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A note on abelian varieties embedded in quadrics. (English) Zbl 1110.14016

A. Van de Ven showed [Ann. Mat. Pura Appl., IV. Ser. 103, 127–129 (1975; Zbl 0304.14025)] using numerical methods that an abelian \(d\)-fold can be embedded in \({\mathbb P}^{2d}\) only if \(d=1\) (smooth cubic curves) or \(d=2\) (Horrocks-Mumford surfaces). This paper proves a similar result for abelian \(d\)-folds in \(2d\)-dimensional quadrics. The only possibility turns out to be \(d=1\) (curves of bidegree \((2,2)\)). Numerical methods are not quite sufficient, though. Using the self-intersection formula and Riemann-Roch reduces to \(d\leq 3\), but the cases \(d=2\) and \(d=3\) are eliminated by using results of Lazarsfeld and of Iyer on line bundles of certain types on abelian varieties.

MSC:

14E25 Embeddings in algebraic geometry
14C99 Cycles and subschemes

Citations:

Zbl 0304.14025
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References:

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