Geometric criteria for $\mathbb A^1$-connectedness and applications to norm varieties

TL;DR

This paper characterizes $A^1$-connectedness of varieties over a field through homotopies connecting generic points to rational points, and demonstrates that symmetric powers and norm varieties are $A^1$-connected, with implications for $R$-triviality.

Contribution

It provides geometric criteria for $A^1$-connectedness and applies these to symmetric powers and norm varieties, establishing new connections between homotopy and rationality.

Findings

$A^1$-connectedness characterized by homotopies from generic to rational points

Symmetric powers of $A^1$-connected varieties are $A^1$-connected

Norm varieties over characteristic zero fields are $A^1$-connected after algebraic closure

Abstract

We show that $\mathbb A^1$-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) $\mathbb A^1$-homotopies. We also show that symmetric powers of $\mathbb A^1$-connected varieties (over an arbitrary field), as well as smooth proper models of them (over an algebraically closed field of characteristic $0$), are $\mathbb A^1$-connected. As an application of these results, we show that the standard norm varieties over a field $k$ of characteristic 0 become $\mathbb A^1$-connected (and consequently, universally $R$-trivial) after base change to an algebraic closure of $k$.