New PDF release: Abelian Varieties: Proceedings of the International

By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Publication through Barth, Wolf, Hulek, Klaus

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

Example text

We apologize for this fact. Surprisingly the singular locus is not affected by it. 1. The conjugacy problem Branch points or singular points of A\,n arise as fixed points of elements of finite order in Γι iTl . Thus, as a first step we have to describe the elements of finite order in Γ\ η and to decide when two of them are conjugate with respect to Γ 1)71 . Gottschling [G01, Go 2] and Ueno [Ue] have determined the conjugacy classes of all elements of finite order in Sp(4,Z). -J. Brasch doing so we have to study, how a conjugacy class of an element in Sp(4, Z ) splits in conjugacy classes of Γι ι Τ ι .

Since φ3) and again S2 = The morphism f2 is an Since E2 is contained in A the exact sequence 0 —> S2 —> A —• E2 —•0 splits. Then it follows A ^ S2 x E2 = Ει χ E2 x £3 and the Theorem is proved. • W e conclude with the following Q u e s t i o n . Is every abelian variety of dimension plane η in (P2)n a product of smooth cubics? Acknowledgements. W e would like to thank Prof. W . Barth for helpful discus- sions. : Abelian Threefolds in Products of Projective Spaces. : Intersection Theory. Ergeb. Math.

P2{S2) is a point, S2 is embedded into P ^ ¥>1(^2) x ^3(^2) χ P^ by according to [3]. In fact S2 — Ει χ E3. isomorphism because ψ2 embeds E2, hence E'2 = E2. Since φ3) and again S2 = The morphism f2 is an Since E2 is contained in A the exact sequence 0 —> S2 —> A —• E2 —•0 splits. Then it follows A ^ S2 x E2 = Ει χ E2 x £3 and the Theorem is proved. • W e conclude with the following Q u e s t i o n . Is every abelian variety of dimension plane η in (P2)n a product of smooth cubics? Acknowledgements.

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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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