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On Fundamental Groups of Galois Closures of Generic Projections

For both groups π top1 (¢ 2 − D) and π topPROOF.1 ( 2 − D) there are surjectivehomomorphisms ψ onto S n that are compatible with ı ∗ . Since δ is trivial inπ top1 ( 2 − D) we conclude that ψ(δ) = 1.By a theorem **of** Oka (cf. [FL, Corollary 8.4]) the short exact sequence0 → A → π top1 (¢ 2 − D)ı ∗→πtop1 ( 2 − D) → 1is central. We know that A is normally generated by δ and hence δ must be acentral element **of** π top (¢ 1 2 − D). Of course δ remains central in every quotient **of**π top (¢ 1 2 − D).We recall the short exact sequences1 → π top1 (X affgal ) → πtop 1 (¢ 2 − D)/ ≪ Γ i 2 ≫ ψ → S n → 1↓ ↓ ı ∗ ||1 → π top1 (X gal ) → π top1 ( 2 − D)/ ≪ Γ i 2 ≫ ψ → S n → 1We already noted that the kernel **of** the surjective homomorphism ı ∗ is generatedby δ. Since ψ(δ) = 1 the loop δ lies in π top1 (Xgal aff ). This yields the first exactsequence.There exist surjective homomorphismsπ top1 (Xgal aff,n−1 ) ↠ πtop 1 (X aff )↓↓π top1 (X gal , S (i)n−1) ↠ π top1 (X)The kernel **of** the upper horizontal homomorphism N is generated by inertiagroups. The kernel **of** the lower horizontal homomorphism is generated by theimage **of** N from above. The kernel **of** the left arrow downwards is generated byδ. Chasing around this diagram we find that the kernel **of** the surjective map fromπ top1 (X aff ) onto π top1 (X) is generated by ¯δ. □In Theorem 6.2 we constructed an isomorphismπ top1 (X affgal )/Caff ∼ =˜K(πtop1 (X aff ), n).Since δ is central it is stable under the S n -action on the right. The same holds truewhen passing to the quotient K(π top1 (X aff ), n). So if we consider K(π top1 (X aff ), n)as a subgroup **of** π top1 (X aff ) n then δ maps to an element **of** the diagonal. AndProposition 6.4 tells us exactly what this element is:72

Proposition 6.5 Under the isomorphism **of** Theorem 6.2 and the surjective map**of** Theorem 4.7 the loop δ maps as followsπ top1 (Xgal aff)/Caff∼ top = ˜K(π 1 (Xgal aff ), n) ↠ K(πtop 1 (X aff ), n)δ ↦→ δ ↦→ (¯δ, ..., ¯δ)where ¯δ is the central element **of** Proposition 6.4.□Again, we can say a little bit more for the abelianisationCorollary 6.6 We keep the notations and assumptions **of** Theorem 6.2. Then wedenote by ¯C proj the image **of** C proj in the abelianised fundamental group **of** X gal .Then there exists an isomorphismH 1 (X gal , ¡ )/ ¯C proj ∼ = H1 (X, ¡ ) ⊕ H 1 (X aff , ¡ ) n−2 .In particular, if π top1 (X aff ) is abelian thenπ top1 (X gal )/C proj ∼ = πtop1 (X) × π top1 (X aff ) n−2 .We note that these isomorphisms are not canonical.To increase readability, we abbreviate H 1 ¡ (−, ) just by H 1 (−).Since abelianisation is not an exact functor we have to proceed by hand:PROOF.1 → 〈δ〉 → π top1 (Xgal aff)→ πtop 1 (X gal ) → 1↓ ↓ ↓〈δ ′ 〉 ↩→ H 1 (X affgal ) ↠ H 1(X gal )where δ ′ denotes the image **of** δ in H 1 (Xgal aff).Let x be an element **of** H 1(X affthat maps to 0 in H 1 (X gal ). We can lift this to an element ˜x **of** π top1 (Xgal aff)thathas to map to a product **of** commutators in π top1 (X gal ) by commutativity **of** thediagram. But this means that ˜x is a product **of** δ s for some integer s times somecommutators. Changing ˜x by commutators we still get a lift **of** x. So we mayassume that ˜x actually equals δ s . Therefore, x is equal to ¯δ s . This shows that wehave an exact sequence1 → 〈¯δ〉 → H 1 (X affgal ) → H 1(X gal ) → 1.We denote by ¯δ the image **of** δ in π top1 (X aff ) we know from Proposition 6.4 that thesubgroup generated by ¯δ inside π top1 (X aff ) is equal to the kernel **of** the projectionπ top1 (X aff )↠π top1 (X). So we obtain another exact sequence1 → 〈¯δ ′ 〉 → H 1 (X aff ) → H 1 (X) → 173gal )

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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

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3 Semidirect products by symmetric

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Remark 3.2 The assumption n ≥ 3 i

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Corollary 3.5 Let n ≥ 2.1. If G i

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The assertions about the torsion an

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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
- 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: Corollary 6.3 Under the assumptions
- 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.