Geometric Criteria for A1 - Connectedness and Applications to Norm Varieties.

Abstract

We show that A1-connectedness of a large class of varieties over a field k can be characterized as the condition that their generic point can be connected to a k-rational point using (not necessarily naive) A 1 - homotopies. We also show that symmetric powers of 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) are A 1 - connected. As an application of these results, we show that the standard norm varieties over a field k of characteristic 0 become A 1 - connected (and consequently, universally R-trivial) after base change to an algebraic closure of k.