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On Fundamental Groups of Galois Closures of Generic Projections

We recall that a surface S is called elliptic if there exists a flat morphism fromS onto a smooth curve C such that the general fibre is a smooth elliptic curve.The singular fibres **of** such a morphism can be singular curves (nodal or cuspidalrational curves) and they can be multiple.Now let S → C be a relatively minimal elliptic surface that has at leastone fibre with singular reduction. Suppose there are exactly k multiple fibresabove points P 1 , ..., P k **of** C with multiplicities m 1 , ..., m k . By results **of** Kodaira,Moishezon and Dolgachev there is an isomorphismπ top1 (S) ∼ = πorb1 (C, {P i , m i })where π orb1 denotes the orbifold fundamental group (cf. Section 4.4 for a definition**of** this group). We refer the reader to [Fr, Chapter 7] for details and references.So if the Kodaira dimension κ **of** a surface is less than 2 we have some ideas**of** how its fundamental group looks like. For surfaces **of** general type the situationis more complicated:1. Smooth surfaces **of** degree ≥ 5 in 3 are simply connected.2. There are quotients **of** the latter surfaces by finite groups giving surfaceswith finite and non-trivial fundamental groups.3. If C 1 and C 2 are two curves **of** genus g 1 ≥ 2 and g 2 ≥ 2, respectively thenC 1 ×C 2 has fundamental group Π g1 ×Π g2 which is non-abelian and infinite.At the moment no pattern in the fundamental groups **of** surfaces **of** general type isknown. Also, it is unclear what these groups can tell us about the classification **of**surfaces **of** general type.We end this section by an example and refer the interested reader to [Hu] forfurther details and references:By the Bogomolov-Miyaoka-Yau inequality a minimal surface **of** general typefulfills K 2 ≤ 9χ where χ denotes the holomorphic Euler characteristic. It isnot so complicated to find surfaces with K 2 ≤ 8χ using complete intersections,fibrations or ramified covers. Moreover, Persson [Per] has given examples **of**minimal surfaces **of** general type with χ = a and K 2 = b for almost all admissiblepairs (a, b) with a ≤ 8b.There where some hints and hopes that surfaces with K 2 ≥ 8χ are uniformisedby non-compact domains. Maybe these surfaces were the analogues**of** the curves **of** genus ≥ 2 that are uniformised by the upper half-plane? Thislead to the so-called “watershed conjecture“:Conjecture 1.1 (Bogomolov et al.) A surface **of** general type with K 2 ≥ 8χ hasan infinite fundamental group.2

Miyaoka [Mi] gave a construction **of** surfaces **of** general type with K 2 ≥ 8χ using**Galois** closures **of** generic projections (cf. Section 2.2 for a precise definition).He also showed that every surface has a finite ramified cover that is a surface **of**general type with K 2 ≥ 8χ.Applying this construction to generic projections from 1 × 1 , Moishezonand Teicher [MoTe1] have shown that there is an infinite number **of** surfaces **of**general type with K 2 ≥ 8χ and trivial fundamental group that are not deformationequivalent. In particular, Conjecture 1.1 is false:Theorem 1.2 (Moishezon-Teicher) There do exist simply connected surfaces **of**general type with K 2 ≥ 8χ.1.3 Complements **of** branch divisorsAnother application **of** fundamental groups are complements **of** branch divisors.Some **of** the following ideas go back to Riemann in the 19th century. We havetaken the presentation from [GH, Chapter 2.3]:Let C be a smooth projective curve **of** genus g ≥ 2. Taking the complete linearsystem to a divisor **of** degree n > 2g we get an embedding **of** C inton−g as acurve **of** degree n. Choosing an arbitrary projection onto1 (linearly embeddedinn−g ) we obtain a ramified coverf : C → 1**of** degree n with a branch divisor B ⊂1 **of** degree 2n + 2g − 2. **On** the otherhand, to give a morphism **of** degree n from C to1 we have to choose a divisorD **of** degree n on C and a section **of** O C (D). So, at least heuristically, a curve **of**genus g ≥ 2 should depend on(2n + 2g − 2) − (n + h 0 (C, O C (D)))= (2n + 2g − 2) − (n + (n − g + 1))= 3g − 3parameters - which is in fact the right number.For x 0 ∈1 − B we define a homomorphismϕ : π top1 ( 1 − B, x 0 ) → S nwhere S n is the symmetric group on n letters: We fix a numbering **of** the n pointsin the fibre f −1 (x 0 ). If we lift a loop based at x 0 inside1 − B to C − f −1 (B)we get a permutation **of** the points in the fibre and hence an element **of** S n .We now assume that f is “generic“ in the sense that the divisor B consists **of**2n+2g −2 distinct points and that there is no point with ramification index bigger3

- 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: 1 A short reminder on fundamental g
- 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
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Every element of H 2 (G) maps to an

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PROOF. We let F d /N ∼ = G and F

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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.