Motives of Certain Hyperplane Sections of Milnor Hypersurfaces

We construct a hyperplane section $Y$ of a Milnor hypersurface associated to a regular semisimple endomorphism $\phi$. Exploiting its structure as a hyperplane section of a projective bundle and its natural torus action, we give a motivic decomposition of $Y$, which encodes both the cellular structure of $Y$ and the arithmetic of the eigenvalues of $\phi$. This decomposition is proven without using the “nilpotence principle”, that is to say there are no “phantoms”.