$\mathbb{A}^1$-connected varieties over non-closed fields

TL;DR

This paper investigates the arithmetic properties of separably $ ext{A}^1$-connected varieties over non-closed fields, establishing the existence of $ ext{A}^1$-curves through rational points and the density of integral points.

Contribution

It proves new results on the existence of $ ext{A}^1$-curves and density of integral points for these varieties over large fields and function fields.

Findings

Existence of $ ext{A}^1$-curves through rational points on certain varieties.

Zariski density of integral points over function fields of curves.

Generalization of Hassett-Tschinkel's theorem.

Abstract

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett-Tschinkel for the Zariski density of integral points over function fields of curves.