Néron-Severi groups of product abelian surfaces

2014/02/10 by Julian Rosen, Rosen, Julian, Ariel Shnidman +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1402.2233

28 pages; revised introduction and added Section 5

openalex publication_date 2014/02/10 · arxiv created 2014/10/12 · arxiv updated 2014/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a natural parameterization of the Néron-Severi group of a product A = E× 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 ℙ⁴, i.e. which come from the Horrocks-Mumford bundle.

Citations

• Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms

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