Birationally superrigid cyclic triple spaces

We prove the birational superrigidity and the nonrationality of a cyclic triple cover of $\mathbb{P}^{2n}$ branched over a nodal hypersurface of degree $3n$ for $n\ge 2$. In particular, the obtained result solves the problem of the birational superrigidity of smooth cyclic triple spaces. We also consider certain relevant problems.