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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

to look at the affine

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 imageof 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 ofπ 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.One 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 complementsof 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

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