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On Fundamental Groups of Galois Closures of Generic Projections

PROOF. If G is abelian (resp. finite, nilpotent, solvable) then so is K(G, n) beinga subgroup **of** G n . If G is perfect then K(G, n) ∼ = G n which is perfect.By the first exact sequence **of** Proposition 3.4 there exists a surjective homomorphismfrom K(G, n) onto G. So if K(G, n) is abelian (resp. finite, nilpotent,perfect, solvable) then so is G being a quotient **of** K(G, n).□We finally give some basic functoriality properties **of** our construction:Proposition 3.8 Let n ≥ 2 be a natural number and let G 1 , G 2 , G be arbitrarygroups.1. If G 1 → G 2 is an injection then so is K(G 1 , n) → K(G 2 , n).2. If G 1 → G 2 is a surjection then so is K(G 1 , n) → K(G 2 , n).3. If G is a semidirect product then so is K(G, n). However, the functorK(−, n) is not exact in the middle as Example 3.11 and Example 3.13 show.4. K(G 1 × G 2 , n) = K(G 1 , n) × K(G 2 , n).5. If G is an abelian group thenK(G, n) tors∼ = K(Gtors , n)K(G, n) ⊗¦§¥ ∼ = K(G ⊗¦§¥ , n)where − tors denotes the torsion subgroup **of** an abelian group.6. For n ≥ 3 the natural homomorphism from K(G, n) ab onto K(G ab , n) is anisomorphism. The assumption n ≥ 3 is needed as Example 3.12 shows.PROOF. We assume that 1 → K → G → Q → 1 is exact. Then also the inducedsequence 1 → K n → G n → Q n → 1 is exact. This induces homomorphisms(notation as in the beginning **of** this section)K n ⋊ θ S n → G n ⋊ θ S n → Q n ⋊ θ S nand induces injections E(K, n)↩→E(G, n) and K(K, n)↩→K(G, n). This provesthe first assertion (we do not need the normality **of** K in G in this step). Thegroup E(Q, n) is generated by S n and commutators [q, σ], q ∈ Q. Since G↠Qis surjective we see that E(G, n)↠E(Q, n) is surjective since we can lift elements**of** S n and commutators. Similarly we see that K(G, n)↠K(Q, n) is surjective.If G is a semidirect product then there exists a split surjection G↠Q. Thismap induces a split surjection K(G, n)↠K(Q, n). Therefore also K(G, n) is asemidirect product.16

The assertions about the torsion and the free part **of** an abelian group followimmediately from Corollary 3.5.The surjection G↠G ab and the universal property **of** abelianisation imply thatthere is a natural surjective homomorphism K(G, n) ab ↠K(G ab , n):1 → K(G, n) → G n → G ab → 1↓ ↓ ab ||1 → K(G ab , n) → (G ab ) n → G ab → 1 .An element **of** the kernel K(G, n) → K(G ab , n) is also an element **of** the kernel**of** G n → (G ab ) n which is [G, G] n . Since we assumed n ≥ 3 every commutator(1, ..., 1, [h 1 , h 2 ], 1..., 1) lies not only in K(G, n) but is even a commutator **of** elements**of** K(G, n), cf. the pro**of** **of** Proposition 3.3. This implies that the kernel**of** K(G, n) → K(G, n) ab is the commutator subgroup **of** K(G, n). Hence thecanonical homomorphism from K(G, n) ab onto K(G ab , n) is an isomorphism forn ≥ 3.□3.3 UniversalityWe assume that we are given a group X and a homomorphism ϕ : S n → Aut(X)with n ≥ 3. Then we form the semidirect product1 → X → X ⋊ ϕ S n → S n → 1.We consider S n as a subgroup **of** the group in the middle via the associated splitting.For a subgroup S ≤ S n we denote [X, S] by X S . Again, X S is a normalsubgroup **of** X and does not change if we pass to an X-conjugate splitting.Proposition 3.9 Let ϕ : S n → Aut(X), n ≥ 3 be a homomorphism and let1 → X → X ⋊ ϕ S n → S n → 1be the split extension determined by ϕ. If we defineY := X Sn /X S(1)n−1then there exists a commutative diagram with exact rows1 → X Sn → X Sn ⋊ ϕ S n → S n → 1↓ ↓ ‖1 → K(Y, n) → K(Y, n) ⋊ θ S n → S n → 1where all homomorphisms downwards are surjective. Moreover, we have an exactsequence1 → ⋂ ni=1 X S (i)n−1→ X Sn → K(Y, n) → 1.17

- 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 and 24: 3 Semidirect products by symmetric
- Page 25 and 26: Remark 3.2 The assumption n ≥ 3 i
- Page 27: Corollary 3.5 Let n ≥ 2.1. If G i
- 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
- 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

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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.