# Degree of irrationality of a very general abelian variety

## TL;DR

This paper investigates the irrationality degree of very general abelian varieties, establishing lower bounds on the degree of rational maps and fibers, extending previous results and improving known bounds.

## Contribution

It extends prior work on the degree of irrationality of abelian varieties by providing new lower bounds for very general cases of higher dimension.

## Key findings

- Lower bounds on fiber dimensions for maps to CH_0(A)
- Minimum degree of dominant rational maps to projective space
- Improved bounds on irrationality degree for very general abelian varieties

## Abstract

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the second-named author which covered the cases $d=1,2$. As a geometric application, we obtain that any dominant rational map from a very general abelian $g$-fold to $\mathbb{P}^g$ has degree at least $(3\dim A+1)/2$ for $g\geq 3$. This improves results of Alzati and the last-named author in the case of a very general abelian variety.