Rational curves on quotients of abelian varieties by finite groups

TL;DR

This paper investigates conditions under which quotients of abelian varieties by finite groups contain rational curves, showing that a specific age condition on automorphisms guarantees their existence.

Contribution

It extends previous results by demonstrating that an age value of 1 suffices to ensure the presence of rational curves on the quotient.

Findings

If $ ext{age}(g^k)=1$, then $A/raket{g}$ has at least one rational curve.

The age condition $0< ext{age}(g^k)<1$ is not necessary for the existence of rational curves.

The work refines criteria for rational curve existence on quotients of abelian varieties.

Abstract

In [3], it is proved that the quotient of an abelian variety $A$ by a finite order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$ is enough to guarantee that $A/\langle g\rangle$ has at least one rational curve.