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On Fundamental Groups of Galois Closures of Generic Projections

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 much**of** 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, surfaces**of** 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 pro**of** 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 **of**the **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

- Page 1: On Fundamental GroupsofGalois Closu
- Page 5: ContentsIntroductioniii1 A short re
- 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 and 50: For an element g of the inertia gro
- Page 51 and 52: eThis automorphism has order e i an
- Page 53 and 54: Hence the homomorphisms from π top
- Page 55 and 56: 5 A generalised symmetric group5.1
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Theorem 5.3 For n ≥ 5 there exist

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Proposition 5.6 (Rowen, Teicher, Vi

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First, assume that i = n. ThenNow a

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We note that for affine subgroups n

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Every element of H 2 (G) maps to an

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PROOF. We let F d /N ∼ = G and F

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5.4 ExamplesWe now compute ˜K(−,

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Theorem 5.22 For a K(G, 1)-complex

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Example 5.24 The homomorphism ψ ma

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6 ConclusionJetzt nehmt den Wein! J

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This can be also formulated as foll

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Severi claimed that a curve with on

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where ϕ denotes the splitting of

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Corollary 6.3 Under the assumptions

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Proposition 6.5 Under the isomorphi

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Remark 6.8 Proposition 6.5 shows us

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We remark that the group on the lef

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E(π top1 (Z), n). By Corollary 3.3

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7.3 Surfaces in 3Let X m be a smoot

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If we denote by Π g the fundamenta

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NotationsVarieties and morphismsf :

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[GR1][GR2][GH][SGA1]H. Grauert, R.