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On Fundamental Groups of Galois Closures of Generic Projections

7 Examples7.1 2Let X :=2 be the complex projective plane.For k ≥ 5 the line bundle L k O© := 2(k) is sufficiently ample by Lemma2.2. Combining Proposition 2.5 with Proposition 2.8 we see that a generic threedimensionallinear subspace **of** H 0 ( 2 , L k ) gives rise to a good generic projectionthat we denote by f k .Proposition 7.1 Let X gal be the **Galois** closure **of** a good generic projection f k .Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = k k2 −1π top1 (X gal )/C proj ∼ = k k2−2 .PROOF. The morphism f k has degree n = deg f k = k 2 . The divisibility index**of** L k in Pic(X) is k and we only have to plug in this data into Theorem 6.11. □Remark 7.2 The results **of** Moishezon and Teicher [MoTe2] show that C aff andC proj are trivial.7.2 1 × 1Let X := 1 × 1 .For a ≥ 5 and b ≥ 5 the line bundle L (a,b) := O© 1 ×© 1(a, b) is sufficientlyample, cf. Lemma 2.2. Combining Proposition 2.5 with Proposition 2.8 we seethat a generic three-dimensional linear subspace **of** H 0 ( 1 ×1 , L (a,b) ) gives riseto a good generic projection that we denote by f (a,b) .Proposition 7.3 Let X gal be the **Galois** closure **of** a good generic projection f (a,b) .Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = gcd(a,b) 2ab−1π top1 (X gal )/C proj ∼ = gcd(a,b) 2ab−2 .PROOF. The morphism f (a,b) has degree n = deg f (a,b) = 2ab. The divisibilityindex **of** L (a,b) in Pic(X) is gcd(a, b) and we only have to plug in this data intoTheorem 6.11.□Remark 7.4 The results **of** Moishezon and Teicher [MoTe1] and [MoTe4] showthat C aff and C proj are trivial.81

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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

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since the product over all componen

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4 A first quotient of π 1 (X gal )

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4.2 The quotient for the étale fun

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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 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 95 and 96: Remark 7.7 Using the results of Moi
- Page 97 and 98: Moishezon and Teicher [MoTe2, Propo
- Page 99 and 100: References[ADKY][BHPV][Bea]D. Aurou
- Page 101: [MoTe3][MoTe4]B. Moishezon, M. Teic