Cyclic covers that are not stably rational

Using methods developed by Kollár, Voisin, ourselves and Totaro, we prove that a cyclic cover of $\mathbb P_{\mathbb C}^n$, $n\geqslant 3$, of prime degree $p$, ramified along a very general hypersurface $f(x_0,\dots , x_n)=0$ of degree $mp$, is not stably rational if $m(p-1) <n+1\leqslant mp$. In dimension 3 we recover double covers of $\mathbb P^3_{\mathbb C}$ ramified along a very general surface of degree 4 (Voisin) and double covers of $\mathbb P^3_{\mathbb C}$ ramified along a very general surface of degree 6 (Beauville). We also find double covers of $\mathbb P^4_{\mathbb C}$ ramified along a very general hypersurface of degree 6. This method also enables us to produce examples over a number field.