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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

7.3 Surfaces in 3Let X m

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 selfintersectionof 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

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