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On Fundamental Groups of Galois Closures of Generic Projections

If τ 1 and τ 2 have exactly one index in common then there is a point with inertiagroup S 3 that is generated by r 1 and r 2 . So there is a triple commutator relationbetween r 1 and r 2 and so also c(r 1 , r 2 ) = 1 holds true in this case.So if Question 2.14 has an affirmative answer for the universal cover ˜X gal **of**X gal then all the c(r 1 , r 2 )’s are equal to 1 and so C proj is trivial.□Again, everything said so far can also be done in the affine setup. We thendefine C aff to be the normal subgroup **of** π top1 (Xgal aff,S n) defined by the c(r 1 , r 2 )’swhere the r i ’s run through the inertia groups corresponding to some universalS n -cover **of** Xgal aff . We then getTheorem 4.7 Let f : X →2 be a good generic projection **of** degree n with**Galois** closure X gal . Then there are surjective homomorphismsπ top1 (X gal ) ↠ π top1 (X gal )/C proj ↠ K(π top1 (X), n)π top1 (Xgal aff)↠ πtop 1 (Xgal aff)/Caff↠ K(π top1 (X aff ), n).If Question 2.14 has an affirmative answer for the universal cover **of** X affgal thenboth C aff and C proj are trivial.If Question 2.14 has an affirmative answer for the universal cover **of** X gal thenat least C proj is trivial.Again, even if C aff is trivial we cannot expect these surjective homomorphisms tobe isomorphisms. We refer to Theorem 6.2 for details.Corollary 4.8 For a good generic projection f : X →surjective and non-canonical homomorphisms2 **of** degree n there areH 1 (X gal , ¡ ) ↠ (H 1 (X, ¡ )) n−1H 1 (X affgal , ¡ ) ↠ ( H 1 (X aff , ¡ ) ) n−1 .PROOF. From Morse theory ([Mil]) it is known that smooth affine and smoothprojective varieties are CW-complexes. So we can apply Hurewicz’s theorem thatH 1 ¡ (−, ) is isomorphic to the abelianised fundamental group.Thus our statement follows from the fact that K(−, n) for n ≥ 3 commuteswith abelianisation by Proposition 3.8 and the computation **of** K(−, n) for abeliangroups given by Corollary 3.5.□42

5 A generalised symmetric group5.1 Definition **of** S n (d)Seht doch hinab! Im Mondschein auf den GräbernHockt eine wild-gespenstische Gestalt!Ein Aff ist’s! Hört ihr, wie sein HeulenHinausgellt in den süßen Duft des Lebens?We let τ k be the transposition (k k + 1) **of** S n . From the theory **of** Coxeter groups(cf. also Section 5.6) it is known that S n admits a presentation asS n = 〈 τ k , k = 1, ..., n − 1 | τ k 2 , (τ k τ k+1 ) 3 , (τ k τ j ) 2 ∀|k − j| ≥ 2 〉 .Let d ≥ 1 and n ≥ 3 be natural numbers. We want to construct a generalisedsymmetric group where we have d copies **of** the transposition (1 2). For this welet s 1 , ..., s d be free generators **of** the free group F d **of** rank d. Then we define thegroup( )˜S n (d) := F d ∗ S (1)n−1 /Rwhere R is the subgroup normally generated by the following elementss i2for i = 1, ..., d(s i · τ 2 ) 3 for i = 1, ..., d(s i · τ k ) 2 for k ≥ 3 and i = 1, ..., d.The reader will identify this group as the d-fold amalgamated sum **of** S n withitself where we amalgamate the subgroup S (1)n−1 in every summand.Every summand has a map (the identity) onto S n that is compatible with thesubgroup that is amalgamated. These homomorphism patch together to a homomorphismψ onto S n . Sending S n via the identity to the first summand we obtaina splitting ϕ **of** ψ.But we still want more relations to hold true: We defineS n (d) := ˜S n (d)/R ′where R ′ is the subgroup normally generated by the following elements:(ϕ(σ)s i ϕ(σ) −1 · s j ) 2(ϕ(σ)s i ϕ(σ) −1 · s j ) 3if σ(1 2)σ −1 and (1 2) are nodal transpositionsif σ(1 2)σ −1 and (1 2) are cuspidal transpositionsThe homomorphisms ψ and ϕ induce homomorphisms on the quotient S n (d) thatwe will call by abuse **of** notation again by ψ and ϕ.43

- 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 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: Hence the homomorphisms from π top
- 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 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.