N\'eron-Severi groups of product abelian surfaces

Julian Rosen; Ariel Shnidman

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arXiv · 1402.2233

N\'eron-Severi groups of product abelian surfaces

Julian Rosen·Ariel Shnidman

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Abstract

We give a natural parameterization of the N\'eron-Severi group of a product $A = E\times E'$ of two elliptic curves in terms of quadratic forms. As an application, we determine (in the non-CM case) whether $A$ contains a smooth curve of any fixed genus. We also determine whether $A$ admits a very ample line bundle of any fixed degree. In particular, we determine which of these abelian surfaces embed in $\mathbb{P}^4$, i.e. which come from the Horrocks-Mumford bundle.

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Julian Rosen, Ariel Shnidman. 2014-02-10. N\'eron-Severi groups of product abelian surfaces. https://arxiv.org/abs/1402.2233

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