N\'eron-Severi groups of product abelian surfaces

TL;DR

This paper characterizes the Néron-Severi groups of product abelian surfaces using quadratic forms and applies this to determine the existence of smooth curves, line bundles, and embeddings into projective space.

Contribution

It provides a natural parameterization of the Néron-Severi group for product abelian surfaces and applies it to geometric embedding problems.

Findings

Classifies when product abelian surfaces contain smooth curves of fixed genus.

Determines the existence of very ample line bundles of fixed degree.

Identifies which abelian surfaces embed into ^4 via the Horrocks-Mumford bundle.

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.