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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

7 Examples7.1 2Let X :=2

7 Examples7.1 2Let X :=2 be the complex projective plane.For k ≥ 5 the line bundle L k O© := 2(k) is sufficiently ample by Lemma2.2. Combining Proposition 2.5 with Proposition 2.8 we see that a generic threedimensionallinear subspace of H 0 ( 2 , L k ) gives rise to a good generic projectionthat we denote by f k .Proposition 7.1 Let X gal be the Galois closure of a good generic projection f k .Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = k k2 −1π top1 (X gal )/C proj ∼ = k k2−2 .PROOF. The morphism f k has degree n = deg f k = k 2 . The divisibility indexof L k in Pic(X) is k and we only have to plug in this data into Theorem 6.11. □Remark 7.2 The results of Moishezon and Teicher [MoTe2] show that C aff andC proj are trivial.7.2 1 × 1Let X := 1 × 1 .For a ≥ 5 and b ≥ 5 the line bundle L (a,b) := O© 1 ש 1(a, b) is sufficientlyample, cf. Lemma 2.2. Combining Proposition 2.5 with Proposition 2.8 we seethat a generic three-dimensional linear subspace of H 0 ( 1 ×1 , L (a,b) ) gives riseto a good generic projection that we denote by f (a,b) .Proposition 7.3 Let X gal be the Galois closure of a good generic projection f (a,b) .Then there are isomorphisms¡¡π top1 (Xgal aff)/Caff∼ = gcd(a,b) 2ab−1π top1 (X gal )/C proj ∼ = gcd(a,b) 2ab−2 .PROOF. The morphism f (a,b) has degree n = deg f (a,b) = 2ab. The divisibilityindex of L (a,b) in Pic(X) is gcd(a, b) and we only have to plug in this data intoTheorem 6.11.□Remark 7.4 The results of Moishezon and Teicher [MoTe1] and [MoTe4] showthat C aff and C proj are trivial.81

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