Motivic Decomposition of a Hyperplane Section of a Milnor Hypersurface Twisted by a Crossed Product Algebra

Abstract

Let $A$ be a central simple algebra over an arbitrary field $F$. Associated to $A$, there is a twisted Milnor hypersurface $X(A)$. Given an element $\alpha \in A$ which generates a Galois extension $L$ of $F$ with $[L: F] = \deg A$, we construct a hyperplane section $Y(A, \alpha)$ of $X(A)$ and give a motivic decomposition for $Y(A, \alpha)$. This generalises work of Xiong and Zainoulline.