Albanese maps and fundamental groups of varieties with many rational points over function fields

TL;DR

This paper studies the properties of Albanese maps and fundamental groups of complex projective varieties with many rational points over function fields, revealing structural constraints and geometric features.

Contribution

It proves that linear quotients of the fundamental group are virtually abelian and that the Albanese map is surjective with connected, simple fibers.

Findings

Linear quotients of the fundamental group are virtually abelian.

The Albanese map is surjective with connected fibers.

The Albanese map has no multiple fibers in codimension one.

Abstract

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.