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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

4 We describe a certain

4 We describe a certain quotient of the fundamental group of the Galois closureof a good generic projection.This is done most naturally within the framework of Galois theory. Wehave therefore given the proof in this setup yielding the result for the étalefundamental group.Given a generic projection from X of degree n with Galois closure X galthere is a short exact sequence1 → π 1 (X gal ) → π 1 (X gal , S n ) → S n → 1 (4)coming from geometry. Here, π 1 (X gal , S n ) is a generalised fundamentalgroup that classifies covers of X gal together with a S n -action.Using inertia groups we see that this short exact sequence partly splits. Thenwe take a naturally defined quotient of this exact sequence to force a splitting.Using the universality result for K(−, n) from Section 3 we then obtainsurjective homomorphismsπ 1 (X gal ) ↠ K(π 1 (X), n)π 1 (X affgal ) ↠ K(π 1(X aff ), n).This yields a proof of what we have said about the image of (1) above.To make this proof also work in the topological setup we have to describehow this generalised fundamental group can be defined topologically. Toachieve this we use ideas of Grothendieck’s [SGA1] and the notion of theorbifold fundamental group.Having introduced this machinery it is not complicated to carry the resultsabove for the étale fundamental groups over to topological fundamentalgroups.5 This is again a purely group theoretical and somewhat technical sectionwhich is important for the main results of Section 6.First we introduce the groups S n (d), d ≥ 1 that generalise the symmetricgroups S n . These groups should be thought of as symmetric groups with dlayers, cf. Section 5.1. It turns out that S n (d) for n ≥ 5 is isomorphic toE(F d−1 , n) where F d−1 is the free group of rank d − 1 and where E(−, n)is as defined in Section 3.For a finitely generated group G and a natural number n ≥ 3 we choosea presentation F d /N of G. Using this presentation we construct a quotientof E(F d , n) that we denote by Ẽ(G, n). Then we show that this quotientvi

depends only on G and n and not on the presentation chosen. There isa split homomorphism from Ẽ(G, n) onto S n yielding a split short exactsequence1 → ˜K(G, n) → Ẽ(G, n) → S n → 1.This construction is related to the one in Section 3 by a central extension0 → H 2 (G, ¡ ) → ˜K(G, n) → K(G, n) → 1where H 2 (G, ¡ ) denotes the second group homology with coefficients inthe integers. Then we give some basic properties of ˜K(G, n) and computeit in some cases.In two appendices we discuss some elementary properties of the secondgroup homology and the connection of Ẽ(−, n) with some finite and someaffine Weyl groups.6 We first recall the algorithm of Zariski and van Kampen to compute thefundamental group of the complement of a curve in the affine or projectivecomplex plane.In Section 4 we introduced a certain quotient to split the short exact sequence(4). We show how to use the groups S n (d) introduced in Section5 as a sort of frame when computing this quotient of π 1 (Xgal aff ). Using theisomorphism of S n (d) with E(F d−1 , n) of Section 5 we see that all relationscoming from a given generic projection lead exactly to a presentation of˜K(π 1 (X aff ), n). Hence we obtain a surjective homomorphismπ top1 (Xgal afftop) ↠ ˜K(π1 (X aff ), n).The kernel of this map is the one needed to split (4). It is closely relatedto connectivity results of the inverse image of the ramification locus R gal off gal in the universal cover of X affgal .Then we study what happens in the projective case. After that we apply ourresults to generic projections from simply connected surfaces and end thissection by some general remarks on symmetric products.7 In this short section we apply our results to good generic projections from2 , 1 × 1 , the Hirzebruch surfaces and surfaces in 3 . For generic projectionsfrom geometrically ruled surfaces we can compute at least our quotientfor the abelianised fundamental group of the Galois closure.We end this section with the discussion of a sufficiently general projectionfrom the Veronese surface of degree 4 in5 . Here it is known that the kernelof (2) is non-trivial.vii

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