Abelian Varieties: Proceedings of the International by Herbert Lange, Wolfgang Barth, Klaus Hulek

By Herbert Lange, Wolfgang Barth, Klaus Hulek

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

Example text

One sees that Δχ G {±11 2 } is equivalent to Δ 2 G {±]1 2 }. This leads to ±ll4,±/5. The remaining cases now give Qi, Q2, QZ : = h(Y,Y),Q4 •= h{Y,-Y). 8. g. ζ = i in the description of the curve C4 ). )). 9. Let Μ G Sp(4, Z) be chosen such that Μ • Ho Π Ho φ 0 holds, then Μ must centralize or normalize IQ . In particular Μ · Ho = Ho . Proof This is a consequence of [Br 2 , II. 12]. • For every isotropy group IsojF of a fixed variety Τ the normal subgroup Iso2 Τ ·,— < Μ G I s o Τ I M2 — fl4 > is generated by quasi-reflections.

Away from the union of curves C\ U C2 ) is only fixed by the corresponding involution. The other component is a connected system of 3 curves and 3 surfaces, which forms a ^-configuration: each pair of surfaces intersects transversally in a unique curve; each pair of curves is contained in a unique surface and intersects transversally in the unique intersection point of the three surfaces. Every curve in the configuration is analytically isomorphic to the modular curve Y (2) and parametrizes certain quotients Ε χ Ε/%2· Locally around the common intersection point of the surfaces the moduli space Ai}2 looks like a 3-dimensional complex vector space modulo the action of the ordinary reflection group of order 23 = 8.

Let A be an abelian threefold over C, embedded in P2 x P2 χ UV Then A is a product E\ χ E2 x F3, where E\, E2 and are smooth plane cubics. Proof. 4 we may assume that φι is not surjective. By Lefschetz hyperplane theorem there are no abelian threefolds in Ψ2 χ P2, since P2 x P2 is simply connected. Thus the image φι(Α) C P ^ must be a curve. Then we have a diagram 0 • Si •A > Fi »0 where F i is an elliptic curve, Si an abelian surface and fi a morphism, which is finite onto its image. 1, Si is then a product of elliptic curves E2 = <^2(Si) and F3 = φζ{3\).

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