Rational connectedness for groups of proper projective similitudes

TL;DR

This paper investigates the rational connectedness of the group of proper projective similitudes associated with quadratic forms over fields of characteristic not 2, providing new conditions and examples related to their structure and properties.

Contribution

It introduces new sufficient conditions for rational connectedness of these groups and constructs examples where this property fails, advancing understanding in quadratic form theory.

Findings

New criteria for rational connectedness based on quadratic form structure

Examples of quadratic forms with non-rationally connected similitude groups

Insights into the relationship between Witt ring powers and group properties

Abstract

For a quadratic form $\varphi$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\bf PSim}^+(\varphi)$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient conditions in terms of structure properties of $\varphi$. We further provide new examples of quadratic forms $\varphi$ belonging to a given power of the fundamental ideal in the Witt ring and such that ${\bf PSim}^+(\varphi)$ is not rationally connected.