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On Fundamental Groups of Galois Closures of Generic Projections

- By Riemann’s existence theorem every finite holomorphic cover Y → X anis algebraic, i.e. there exists a finite étale cover p : Y → X by an algebraicscheme Y such that the associated analytification is isomorphic to Y →X an .This implies that there is an equivalence **of** categories between the category **of**holomorphic covers **of** X an and the category **of** topological covers **of** X an . Bothcategories can be described in terms **of** discrete sets with an action **of** the fundamentalgroup π top1 (X an , x 0 ) on it where x 0 is a point **of** X an .As explained in [SGA1] there exists a pr**of**inite group πét 1 (X, x 0) that classifiesconnected and finite étale covers **of** X.By the above the category **of** finite étale covers **of** the scheme X is equivalentto the full subcategory **of** finite holomorphic covers **of** the category **of** holomorphiccovers **of** X an . So there is a natural homomorphism π top1 (X an , x 0 ) → πét 1 (X, x 0)that identifies the finite quotients **of** both groups. Hence this map induces anisomorphismπ̂ top1 (X an , x 0 ) ∼ = π ét1 (X, x 0)where ̂ denotes the pr**of**inite completion **of** a group.The homomorphism **of** a group to its pr**of**inite completion is in general notsurjective as the ¡ example → ̂ shows. However, the image **of** a group inside¡its pr**of**inite completion is always dense with respect to the pr**of**inite topology.We recall that a group G is called residually finite if the natural homomorphismfrom G to its pr**of**inite completion Ĝ is injective. There do exist finitely presentedgroups that are not residually finite, e.g. Higman’s 4-group [Se, Chapter I.1.4].Serre asked in loc. cit. whether there are complex algebraic varieties that havenon-residually finite fundamental groups. The facts are as follows:1. If X is a smooth complex projective algebraic curve then π top1 (X, x 0 ) beinga subgroup **of** SL 2 (£ ) is residually finite (cf. [LS, Proposition III.7.11] andSection 1.1).2. If X is a smooth complex affine algebraic curve then π top1 (X, x 0 ) is a freegroup and hence residually finite.3. Toledo [To] constructed smooth complex projective algebraic surfaces withfundamental groups that are not residually finite.So having proven Theorem 4.3 there is no way to deduce from it the correspondingstatement for topological fundamental groups since we are dealing with algebraicsurfaces and so the fundamental groups involved may not be residually finite.24

4.2 The quotient for the étale fundamental groupFor the following quite elementary treatment **of** étale fundamental groups in terms**of** **Galois** groups and the arguments on inertia groups used we refer the reader tothe book **of** H. Popp [Popp] for pro**of**s and details.We let f : X →2 be a good generic projection **of** degree n with **Galois**closure f gal : X gal →2 . We define the following function fields:L := function field **of** X galK := function field **of** Xk := function field **of** 2 .We will assume that they are all contained in a fixed algebraically closed field Ω.We have already seen in Proposition 2.7 that the **Galois** group **of** L/k is isomorphicto the symmetric group S n . We may assume that K is the fixed field **of** S (1)n−1in the notation **of** Definition 2.10.We denote by K nr the maximal unramified extension **of** K i.e. the compositum**of** all finite field extensions inside Ω **of** K such that the normalisation **of** X inthese fields is étale over X. We similarly denote by L nr the maximal unramifiedextension **of** L and by k nr the maximal unramified extension **of** k. Of course, wehave k nr = k. But for later generalisations it is better to use this fact as late aspossible.More or less by definition **of** the étale fundamental groups there are isomorphisms**of** pr**of**inite groupsπét 1 (X gal)∼ = Gal(L nr /L) and πét 1 (X) ∼ = Gal(K nr /K).To be more precise, there is an isomorphism **of** πét 1 (X, Spec Ω) with the oppositegroup Gal(K nr /K) that depends on the choice **of** the embedding **of** K into Ω.Of course there are similar dependencies for πét 1 (X gal) and πét 1 ( 2 ). So we fixSpec Ω as base point for all étale fundamental groups occurring in this section.Since we fixed Ω and embeddings **of** the fields k, K, L, k nr , K nr and L nr into Ωwe will not mention base points and identify the étale fundamental groups withtheir corresponding **Galois** groups with these choices understood. We refer toSection 4.3 for more details on these choices.Both extensions L/k and L nr /L are **Galois**. It is easy to see that L nr /k alsois a **Galois** extension: The **Galois** closure **of** L nr /k would have to be unramifiedover L i.e. must be contained in L nr . Hence there is a short exact sequence1 → Gal(L nr /L) → Gal(L nr /k) → Gal(L/k) → 1with Gal(L/k) ∼ = S n .25

- 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: 4 A first quotient of π 1 (X gal )
- 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

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We remark that the group on the lef

- Page 91:
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.