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

On Fundamental Groups of Galois Closures of Generic Projections

Question 2.14 Is it true that for distinct transpositions τ 1 and τ 2 every irreduciblecomponent **of** p −1 (R τ1 ) intersects every irreducible component **of** p −1 (R τ2 ) ?In this thesis we want to compute the fundamental groups π top1 (X gal ) andπ top1 (Xgal aff ). The main result (Theorem 6.2) is that there is always a surjectivehomomorphismπ toptop) ↠ ˜K(π1 (X aff ), n)1 (Xgal affwhere ˜K(−, n) is the group-theoretic construction defined in Section 5.3.Now, if Question 2.14 has an affirmative answer for all topological covers **of**then the kernel **of** this surjective homomorphism is trivial.X affgal10

3 Semidirect products by symmetric groups3.1 Definition **of** K( − ,n) and E( − ,n)Ein voller Becher Weins zur rechten ZeitIst mehr wert, als alle Reiche dieser Erde!Dunkel ist das Leben, ist der Tod.Let G be an arbitrary group and n ≥ 2 be a natural number. We denote by θ thepermutation representation **of** the symmetric group S n on G n given byθ : S n → Aut(G n )σ ↦→ ( θ(σ) : (g 1 , ..., g n ) ↦→ (g σ −1 (1), ..., g σ −1 (n)) )Then we form the split extension **of** groups with respect to θ1 → G n → G n ⋊ θ S n → S n → 1and denote by s : S n → G n ⋊ θ S n the associated splitting.We define the subgroup E(G, n) **of** G n ⋊ θ S n to be the group generated byall conjugates **of** s(S n ) and define K(G, n) to be the intersection G n ∩ E(G, n).Hence we get a split extension1 → K(G, n) → E(G, n) → S n → 1.We give another characterisation **of** these groups in Proposition 3.3.More generally, let S be a subgroup **of** S n . Then we defineE(G, n) S := 〈⃗gs(σ)⃗g −1 |⃗g ∈ G n , σ ∈ S〉 ≤ E(G, n)K(G, n) S := E(G, n) S ∩ K(G, n) K(G, n) .These subgroups remain the same when passing to a G n -conjugate splitting. Wewill therefore suppress s in future. Clearly, K(G, n) S is always a normal subgroup**of** G n and K(G, n).In the notation introduced in Definition 2.10 we have the following equalitiesand isomorphisms:E(G, n) Sn = E(G, n)E(G, n) (i) S n−1∼ = E(G, n − 1) for n ≥ 3and similarly for K(−, n).Later on we have to deal with subgroups **of** K(G, n) that are generated byK(G, n)-conjugates **of** a subgroup S **of** S n rather than G n -conjugates. Fortunately,we have the following11

- 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: Proposition 2.12 Let L be a suffici
- 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 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 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.