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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

6 Conclusion 636.1 The

6 Conclusion 636.1 The algorithm of Zariski and van Kampen . . . . . . . . . . . . . 636.2 On the fundamental group of Xgal aff . . . . . . . . . . . . . . . . . 676.3 Adding the line at infinity . . . . . . . . . . . . . . . . . . . . . . 716.4 Generic projections from simply connected surfaces . . . . . . . . 756.5 A purely topological description of the Galois closure . . . . . . . 767 Examples 817.1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.2 1 × 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.3 Surfaces in 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.4 Hirzebruch surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 827.5 Geometrically ruled surfaces . . . . . . . . . . . . . . . . . . . . 837.6 An instructive counter-example . . . . . . . . . . . . . . . . . . . 84Notations 86References 87ii

IntroductionSchon winkt der Wein im gold’nen Pokale,Doch trinkt noch nicht, erst sing’ ich euch ein Lied!Das Lied vom KummerSoll auflachend in die Seele euch klingen.Fundamental groups are birational invariants of smooth algebraic varieties and fora classification it is important to know them. Also it is interesting to see how muchof the classification is encoded in them.These groups are known for smooth and complex quasi-projective curves.For smooth and complex projective curves the fundamental group determines thecurve up to deformation of the complex structure. For surfaces the situation ismuch more complicated. The classification of surfaces is still not complete andmainly surfaces of general type are still not well understood. In particular, surfacesof general type with K 2 ≥ 8χ seemed to be mysterious and were hard toconstruct. Bogomolov and others conjectured that these surfaces have infinitefundamental groups.Miyaoka considered generic projections from smooth projective surfaces tothe projective plane and studied the Galois closures of these projections. He wasable to construct many surfaces of general type with K 2 ≥ 8χ via this method.Moishezon and Teicher showed that there are generic projections from 1 ×1 such that the corresponding Galois closures are simply connected and fulfillK 2 ≥ 8χ. These were the first counter-examples to the conjecture mentionedabove. Their proof involved a certain amount of computations and was based ondegeneration techniques and braid monodromy factorisations.In this thesis we attack the problem of determining the fundamental group ofthe Galois closure of a generic projection via determining some “obvious“ contributionscoming from X. So let f : X →2 be a generic projection of degreen and let X gal be the corresponding Galois closure. It is known that X gal embedsinto X n which induces a homomorphism of fundamental groupsπ 1 (X gal ) → π 1 (X) n . (1)If we denote by K(G, n) the kernel of the homomorphism from G n onto G ab thenthe image of (1) is precisely K(π 1 (X), n). We prove this by purely algebraicmethods. In particular, we obtain this result also for étale fundamental groupsand generic projections defined over algebraically closed fields of characteristic≠ 2, 3.Over the complex numbers there is the algorithm of Zariski and van Kampento determine the fundamental group of the complement of a curve in the affine orprojective plane. Since the monodromy at infinity is a little bit tricky, it is easieriii

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