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By Herbert Lange, Wolfgang Barth, Klaus Hulek

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Extra resources for Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993

Example text

The generic branch point (i. e. in the complement of C\ U Ci) is only fixed by the respective involution, which has the surface as its fixed variety. We remind that the case η = 3 was already treated in [HKW2]. For the reader, who is interested in this case, we repeat the statement in section 4. 15. The singular locus of Ait2 consists of two disjoint curves C\ and C2 each isomorphic to the modular curve X (2), which are contained in the Humbert surface Tio parametrizing products of elliptic curves with product polarization.

Before we state the theorem, which describes the branch and singular locus in A\ Ai iU with the same letter. 9. Let η > 4 be an arbitrary integer. The singular locus of «4ι,η consists of the two disjoint curves C\ and C2 each isomorphic to the modular curve Χ (η), which are contained in the Humbert surface Τίο .

Bauer and T. Szemberg Case II: a > 2 and β > 2. Let Fi be a general fibre of ψ\. Then we obtain [Fi] = [A] • h\ = a2zh\h22hz + aZ2h\h23h2. Furthermore, we have [Fi] = d\ · [Fi], hence a2Z = di • a > 6 · 2 = 12 and also 032 > 1 2 . Arguing in the same way with the projections φ2 and φζ we obtain a,ij > 12 for i,j = 1,2,3, i Φ j. 1 then yields α - 1 8 3 < a(a - 27) = ϋ ( 9 ~ a kj) ^ _216 ' (i,j,fc)es3 a contradiction. We conclude that not all of the projections ψι, φ 2 and surjective. can be • 3.

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