentioned in this context.We now follow [Ch] and [Mo] to describe this algorithm: We choose a genericline l in ¢ 2 , i.e. a line intersecting C in d distinct points. The inclusion mapsinduce group homomorphisms(¢(¢π top1 2 − C) → π top1 ( 2 − C)π top1 (l − l ∩ C) → π top1 2 − C).Both homomorphisms are surjective. A modern pro**of** for this is for example givenby [N, Proposition 2.1] and its corollaries.The underlying topological space **of** l − l ∩ C can be identified with £ 2 withd points cut out. Hence its fundamental group is the free group **of** rank d. To get asystem **of** d generators we may proceed as follows: We let u 0 be the base point forthe fundamental group **of** l − l ∩ C. We let w 1 ,...,w d be the points **of** l ∩ C. Nextwe choose paths γ i from u 0 to w i for all i = 1, ..., d and assume that distinct γ i ’smeet only in u 0 . Next we shorten the γ i ’s such that they stop before reaching theirw i ’s. Putting a little circle around w i at the end **of** the so shortened γ i ’s we obtainloops Γ i that lie in l − l ∩ C. Loops like this are usually called simple loops andwe already met them in Section 4.4. These Γ i ’s freely generate the fundamentalgroup **of** l − l ∩ C:π top1 (l − l ∩ C, u 0 ) = 〈Γ i , i = 1, ..., d〉 ∼ = Fd .We consider the closure ¯l **of** l inside2 and denote by ∞ := ¯l − l the point atinfinity. We may put an orientation on the Γ i ’s and order them in such a way that63

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On Fundamental GroupsofGalois Closu

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ContentsIntroductioniii1 A short re

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IntroductionSchon winkt der Wein im

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is defined by a line bundle L on X.

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depends only on G and n and not on

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1 A short reminder on fundamental g

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Miyaoka [Mi] gave a construction of

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2 Generic projections and their Gal

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2.2 Galois closures of generic proj

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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
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- Page 31 and 32: For j ≠ i, j ≥ 2 the group X τ
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- 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: Example 5.24 The homomorphism ψ ma
- 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
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- 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.