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By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

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

Example text

Brasch The moduli space A\, n has at worst quotient singularities. One can view the quotient map H2 —• Ai 7 the moduli space of abelian varieties of complex dimension g is of general type [Tai, Mu].

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 . Every singularity on C\ resp. C2 is locally isomorphic to a product of the modular curve X (η) and the rational triple point Ast 1 resp. e. in the complement of Cx U C2 ) is only fixed by the respective involution. -J. Brasch 42 Proof. Singular points or branch points arise as fixed points of elements of finite order in Γ ι ) η . 2 with non-trivial isotropy group Iso Z.

Let χ be an integer prime to n. By eventually substituting χ by χ + η we may assume that Branch points in moduli spaces of certain abelian surfaces 37 x is odd. This implies the existence of integers r, t with xr — Atn2 = 1. Setting we observe that the matrix ( 0 χ χ - ι ) is contained in the image. Combining everything the image is the group of upper triangular matrices for η odd and in the remaining case 'η even' a subgroup of index 2 as claimed. 5. Every involution in ΓιιΤΙ is conjugate with respect to Γι >η to exactly one of the following matrices in dependence on η mod 4.

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Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 by Herbert Lange, Wolfgang Barth, Klaus Hulek


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