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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

G\Y. After removing the

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 iof 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 ofX 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 ofspaces 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})רdim 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 ofD. 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 setof 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

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