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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

We recall that a surface

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 definitionof 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 ideasof 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 ofsurfaces 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 ofminimal 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 analoguesof 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 usingGalois 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 ofgeneral 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 ofgeneral 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 ofgeneral 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 → 1of 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 ofgenus 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 of2n+2g −2 distinct points and that there is no point with ramification index bigger3

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