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On Fundamental Groups of Galois Closures of Generic Projections

G\Y. After removing the ramification locus **of** G\Y → G\X which has realcodimension two this space remains connected since we assumed that our spacesare normal, cf. [GR2, Chapter 7.4.2]. This connectivity result implies that thehomomorphismπ top1 (G\X − D, q(x 0 )) ↠ π top1 (X, G, x 0 ) (∗)is surjective. It remains to compute its kernel.However, first we want to define a surjective homomorphismψ : π top1 (G\X − D, q(x 0 )) ↠ G.For this we lift a loop γ in the group on the left to a path in X starting at x 0 . Thislift ends at a point g · x 0 where the element g ∈ G is unique. This defines thehomomorphism we are looking for. Of course if we take the pull-back via themorphism X → G\X we are in the situation **of** Section 4.3 where we defined ahomomorphism π top1 (X, G, x 0 )↠G in a similar way via lifting elements **of** thegroup on the left to the point x 0 × 1 **of** X × G. Chasing through the diagrams wesee that the homomorphisms onto G are compatible with the homomorphism (∗).For the divisor D i we define the following loop Γ i in Z := G\X − D: Wechoose a point w i on D i that is a smooth point **of** D. We let γ i be a path connectingq(x 0 ) to w i inside Z. We shorten γ i a little bit before reaching w i . Then we puta little circle around w i starting at the end **of** γ i . This defines a loop Γ i based atq(x 0 ). Such a loop is usually called a simple loop.If we lift this loop to a path based at x 0 ∈ X it “winds around“ a component R i**of** q −1 (D i ): We choose a small neighbourhood U(w i ) **of** the point w i ∈ D i thatwe have chosen above. We let V (w i ) be the connected component **of** q −1 (U(w i ))such that the lift **of** Γ i to x 0 meets V (w i ). The map q : X → G\X looks in localcoordinates likeV (w i ) → U(w i )e(z 1 , z 2 , ...) ↦→ (z i 1 , z 2 , ...)where e i is the ramification index **of** D i . The reason for this is that locally aroundw i the map q is a branched **Galois** cover with group ¡ e iand branch locus D i .In these coordinates R i is given by the equation z 1 = 0. The automorphism **of**X induced by the lift **of** Γ i to x 0 clearly is the map x ↦→ ψ(Γ i ) · x. It is clear fromthis local description that R i must be fixed by ψ(Γ i ).We let ˜p : ˜X → X be a universal cover **of** X. We choose a point ˜x0 lyingabove x 0 ∈ X. Lifting Γ i to ˜x 0 we get a path that “winds“ around a component˜R ′ i **of** ˜p −1 (R i ). It corresponds to an automorphism **of** ˜X that fixes ˜R ′ i. Via basechange to X × G → X we get exactly an element that corresponds to the inertiaautomorphism **of** ˜R ′ i corresponding to ψ(Γ i) as described in Section 4.3.38

eThis automorphism has order e i and so the image **of** Γ i i under (∗) must beetrivial. In particular, the subgroup normally generated by the Γ i i ’s lies in thekernel **of** (∗). We stop for a moment to define a new object:With respect to the Γ i and e i we define the orbifold fundamental group withrespect to G\X, D i , e i to be the quotientπ orb1 (G\X, {D i , e i }, q(x 0 )) := π top1 (G\X − D, q(x 0 )) / ≪ Γ ie i≫ .If we choose different set **of** loops Γ ′ i around the D i’s as described above thenthey are conjugate to the original Γ i ’s and so this set generates the same normalsubgroup. Hence this quotient is well-defined.This orbifold fundamental group is the opposite automorphism group **of** sometopological cover ˜c ′ : Ỹ → G\X − D. By what we have said above the homomorphism(∗) factors through the orbifold fundamental group and so Ỹ dominates˜X − (q ◦ ˜p) −1 (D). For a smooth point w i on D i we let U(w i ) be an admissibleneighbourhood, i.e. a neighbourhood such that ˜c ′−1 (U(w i )) is a disjoint union **of**spaces that are homeomorphic to U(w i ). We assume that D i is smooth in U(w i )so that U(w i )−D i is homeomorphic to (¨ −{0})×¨dim X−1 . This means that thefundamental group **of** U(w i ) − D i is isomorphic to ¡ . It is generated by a loop Γ ias described above. Looking at this locally we can extend Ỹ → ˜X − (q ◦ ˜p) −1 (D)to some map ¯Ỹ → ˜X − (q ◦ ˜p) −1 (S) where S is the finite set **of** singularities **of**D. Since both spaces are locally homeomorphic this is a topological cover map.The space ˜X is normal and simply connected. Since (q ◦ ˜p) −1 (S) is a discrete set**of** real codimension 4 also ˜X − (q ◦ ˜p) −1 (S) is simply connected. Since ¯Ỹ is aconnected topological cover **of** ˜X − (q ◦ ˜p) −1 (S) they must be homeomorphic.Then there is only one way to complete this to a cover **of** ˜X: namely to take thetrivial cover **of** ˜X. So we conclude that Ỹ is homeomorphic to ˜X − (q ◦ ˜p) −1 (D)and this means that the homomorphism (∗) induces an isomorphismπ orb1 (G\X, {D i , e i }, q(x 0 )) ∼ = πtop1 (X, G, x 0 ).We already noted above that both groups possess surjective homomorphisms ontoG that are compatible under this isomorphism.4.5 The quotient in the topological setupWe let X be smooth projective surface over the complex numbers and f : X → 2be a good generic projection **of** degree n. From Proposition 2.7 we know that S nacts on X gal . With respect to this action and the action **of** the subgroup S (1)n−1 we39

- 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 and 48: In particular, for H = 1 we obtain
- Page 49: For an element g of the inertia gro
- 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.