Geometric criteria for 𝔸¹-connectedness and applications to norm varieties

We show that A 1 \mathbb {A}^1 -connectedness of a large class of varieties over a field k k can be characterized as the condition that their generic point can be connected to a k k -rational point using (not necessarily naive) A 1 \mathbb {A}^1 -homotopies. We also show that symmetric powers of A 1 \mathbb {A}^1 -connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic 0 0 ) are A 1 \mathbb {A}^1 -connected. As an application of these results, we show that the standard norm varieties over a field k k of characteristic 0 0 become A 1 \mathbb {A}^1 -connected (and consequently, universally R R -trivial) after base change to an algebraic closure of k k .