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On Fundamental Groups of Galois Closures of Generic Projections

If we fix an isomorphism between the G-fundamental groups **of** X with respect tox 0 and x 1 there is the following compositionI x1 ↩→ π top1 (X, G, x 1 ) ∼ = πtop1 (X, G, x 0 ) ↠ G.Even though the isomorphism in the middle is only well-defined up to conjugationby an element **of** π top1 (X, x 0 ) the whole composition always coincides with theinclusion map **of** I x1 into G.In terms **of** automorphisms **of** the universal G-cover ˜p G : ˜X × G → X wefix a point ˜x 1 on the fibre ˜p −1 G (x 1 ). Given an element g **of** I x1 there is a uniqueautomorphism ϕ g **of** ˜X × G that sends ˜x 1 to g · ˜x 1 . However, this automorphismreally depends on the choice **of** ˜x 1 .We now let R be a path connected subset **of** X that contains the point x 1 . Ifwe forget the G-action for a moment then ˜p −1 G (R) is a disconnected topologicalcover **of** R if G is non-trivial. We let ˜R be a component **of** ˜p −1 G (R) on ˜X × {1}.The group I R acts on ˜R simply by interchanging the |I R | different but homeomorphiccomponents. We choose a point **of** ˜R above x 1 to obtain an isomorphism**of** π top1 (X, G, x 0 ) with Aut(˜X × G). In this special situation we see that an automorphismcorresponding to an element **of** I R depends only on ˜R and not on theparticular point lying above x 1 . Hence it makes sense to talk about an automorphism**of** the universal G-cover that is the inertia automorphism **of** a component**of** ˜p −1 G (R).We finally want to stress that in general there is no natural way **of** relatingelements **of** G to cover automorphisms **of** ˜X×G or elements **of** the G-fundamentalgroup **of** X since the G-action usually does not respect the fibres. It is only inertiathat makes this possible.Given a G-cover p : Y → X there is always an injection **of** inertia groupsI y ⊆ I p(y) for all points y ∈ Y. Given a cover Z → G\X we can form the fibreproduct with X and obtain a G-cover p ′ : Z × G\X X → X. All points z on thisfibre product fulfill I z = I p ′ (z). Conversely, if p : Y → X is a G-cover thatfulfills I y = I p(y) for all points y **of** Y then the quotient by G defines a coverG\p : G\Y → G\X. Hence there is a one-to-one correspondence{ }{ } G-covers p : Y → X such thatcovers **of** G\X ↔I y = I p(y) for all y ∈ YSince this remains true if we assume connectivity on both sides **of** this correspondencethe induced homomorphism **of** fundamental groupsπ top1 (X, G, x 0 ) ↠ π top1 (G\X, ¯x 0 )is surjective. Here, ¯x 0 denotes the image **of** x 0 under the quotient map X → G\X.36

For an element g **of** the inertia group I x1 we denote by ı g the image **of** g inπ top1 (X, G, x 1 ) as constructed above. We let p : Y → X be a G-cover withI y = I p(y) for all points **of** Y. If we fix a point ˜x on ˜X then there is a uniquecover automorphism ϕ g **of** ˜X × G such that ϕ g · ˜x = ˜x · ı g . By our assumptionson the inertia groups **of** Y this automorphism ϕ g will act trivially on Y. So thesubgroup N normally generated by all inertia elements lies in the kernel **of** thehomomorphism from π top1 (X, G, x 0 ) onto π top1 (G\X, ¯x 0 ). Conversely, the quotient**of** the universal G-cover by N is a G-cover q : Z → X with I z = I q(z) for all pointsz **of** Z. Hence π top1 (X, G, x 0 )/N is a quotient **of** π top1 (G\X, ¯x 0 ). But this meansthat N is precisely the kernel we are looking for. So we obtain a short exactsequence1 → N → π top1 (X, G, x 0 ) → π top1 (G\X, ¯x 0 ) → 1.As a special case we obtain the following: If G acts without fixed points on X thenX → G\X is a regular cover with group G, there are no non-trivial inertia groupsand we just get the well-known short exact sequence1 → π top1 (X, x 0 ) → π top1 (G\X, ¯x 0 ) → G → 1.4.4 Loops and the orbifold fundamental groupThe material **of** this section should be well-known. However, the author could notfind a reference for it.As in the previous section we let X be normal irreducible complex analyticspace and G be a finite group **of** automorphisms **of** X. We keep all notationsintroduced so far.We will always assume that the quotient space G\X is smooth, i.e. a complexmanifold. By purity **of** the branch locus the branch locus D **of** q : X → G\X is adivisor, cf. [GR1, Satz 4]. We denote by D i , i = 1, ..., r the irreducible components**of** this divisor, cf. [GR2, Chapter 9.2.2]. We denote by e i the ramificationindex **of** D i .The inertia groups **of** the components **of** q −1 (D i ) for fixed i are conjugatesubgroups **of** G. These components are divisors and so their inertia groups must becyclic. More precisely, every inertia group **of** a component **of** q −1 (D i ) is abstractlyisomorphic to the cyclic group ¡ e i.Given a G-cover p : Y → X we form the quotient G\p : G\Y → G\X.Outside ⋃ i D i this is a topological cover. This defines a homomorphism fromthe fundamental group **of** G\X − D to the G-fundamental group **of** X. If Y isconnected as a G-cover then its quotient is also connected. We assumed X tobe normal so also Y must be normal and so the same is true for the quotient37

- 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: In particular, for H = 1 we obtain
- 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.