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- Generic,
- Subgroup,
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- Element,
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- Homomorphism,
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On Fundamental Groups of Galois Closures of Generic Projections

We already said in Section 4.1 that C is equivalent to the category **of** topologicalcovers **of** X. We identify C with C {1} where {1} denotes the trivial group.We recall that an object Y **of** a category in which coproducts exist is calledconnected if it is not isomorphic to a coproduct Y 1∐Y2 where Y 1 and Y 2 areobjects **of** this category not isomorphic to the initial object.For an arbitrary discrete group π 1 we define the following category:C(π 1 ) π 1 -setsThe objects are discrete sets with a left or right action **of** the group π 1 .The morphisms are π 1 -equivariant maps between these sets.We warn the reader that when discussing the fundamental group π 1 in algebraicgeometry one **of**ten considers sets with left π 1 -actions whereas in topology oneusually considers sets with right π 1 -actions. Therefore the author decided to berather pedantic about this point, especially after he was trapped when he was notpaying attention to it.We choose a universal cover ˜p : ˜X → X **of** X in the sense **of** topology. Wedenote by Aut(˜X) the group **of** deck transformations **of** ˜X over X. Then we defineπ top1 (X, ˜X) to be the opposite group to Aut( ˜X). Given a (connected) cover p :Y → X the group Aut(˜X) acts from the right on the set **of** cover morphismsHom(˜X, Y). Hence there is a left π top1 (X, ˜X)-action on this set. This defines afibre functor from the category C to the category **of** sets with a left π top1 (X, ˜X)-action and makes C into a **Galois** category.Here we have to relax Grothendieck’s terminology a little bit: We also allowquotients by discrete groups rather than only finite ones. Also we assume that thefibre functor maps to the category **of** discrete sets with a group action **of** a discretegroup rather than only to the category **of** finite sets together with a continuousaction **of** a pr**of**inite group.Conversely, given a fibre functor F there is always a group π top1 (X, F ) calledthe automorphism group **of** the functor F . A map between two covers Y 1 and Y 2over X is uniquely determined by the π top1 (X, F )-equivariant map from F (Y 1 )to F (Y 2 ). The main content **of** **Galois** theory and the theory **of** the fundamentalgroup in this setup is that a fibre functor induces an equivalence **of** categoriesbetween C and C(π 1 ).The connection with the fundamental group defined via loops is as follows:We let F x0 (Y) := p −1 (x 0 ) be the fibre **of** p : Y → X above a point x 0 **of**X. Lifting loops based at x 0 to paths in Y defines a right action **of** the “loop“-fundamental group π top1 (X, x 0 ) on the set F x0 (Y). Now we fix a point ˜x 0 on thefibre F x0 (˜X) **of** a universal cover. Then we compare the left Aut( ˜X)-action withthe right π top1 (X, x 0 )-action in this point: For every automorphism ϕ there is aunique element γ in the fundamental group such that ϕ · ˜x 0 = ˜x 0 · γ. This defines32

an isomorphism between these two groups that depends on the choice **of** ˜x 0 . Infact, given another point ˜x ′ 0 **of** F x 0(˜X) there is a unique element α ∈ π top1 (X, x 0 )such that ˜x ′ 0 = ˜x 0 · α. We then computeϕ · ˜x ′ 0 = ϕ · (˜x 0 · α) = (ϕ · ˜x 0 ) · α = (˜x 0 · γ) · α = ˜x ′ 0 · (α−1 γα).That is, with respect to ˜x ′ 0 we obtain an isomorphism **of** π top1 (X, x 0 ) with Aut(˜X)that differs from the isomorphism with respect to ˜x 0 by conjugation with α.If we fix a point ˜x 0 **of** ˜X in the fibre ˜p −1 (x 0 ) we can identify Hom(˜X, Y) withthe set F x0 (Y) by associating to a morphism ϕ : ˜X → Y the point ϕ(˜x 0 ). Underthis identification the right action **of** π top1 (X, x 0 ) on F x0 (Y) becomes a left actionon Hom(˜X, Y) and it is this point where the group actions change their side whenpassing from topology to algebraic geometry and vice versa.For a cover p : Y → X we recover the group **of** its automorphisms as follows:The group Aut(˜X) acts on Hom(˜X, Y). We choose a point on this latter set,i.e. we choose a map from ˜X to Y, and denote by H the subgroup **of** Aut( ˜X)stabilising this point. This identifies Y with the quotient H\ ˜X. An element **of**Aut(˜X) induces an automorphism **of** Y if and only if it normalises H. Since theelements acting trivially on Y are precisely those **of** H we get an isomorphismbetween the group **of** cover automorphisms **of** Y over X and NH/H where NHdenotes the normaliser **of** H in Aut( ˜X).The same can be done for covers with a G-action. So we assume a finite groupG **of** automorphisms acts from the left on X. The following constructions werealready sketched in [SGA1, Remarque IX.5.8] and we will fill out some **of** thedetails:For a connected C-cover p : Y → X we define the following C G -cover:and a left G-action on Y × G viaY × G → X(y, h) ↦→ h · p(y)G × (Y × G) → Y × Gg , (y, h) ↦→ (y, gh)This clearly is a connected object **of** C G . The object so associated to Y is the sameas the fibre product **of** Y with (X × G) with G-action as described above over X.Every connected G-cover **of** X × G is dominated by a G-cover **of** the formY × X (X × G) where Y → X is a connected topological cover. Indeed, forgettingthe G-action, a connected G-cover **of** X × G becomes a cover **of** X consisting **of**exactly |G| components. If we choose Y to dominate each **of** these componentsit is not complicated to obtain a G-morphism from Y × G onto the G-cover **of**X × G we started with.33

- 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: zero. Hence the proof also works al
- 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 :

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[GR1][GR2][GH][SGA1]H. Grauert, R.