$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

Abstract

Abstract. In this paper, we proved two results regarding the arithmetics of separably A 1 -connected varieties of rank one. First we proved over a large field, there is an 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.