- Text
- Generic,
- Subgroup,
- Fundamental,
- Element,
- Proposition,
- Generated,
- Homomorphism,
- Finite,
- Quotient,
- Projection,
- Galois,
- Closures,
- Projections

On Fundamental Groups of Galois Closures of Generic Projections

obtain the following two short exact sequences1 → π top1 (X gal ) → π top1 (X gal , S n ) → S n → 1|| ↑ ↑1 → π top1 (X gal ) → π top1 (X gal , S (1)n−1 ) → S(1) n−1 → 1The arrows upwards are injective. We now fix a universal cover ˜X gal **of** X gal andalso do not mention base points unless it is important for our considerations.The quotient S n \X gal is isomorphic to2 and the inertia groups generate thekernel **of** the homomorphism from π top1 (X gal , S n ) onto π top1 (S n \X gal ). Since thislatter group is trivial it follows that the inertia groups generate π top1 (X gal , S n ).The quotient S (1)n−1\X gal is isomorphic to X. So the kernel **of** the surjectivehomomorphism onto the fundamental groups **of** X is the the subgroup normallygenerated by the inertia groups contained in π top1 (X gal , S (1)n−1).By Proposition 2.12 the ramification divisor R gal **of** f gal : X gal →2 is theunion **of** the curves R τ where τ runs through the transpositions **of** S n . We denoteby ˜p : ˜X gal → X gal the universal cover **of** X gal . Then we let ˜R τ be a connectedcomponent **of** ˜p −1 (R τ ). We have seen in the previous section that there is a uniqueinertia automorphism **of** the universal S n -cover ˜X gal × S n that sends ˜R τ × {1}to ˜R τ × {τ}. Since the inertia group **of** R τ is ¡ 2 this automorphism is the onlynon-trivial inertia automorphism **of** ˜R τ .We let τ 1 and τ 2 be two transpositions **of** S n and choose two components˜R τ1 and ˜R τ2 **of** ˜p −1 (R τ1 ) and ˜p −1 (R τ2 ), respectively. For the non-trivial inertiaelements r 1 and r 2 **of** their inertia groups we set (cf. Definition 2.11)⎧⎪⎨1 if τ 1 = τ 2c(r 1 , r 2 ) := r 1 r 2 r1 −1⎪⎩r 2 −1 if τ 1 and τ 2 are disjointr 1 r 2 r 1 r −1 2 r −1 −1 1 r 2 if τ 1 and τ 2 are cuspidal.Then we define C proj to be the subgroup normally generated by all the c(r 1 , r 2 )’sinside π top1 (X gal , S n ) where the τ i ’s run through all transpositions **of** S n and ther i ’s run through all inertia groups **of** all components **of** the ˜p −1 (R τi )’s.Lemma 4.5 The subgroup C proj is contained in π top1 (X gal ) and in the followingkernels:ker( π top1 (X gal , S n ) ↠ π top1 ( 2 ) )ker( π top1 (X gal , S (1)n−1 ) ↠ πtop 1 (X) )The pro**of** is completely analogous to the pro**of** **of** Lemma 4.1 and therefore leftto the reader.□40

Hence the homomorphisms from π top1 (X gal , S n ) onto π top1 ( 2 ) and the mapfrom π top1 (X gal , S (1)n−1) onto π top1 (X) factor over the quotient by C proj . Moreover,we get the following two short exact sequences1 → π top1 (X gal ) → π top1 (X gal , S n ) → S n → 1↓ ↓ ||1 → π top1 (X gal )/C proj → π top1 (X gal , S n )/C proj → S n → 1 (∗)where the arrows downwards are surjective.Proposition 4.6 We can split the short exact sequence (∗) using inertia groups.With respect to this splitting there are the following isomorphisms(π top1 (X gal )/C proj ) / (π top1 (X gal )/C proj ) Sn∼ = πtop1 ( 2 ) = {1}(π top1 (X gal )/C proj ) / (π top1 (X gal )/C proj ) ∼(1) S = πtop1 (X)n−1where the notations are the ones introduced in Section 3.1If Question 2.14 has an affirmative answer for the universal cover ˜X gal **of** X galthen the group C proj is trivial.PROOF. The pro**of** is analogous to the one **of** Proposition 4.2:For every transposition (1 k) we choose a component **of** ˜p −1 (R (1 k) ) and denoteby r k the non-trivial element **of** its inertia group. We denote by ¯r k the image **of** r kinside π top1 (X gal , S n )/C proj . As in the pro**of** **of** Proposition 4.2 we conclude thatthese ¯r k ’s fulfill the Coxeter relations **of** the symmetric group and so they provideus with a splitting s : S n → π top1 (X gal , S n )/C proj .As in the pro**of** **of** Proposition 4.2 there are the following equalities for thekernel N **of** the homomorphism from π top1 (X gal , S n ) onto π top1 ( 2 ):andN = (π top1 (X gal )/C proj ) Sn · s(S n )N ∩ (π top1 (X gal )/C proj ) Sn = (π top1 (X gal )/C proj ) SnApplying the second isomorphism theorem **of** group theory we obtain the firststatement. Again, we leave the second identity to the reader.Now suppose that the components **of** ˜p −1 (R gal ) fulfill the connectivity properties**of** Question 2.14 with respect to the universal cover ˜p : ˜X gal → X gal .For two disjoint transpositions τ 1 and τ 2 we choose components ˜R 1 and ˜R 2 **of**˜p −1 (R 1 ) and ˜p −1 (R 2 ), respectively. We let r 1 and r 2 be the non-trivial elements **of**their inertia groups. We know that these components intersect in a point z. Thereis an inclusion ¡ **of** 2 ¡ × 2 into π top1 (X gal , S n , ˜p(z)). This group is generated byr 1 and r 2 . Hence these two elements commute and c(r 1 , r 2 ) is equal to 1.41

- 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: eThis automorphism has order e i an
- 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 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.