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On Fundamental Groups of Galois Closures of Generic Projections

For a pro**of** we refer to [Fa, Section 4] or [MoTe1, Chapter 0].Since the existence **of** singularities on the branch curve **of** a generic projectionplays an important rôle later on we remark thatLemma 2.9 Let f : X →line bundle L.2 be a generic projection given by a sufficiently ample1. There is at least one cusp on the branch curve **of** f.2. There exists a positive integer m 0 (depending on L) such that for all m ≥m 0 the branch curve **of** a generic projection with respect to L ⊗m has at leastone simple double point.PROOF. The degree **of** a generic projection f is equal to the self-intersectionnumber **of** L which is at least 5. By a theorem **of** Gaffney and Lazarsfeld (quotedas [FL, Theorem 6.1]) there exists a closed point x on X with ramification indexat least 3. The image f(x) **of** x lies on the branch curve D **of** f and D necessarilyhas a cusp in such a point.The number **of** simple double points **of** a branch curve **of** a generic projectionwith respect to the line bundle L ⊗m is a polynomial **of** degree 4 as a function **of**m tending to +∞ as m tends to +∞. Hence there exists a positive integer m 0 asstated above.□2.3 Questions on connectivityDefinition 2.10 We let S n be the symmetric group on n letters. Then we denoteits subgroup **of** permutations fixing the letter i by S (i)n−1 .Definition 2.11 For a permutation **of** S n we define its support to be the largestsubset **of** {1, ..., n} on which it acts non-trivially. We say that two permutations aredisjoint or nodal if their supports are disjoint. In the case where their supportsintersect in exactly one element we say that they are cuspidal.We let L be a sufficiently ample line bundle on the smooth projective surfaceX. We let E be a three-dimensional linear subspace **of** H 0 (X, L) belonging tothe G ′ given by Proposition 2.7. We let f = f E : X →2 be the correspondinggeneric projection **of** degree n and denote by f gal : X gal →2 its **Galois** closure.We denote by R gal ⊂ X gal the ramification divisor **of** f gal . We know fromProposition 2.7 that the symmetric group S n acts on X gal . For a transposition τ**of** S n we consider the following components **of** R gal :Then there is the following resultR τ := Fix(τ) := {x ∈ X gal , τx = x}.8□

Proposition 2.12 Let L be a sufficiently ample line bundle on X and let f =f E : X → 2 be a generic projection coming from a three-dimensional linearsubspace E ∈ G ′ with G ′ as in Proposition 2.7. We furthermore assume that thebranch curve **of** f has a simple double point. Then1. The R τ ’s defined above are smooth and irreducible curves.2. The ramification locus R gal **of** f gal is the union **of** the R τ ’s where τ runsthrough the transpositions **of** S n .3. If τ 1 and τ 2 are disjoint transpositions then R τ1 and R τ2 intersect transversely.These intersection points lie over simple double points **of** D andthere is no other component **of** R gal through such points.4. If τ 1 and τ 2 are cuspidal transpositions then R τ1 and R τ2 intersect transversely.These intersection points lie over cusps **of** D and the only othercomponent **of** R gal through such points is R τ1 τ 2 τ 1−1 = R τ2 τ 1 τ 2−1.For a pro**of** we refer to [Fa, Lemma 1] and [Fa, Section 4]. We note that a lessprecise statement without pro**of** was already made by Miyaoka [Mi]. □Definition 2.13 Let L be a sufficiently ample line bundle on a smooth projectivesurface X. We call a generic projection f = f E : X → 2 associated to athree-dimensional linear subspace E ∈ G ′ with G ′ as in Proposition 2.7 a goodgeneric projection if the branch curve **of** f has a simple double point.By Lemma 2.2 the tensor product **of** five very ample line bundles is sufficientlyample. Twisting a sufficiently ample line bundle with itself at least m 0 times withm 0 as in Lemma 2.9 we arrive at a line bundle L ′ such that there is an open densesubset **of** ¤ (3, H 0 (X, L ′ )) giving rise to good generic projections.It is in this sense that a “sufficiently general“ three-dimensional linear subspace**of** the space **of** global sections **of** an ample line bundle gives rise to a goodgeneric projection for “almost all“ ample line bundles.We let f : X →2 be a good generic projection **of** degree n with **Galois**closure f gal : X gal → 2 . Let l be a generic line in 2 , i.e. a line intersecting Din deg D distinct points. We then define¢2:=2 − l,X aff := f −1 (¢ 2 ),X affgal := f gal −1 (¢ 2 ).We let p : Y aff → Xgal affaffbe a topological cover **of** Xgal or p : Y → X gal be atopological cover **of** X gal . Then for all transpositions τ **of** S n the inverse imagep −1 (R τ ) is a disjoint union **of** smooth and irreducible curves.9

- 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 and 16: Miyaoka [Mi] gave a construction of
- Page 17 and 18: 2 Generic projections and their Gal
- Page 19: 2.2 Galois closures of generic proj
- 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
- Page 69 and 70: 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.