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By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

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

Sample text

3 gives a diagram 0 • Ει •A • Si •0 where the upper row is an exact sequence of abelian varieties, Ει being an elliptic curve and Si an abelian surface, and fi is a finite morphism of degree di, say. 3, we have 1 3 6. Since φι(Ει) is a point, we have [Ei] - ahlh%h3 + ßh\h\h2 with α, β >0. Claim: We have α / 1 and β Φ 1· Proof: By symmetry it is enough to consider a. Applying the projection formula we get a = Ει • h3 = (φ3)*(Ει) • C7Pa(l) = d e g ( ^ 3 | ^ i ) · d e g ^ C ^ i ) · If we had a = 1, then the morphism φ3\Ει : Ει —> φ3(Ει) would be of degree 1 onto a line in P2> which of course is impossible.

G. [Br 2, II. 2]). Therefore we have merely to consider the action of the groups generated by S or Τ. For the action locally around a fixed point ZQ = we will introduce coordinates around a small neighbourhood of the origin in C 3 by setting Ζ = T3+z) • This gives by a straightforward calculation: S-Z 'higher order terms' T Z 'higher order terms'. To determine the type of singularity we may look at the tangent space and thus are allowed to neglect higher order terms. Thus, linearizing the action one gets: 5: (Χ,Υ,Ζ) Τ: {Χ,Υ,Ζ) ~ (ρίχ,-ρΥ,Ζ) (-X,-iY,Z).

For the action locally around a fixed point ZQ = we will introduce coordinates around a small neighbourhood of the origin in C 3 by setting Ζ = T3+z) • This gives by a straightforward calculation: S-Z 'higher order terms' T Z 'higher order terms'. To determine the type of singularity we may look at the tangent space and thus are allowed to neglect higher order terms. Thus, linearizing the action one gets: 5: (Χ,Υ,Ζ) Τ: {Χ,Υ,Ζ) ~ (ρίχ,-ρΥ,Ζ) (-X,-iY,Z). Obviously the action in the Z-direction along the fixed curve of S resp.

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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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