Rational connectedness for groups of proper projective similitudes

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.