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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Under the isomorphism

Under the isomorphism given in the example the “extra“ reflection maps to theelement (1, 0, ..., 0, −1)(1 n) of E(¡ , n)The upper chain forms a subgraph of type A n−1 inside ˜D n . This defines asubgroup isomorphic to S n inside W (˜D n ). We define a split surjectionψ : W (˜D n ) ↠ S nbeing the identity when restricted to the subgroup S n and sending a remainingreflection to the image of the respective reflection “lying above“ it in the graph˜D n . We leave it to the reader to show that we get theExample 5.27 The homomorphism ψ makes ker ψ into K(D ∞ , n) and induces anisomorphismW (˜D n ) ∼ = E(D∞ , n)where D ∞ denotes the infinite dihedral group.62

6 ConclusionJetzt nehmt den Wein! Jetzt ist es Zeit, Genossen!Leert eure gold’nen Becher zu Grund!Dunkel ist das Leben, ist der Tod!6.1 The algorithm of Zariski and van KampenLet C be a reduced but not necessarily smooth or irreducible projective curve ofdegree d in the complex projective plane. We choose a generic line ˜l ⊂2 , i.e.a line that intersects C in d distinct points. We ¢ set 2 :=2 − ˜l and denote theintersection C ¢ ∩ 2 again by C. We are interested in computing the fundamentalgroupsπ top1 ( 2 − C) and π top (¢ 1 2 − C).An algorithm that yields presentations of these groups is given in van Kampen’sarticle [vK]. The result was known to Zariski before and also Enriques, Lefschetzand Picard should be mentioned in this context.We now follow [Ch] and [Mo] to describe this algorithm: We choose a genericline l in ¢ 2 , i.e. a line intersecting C in d distinct points. The inclusion mapsinduce group homomorphisms(¢(¢π top1 2 − C) → π top1 ( 2 − C)π top1 (l − l ∩ C) → π top1 2 − C).Both homomorphisms are surjective. A modern proof for this is for example givenby [N, Proposition 2.1] and its corollaries.The underlying topological space of l − l ∩ C can be identified with £ 2 withd points cut out. Hence its fundamental group is the free group of rank d. To get asystem of d generators we may proceed as follows: We let u 0 be the base point forthe fundamental group of l − l ∩ C. We let w 1 ,...,w d be the points of l ∩ C. Nextwe choose paths γ i from u 0 to w i for all i = 1, ..., d and assume that distinct γ i ’smeet only in u 0 . Next we shorten the γ i ’s such that they stop before reaching theirw i ’s. Putting a little circle around w i at the end of the so shortened γ i ’s we obtainloops Γ i that lie in l − l ∩ C. Loops like this are usually called simple loops andwe already met them in Section 4.4. These Γ i ’s freely generate the fundamentalgroup of l − l ∩ C:π top1 (l − l ∩ C, u 0 ) = 〈Γ i , i = 1, ..., d〉 ∼ = Fd .We consider the closure ¯l of l inside2 and denote by ∞ := ¯l − l the point atinfinity. We may put an orientation on the Γ i ’s and order them in such a way that63

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