# Degree of irrationality of very general abelian surfaces

## TL;DR

This paper proves that the degree of irrationality for very general polarized abelian surfaces is uniformly bounded by 4, regardless of polarization degree, challenging previous conjectures and extending understanding of their geometric complexity.

## Contribution

It establishes a uniform upper bound of 4 for the degree of irrationality of very general polarized abelian surfaces, regardless of polarization degree.

## Key findings

- Degree of irrationality is bounded above by 4 for very general abelian surfaces.
- Disproves part of a previous conjecture on irrationality bounds.
- Shows the bound is independent of polarization degree.

## Abstract

The degree of irrationality of a projective variety $X$ is defined to be the smallest degree rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces $(A, L)$ of degree $d$. Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by $4$, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld and Ullery.