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On Fundamental Groups of Galois Closures of Generic Projections

to look at the affine situation first, i.e. to look at the fibre **of** f and f gal over agenerically chosen affine plane in2 . We refer to these fibres as X aff and Xgal aff,respectively.In the affine situation there is also a surjective homomorphism from π 1 (Xgal aff)onto K(π 1 (X aff ), n) as in (1). Using the algorithm **of** Zariski and van Kampen weobtain a quotientπ 1 (X affgal) ↠ ˜K(π 1 (X aff ), n). (2)Here, ˜K(G, n) is a purely group theoretical construction that can be defined forevery finitely generated group G and every natural number n ≥ 3. It is related toK(G, n) via a short exact and central sequence0 → H 2 (G, ¡ ) → ˜K(G, n) → K(G, n) → 1 (3)where H 2 ¡ (G, ) denotes the second group homology with integral coefficients.Even though the computation **of** K(G, n) for a given group G is usually not socomplicated it is quite hard to say something about ˜K(G, n) and therefore aboutabout the quotient (2) **of** π 1 (Xgal aff ) in general.Also we deduce from (3) that the quotient **of** π 1 (Xgal aff) computed by (2) isusually larger than the one given by (1).It remains to determine the kernel **of** the homomorphism (2). We show that itis a naturally defined subgroup that can be formulated independent **of** the specificsituation. We denote by R gal ⊂ X gal the ramification locus **of** f gal . This divisor isample but it is not irreducible. Then the kernel **of** (2) is trivial if the inverse image**of** R gal in the universal cover **of** X affgalhas certain connectivity properties. Thus ifthese hold true then π 1 (Xgal aff)is isomorphic to ˜K(π 1 (X aff ), n).It is interesting to see that in all known examples (except the projection fromthe Veronese surface **of** degree 4 - but this surface has to be excluded in manysituations **of** classical algebraic geometry) computed by Moishezon, Teicher andothers the kernel **of** (2) actually is trivial. Whether this is a coincidence or ageneral phenomenon does not seem to be clear.The author would like to note that he originally believed that the quotient **of**π 1 (Xgal aff ) he wanted to construct using the algorithm **of** Zariski and van Kampenwas K(π 1 (X aff ), n) and so a subgroup **of** π 1 (X aff ) n . The appearance **of** (3) andthe second homology group was quite some surprise and seems still to be rathermysterious.**On**e application where it is actually easy to compute the quotient given by (2)is the case when we start with a simply connected surface X. In this case we canalso say something about π 1 (X gal ): Namely, suppose that the generic projectioniv

is defined by a line bundle L on X. The degree n **of** f is precisely the selfintersectionnumber **of** L. If we denote by d the divisibility index **of** L in thePicard group **of** X then our quotients take the form¡¡π 1 (Xgal aff)↠ d n−1π 1 (X gal ) ↠ d n−2 .Detailed description **of** the sections1 We have a glimpse on fundamental groups **of** complex algebraic curves andsurfaces. After that we give a rather sketchy motivation why complements**of** divisors on 1 and 2 (may) give some insight into the classificationproblem **of** algebraic curves and surfaces. Also, generic projections andtheir **Galois** closures enter the picture.2 We introduce the notion **of** a good generic projection that is a little bit morerestrictive than the usual notion **of** a generic projection.After that we recall some general facts on **Galois** closures **of** (good) genericprojections. Important for this thesis are the results on the geometry **of** theramification loci due to Miyaoka and Faltings.For a good generic projection f : X → 2 we let f gal : X gal → 2be its **Galois** closure. We let l be a generic line in2 and ¢ let 2 be thecomplement2 − l. Then we define the following objects:projective situation: f : X → 2f gal : X gal → 2¢¢affine situation: f : X aff := X − f −1 (l) →2f gal : Xgal aff gal − f −1 gal (l) →23 For a given group G and a natural number n ≥ 3 we define K(G, n) tobe the kernel **of** the homomorphism from G n onto G ab . The action **of** thesymmetric group S n on n letters on G n given by permuting the factorsrespects K(G, n). We then form the semidirect product **of** K(G, n) by S nvia this action:1 → K(G, n) → E(G, n) → S n → 1.We give some **of** the basic properties **of** K(G, n), prove a universality result,and compute some examples.v

- Page 1: On Fundamental GroupsofGalois Closu
- Page 5 and 6: ContentsIntroductioniii1 A short re
- Page 7: IntroductionSchon winkt der Wein im
- Page 11 and 12: depends only on G and n and not on
- Page 13 and 14: 1 A short reminder on fundamental g
- Page 15 and 16: Miyaoka [Mi] gave a construction of
- 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
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Proposition 5.6 (Rowen, Teicher, Vi

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First, assume that i = n. ThenNow a

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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.