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On Fundamental Groups of Galois Closures of Generic Projections

than 2. Knowing B and the homomorphism ϕ we can reconstruct C out **of** thesedata since ϕ tells us how to cut and glue different copies **of** 1 − B to get C.The ideas outlined above may generalise in some way to surfaces:We let D ⊂ 2 be the branch divisor **of** a generic projection f : S → 2 **of**degree n ≥ 5. In a similar fashion as above we define a homomorphismπ top1 ( 2 − D, x 0 ) → S nand recover S out **of** these data by [Ku, Proposition 1]. Moreover, there is eventhe followingConjecture 1.3 (Chisini) Assume that D ⊂2 is the branch divisor **of** a genericprojection **of** degree ≥ 5. Then there is a unique generic projection having D asbranch divisor.For the pro**of** **of** this conjecture in some important cases and the work **of** Kulikovand Moishezon on it we refer to [Ku].There are still discrete invariants missing to distinguish between different components**of** the moduli space **of** minimal surfaces **of** general type with fixed χ andK 2 . The results above suggest that it may be possible to get such invariants out **of**π top1 ( 2 − D) where D is the branch curve **of** a generic projection.So it may be that generic projections turn out to be important for the classification**of** algebraic surfaces **of** general type.4

2 **Generic** projections and their **Galois** closures2.1 Sufficiently ample line bundlesHerr dieses Hauses!Dein Keller birgt die Fülle des goldenen Weins!Hier, diese Laute nenn’ ich mein!Die Laute schlagen und die Gläser leeren,das sind die Dinge, die zusammen passen.Let X be a smooth projective surface over the complex numbers.Definition 2.1 We call a line bundle L on X sufficiently ample if1. L is an ample line bundle with self-intersection number at least 5,2. for every closed point x ∈ X the global sections H 0 (X, L) generate thefibreL x /m 4 x · L,3. for any pair {x, y} **of** distinct closed points **of** X the global sections **of** Lgenerate the direct sumL x /m 3 x · L ⊕ L y /m 3 y · L,4. for any triple {x, y, z} **of** distinct closed points **of** X the global sections **of**L generate the direct sumL x /m 2 x · L ⊕ L y /m 2 y · L ⊕ L z /m 2 z · L.To produce such line bundles later on we will use the following lemma that alreadyappeared as a remark in [Fa, Section 2]:Lemma 2.2 If a line bundle is the tensor product **of** at least five very ample linebundles it is sufficiently ample.PROOF. Let L i , i = 1, ..., 5 be very ample line bundles and M their tensorproduct. Since the intersection **of** L i with L j for all i, j is a positive integer itfollows that the self-intersection **of** M is at least 25 and so even bigger than 5.For each closed point x ∈ X the global sections **of** each L i generate the fibreL i,x /m 2 x since L i is very ample. It follows that the global sections **of** L i ⊗ L jgenerate the fibre (L i ⊗ L j ) x /m 3 x and that the global sections **of** L i ⊗ L j ⊗ L kgenerate the fibre (L i ⊗ L j ⊗ L k ) x /m 4 x . 5

- Page 1: On Fundamental GroupsofGalois Closu
- Page 5 and 6: ContentsIntroductioniii1 A short re
- Page 7 and 8: IntroductionSchon winkt der Wein im
- 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: Miyaoka [Mi] gave a construction of
- 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
- Page 57 and 58: Theorem 5.3 For n ≥ 5 there exist
- Page 59 and 60: Proposition 5.6 (Rowen, Teicher, Vi
- Page 61 and 62: First, assume that i = n. ThenNow a
- Page 63 and 64: We note that for affine subgroups n
- Page 65 and 66: Every element of H 2 (G) maps to an
- Page 67 and 68:
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.