Symmetric powers of Severi-Brauer varieties

A Severi-Brauer variety over a field $k$ is an algebraic variety $X/k$ that becomes isomorphic to the projective space $\mathbb{P}^n$ after the base change to a separable closure $k^{sep}$. I will show that $(n+1)$-th symmetric power of any $n$-dimensional Severi-Brauer varietiy is rational. We will also study some facts about their arbitrary symmetric powers and Grassmanians. The talk is due to the article of János Kollár and is going to be elementary (only some basic facts from algebraic geometry will be used).