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On Fundamental Groups of Galois Closures of Generic Projections

Given an arbitrary connected G-cover Z → X we can form its fibre productwith X × G. Then we can find a connected component Z ′ (with respect to C G ) **of**this fibre product that dominates Z. Since Z ′ is a G-cover **of** X × G it is dominatedby ˜X × G. From this we conclude that ˜X × G is a universal cover **of** C G .We define ˜p G : ˜X × G → X with the G-action as above and denote its group**of** G-automorphisms by Aut(˜X × G). For every point x 0 **of** X, this latter groupacts from the left on the fibre ˜p G −1 (x 0 ).Definition 4.4 For a point x 0 **of** X and a group G that acts by automorphisms onX we denote the opposite group **of** Aut(X × G) by π top1 (X, G, x 0 ) and call it theG-fundamental group **of** X.As in the case **of** the classical fundamental group to give a G-cover is the same asto give a discrete set with a right action **of** π top1 (X, G, x 0 ) on it.Given a subgroup H **of** G the fundamental groups classifying covers withactions **of** H and G are related as follows: We fix a system R **of** representatives**of** G/H. We will assume that the class **of** H is represented by the unit element **of**G. For a connected C H -cover p : Y → X we define the following C G -cover:and a left G-action on Y × R viaY × R → X(y, r) ↦→ r · p(y)G × (Y × R) → Y × Rg , (y, r) ↦→ (h g y, r g h)where g = r g h g is the unique decomposition **of** an element **of** G into a product**of** an element **of** H and an element **of** R. This clearly is a connected object **of**C G . The object so associated to Y is the same as the fibre product **of** Y withq : X × R → X with the G-action described above. This is an exact functor fromC H to C G and hence defines an injective homomorphism **of** fundamental groupsπ top1 (X, H, x 0 ) ↩→ π top1 (X, G, x 0 ).With respect to the action **of** π top1 (X, G, x 0 ) on the fibre q −1 (x 0 ) the image **of** thishomomorphism is the stabiliser **of** the point (x 0 , 1) **of** X × R.Given an element γ **of** π top1 (X, G, x 0 ) it acts on the fibre q −1 (x 0 ) **of** the G-cover q : X × R → X by sending (h −1 x 0 , h) to (h −1 r(γ) −1 x 0 , r(γ)h). Moreover,if H is a normal subgroup **of** G then the map that sends γ to r(γ) defines a homomorphismfrom π top1 (X, G, x 0 ) to G/H. This homomorphism is surjective sincewe can lift the map x ↦→ g·x to the universal cover as explained in [Di, Satz I.8.9].Hence there exists a short exact sequence1 → π top1 (X, H, x 0 ) → π top1 (X, G, x 0 ) → G/H → 1.34

In particular, for H = 1 we obtain the short exact sequence1 → π top1 (X, x 0 ) → π top1 (X, G, x 0 ) → G → 1. (∗)To obtain an isomorphism **of** π top1 (X, G, x 0 ) with Aut(˜X × G) we have to choosea point on the fibre ˜p −1 G (x 0 ).If we choose another base point, say x 1 on X then the G-fundamental groupswith respect to two x i ’s are isomorphic. However, such an isomorphism dependson the choice **of** points ˜x i in the respective fibres ˜p −1 G (x i ), i = 0, 1. We willassume that the two ˜x i ’s lie on the same topological component **of** the universalG-cover ˜X × G. This means that we choose a path connecting x 0 to x 1 . Then theisomorphismπ top1 (X, G, x 0 ) ∼ = πtop1 (X, G, x 1 )is well-defined up to conjugation by an element **of** π top1 (X, x 0 ) and the two homomorphismsonto G coming from the short exact sequence (∗) are compatibleunder this isomorphism.For a G-cover p : Y → X and a closed subset **of** A **of** Y we call the subgroup**of** G fixing A pointwise the inertia group **of** A (in G):I A := {g ∈ G | ga = a, ∀a ∈ A}.The possibly larger subgroup **of** G fixing A but not necessarily pointwise is calledthe decomposition group **of** A (in G):D A := {g ∈ G | g(A) = A}.The inertia group is always a normal subgroup **of** the decomposition group.We choose a point x 1 on X and let ˜p G : ˜X × G → X be the universal G-cover. Then the inertia group I x1 acts on the fibre ˜p −1 G (x 1 ). We choose a point˜x 1 on this fibre. Then we compare the left action **of** I x1 with the right action **of**π top1 (X, G, x 1 ) in this point ˜x 1 . This associates to each element **of** I x1 an element**of** π top1 (X, G, x 1 ). Given another point ˜x ′ 1 above x 1 there is a G-automorphism ϕthat sends ˜x 1 to ˜x ′ 1. We assume that g · ˜x 1 = ˜x 1 · γ g for an element g **of** I x1 . Sinceϕ is G-equivariant we computeg · ˜x ′ 1 = g · (ϕ(˜x 1)) = ϕ(g · ˜x 1 ) = ϕ(˜x 1 · γ g ) = ϕ(˜x 1 ) · γ g = ˜x ′ 1 · γ g.Hence γ g does not depend on the choice **of** a point in the fibre above x 1 and itacts like multiplication by g on all points on this fibre. This means that there is anatural injective homomorphismI x1 ↩→ π top1 (X, G, x 1 ).35

- 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: an isomorphism between these two gr
- 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

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NotationsVarieties and morphismsf :

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