- Text
- Generic,
- Subgroup,
- Fundamental,
- Element,
- Proposition,
- Generated,
- Homomorphism,
- Finite,
- Quotient,
- Projection,
- Galois,
- Closures,
- Projections

On Fundamental Groups of Galois Closures of Generic Projections

7.3 Surfaces in 3Let X m be a smooth surface **of** degree m ≥ 2 in 3 .For k ≥ 5 the line bundle L k := O© 3(k)| Xm is sufficiently ample, cf. Lemma2.2. Combining Proposition 2.5 with Proposition 2.8 we see that a generic threedimensionallinear subspace **of** H 0 (X m , L k ) gives rise to a good generic projectionthat we denote by f k : X m → 2 .Proposition 7.5 Let X gal be the **Galois** closure **of** a good generic projection f k .Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = k mk2 −1π top1 (X gal )/C proj ∼ = k mk2−2 .PROOF. The morphism f k has degree n = deg f k = mk 2 .Lefschetz’s theorem on hyperplane sections tells us that the surface X m issimply connected. We let C be a smooth section **of** O© 3(1)| Xm . The surfaceX m − C is simply connected by [N, Example 6.8]. So the divisibility index **of** L 1equals 1 for and hence this index is equal to k for L k .Applying Theorem 6.11 we get the result.7.4 Hirzebruch surfacesLet X := e :=We denote by F the class **of** a fibre **of** X →© 1(O© 1 ⊕ O© 1(−e)) with e ≥ 2 be the e.th Hirzebruch surface.□1 and by H the class **of** thetautological bundle O e(1) in Pic( e). We refer to [Hart, Section V.2] for detailson the intersection theory and the canonical line bundle **of** Hirzebruch surfaces.For a > 0 and b > ae the line bundle L (a,b) := O e(aH + bF ) on e is ampleby [Hart, Theorem V.2.17]. We assume that L (a,b) is sufficiently ample which canbe achieved by taking a tensor product **of** at least five very ample line bundles cf.Lemma 2.2. If Proposition 2.8 assures the existence **of** simple double points thenwe denote by f (a,b) : e →2 the good generic projection associated to a genericthree-dimensional linear subspace **of** H 0 e, L ( (a,b) ).Proposition 7.6 Assume that f (a,b) : e →2 is a good generic projection. Welet X gal be the **Galois** closure **of** f (a,b) . Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = gcd(a,b) 2ab+ea2 −1π top1 (X gal )/C proj ∼ = gcd(a,b) 2ab+ea2−2 .PROOF. The morphism f (a,b) has degree n = deg f (a,b) = 2ab + ea 2 . The divisibilityindex **of** L (a,b) in Pic(X) is gcd(a, b) and we only have to plug in this datainto Theorem 6.11.□82

Remark 7.7 Using the results **of** Moishezon, Teicher and Robb [MoTeRo] we seethat C aff and C proj are trivial.7.5 Geometrically ruled surfacesWe let C be a smooth projective curve **of** genus g and we let E be a rank 2 vectorbundle on C. We assume that H 0 (C, E) ≠ 0 but that for all line bundles L withnegative degree the bundle E ⊗ L has no non-trivial global sections.Then we define π : X := (E) → C to be the projectivisation **of** E ande := − deg E. This is a geometrically ruled surface over C with invariant e.Conversely, by [Hart, Proposition V.2.8] every geometrically ruled surface overa curve is the projectivisation **of** a rank 2 vector bundle that fulfills the aboveassumptions on the global sections.The Picard group **of** X is isomorphic ¡ to ⊕ Pic(C). It is generated by thepull-back **of** Pic(C) and by the class C 0 **of** a section **of** π with O X (C 0 ) isomorphicto the tautological line bundle O X (1) on X. We choose a natural number k > 0and a line bundle L C on C **of** degree deg L C > ke. Then we define the line bundleL X on X to beL X := O X (C 0 ) ⊗k ⊗ π ∗ (L C ).This line bundle is ample by [Hart, Proposition V.2.20] and [Hart, PropositionV.2.21]. We assume that L X is sufficiently ample which can be achieved by takingthe tensor product **of** at least five very ample line bundles, cf. Lemma 2.2.If Proposition 2.8 assures the existence **of** simple double points then we denoteby f LX : X → 2 the good generic projection associated to a generic threedimensionallinear subspace **of** H 0 (X, L X ). The degree **of** f LX equals the selfintersection**of** L Xn := deg f LX = 2k deg L C − ek 2 .Also we denote byd(L X ) := max{m ∈ ¡ | ∃M ∈ Pic(X), M ⊗m ∼ = L X }the divisibility index **of** L X in Pic(X). This number divides the greatest commondivisor gcd{k, deg L C }.Proposition 7.8 Let X be a geometrically ruled surface over a curve **of** genus gand let L X be the line bundle considered above.We assume that L X is sufficiently ample and that f LX : X → 2 is a goodgeneric projection. We let X gal be the **Galois** closure **of** f LX . Then there areisomorphisms¡ ¡¡ ¡π top1 (Xgal aff)ab/ ¯C aff ∼ = d(L X ) n−1 2g(n−1)⊕π top1 (X gal ) ab / ¯C proj ∼ = d(L X ) n−2 ⊕ 2g(n−1) .83

- 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 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
- Page 69 and 70: 5.4 ExamplesWe now compute ˜K(−,
- Page 71 and 72: Theorem 5.22 For a K(G, 1)-complex
- Page 73 and 74: Example 5.24 The homomorphism ψ ma
- Page 75 and 76: 6 ConclusionJetzt nehmt den Wein! J
- Page 77 and 78: This can be also formulated as foll
- Page 79 and 80: Severi claimed that a curve with on
- Page 81 and 82: where ϕ denotes the splitting of
- Page 83 and 84: Corollary 6.3 Under the assumptions
- Page 85 and 86: Proposition 6.5 Under the isomorphi
- Page 87 and 88: Remark 6.8 Proposition 6.5 shows us
- Page 89 and 90: We remark that the group on the lef
- Page 91: E(π top1 (Z), n). By Corollary 3.3
- Page 96 and 97: If we denote by Π g the fundamenta
- Page 98 and 99: NotationsVarieties and morphismsf :
- Page 100 and 101: [GR1][GR2][GH][SGA1]H. Grauert, R.