Degree of irrationality of a very general abelian variety

Abstract

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\le \dim A$. We show that if the map $A^k\to C{H}_0(A)$ has a $d$-dimensional fiber then $k\ge d+(\dim A+1)/2$. 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 $g\ge 3$. This improves results of Alzati and the last-named author in the case of a very general abelian variety.

9 pages