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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

than 2. Knowing B and

than 2. Knowing B and the homomorphism ϕ we can reconstruct C out of thesedata since ϕ tells us how to cut and glue different copies of 1 − B to get C.The ideas outlined above may generalise in some way to surfaces:We let D ⊂ 2 be the branch divisor of a generic projection f : S → 2 ofdegree n ≥ 5. In a similar fashion as above we define a homomorphismπ top1 ( 2 − D, x 0 ) → S nand recover S out of these data by [Ku, Proposition 1]. Moreover, there is eventhe followingConjecture 1.3 (Chisini) Assume that D ⊂2 is the branch divisor of a genericprojection of degree ≥ 5. Then there is a unique generic projection having D asbranch divisor.For the proof of this conjecture in some important cases and the work of Kulikovand Moishezon on it we refer to [Ku].There are still discrete invariants missing to distinguish between different componentsof the moduli space of minimal surfaces of general type with fixed χ andK 2 . The results above suggest that it may be possible to get such invariants out ofπ top1 ( 2 − D) where D is the branch curve of a generic projection.So it may be that generic projections turn out to be important for the classificationof algebraic surfaces of general type.4

2 Generic projections and their Galois closures2.1 Sufficiently ample line bundlesHerr dieses Hauses!Dein Keller birgt die Fülle des goldenen Weins!Hier, diese Laute nenn’ ich mein!Die Laute schlagen und die Gläser leeren,das sind die Dinge, die zusammen passen.Let X be a smooth projective surface over the complex numbers.Definition 2.1 We call a line bundle L on X sufficiently ample if1. L is an ample line bundle with self-intersection number at least 5,2. for every closed point x ∈ X the global sections H 0 (X, L) generate thefibreL x /m 4 x · L,3. for any pair {x, y} of distinct closed points of X the global sections of Lgenerate the direct sumL x /m 3 x · L ⊕ L y /m 3 y · L,4. for any triple {x, y, z} of distinct closed points of X the global sections ofL generate the direct sumL x /m 2 x · L ⊕ L y /m 2 y · L ⊕ L z /m 2 z · L.To produce such line bundles later on we will use the following lemma that alreadyappeared as a remark in [Fa, Section 2]:Lemma 2.2 If a line bundle is the tensor product of at least five very ample linebundles it is sufficiently ample.PROOF. Let L i , i = 1, ..., 5 be very ample line bundles and M their tensorproduct. Since the intersection of L i with L j for all i, j is a positive integer itfollows that the self-intersection of M is at least 25 and so even bigger than 5.For each closed point x ∈ X the global sections of each L i generate the fibreL i,x /m 2 x since L i is very ample. It follows that the global sections of L i ⊗ L jgenerate the fibre (L i ⊗ L j ) x /m 3 x and that the global sections of L i ⊗ L j ⊗ L kgenerate the fibre (L i ⊗ L j ⊗ L k ) x /m 4 x . 5

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