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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

For a

For a proof 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 ofm 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 τof 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 proof we refer to [Fa, Lemma 1] and [Fa, Section 4]. We note that a lessprecise statement without proof 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 subspaceof 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 Galoisclosure 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

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