Geodesic
Cyclic covers that are not stably rational
Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 665-677.

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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) $. 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.
Keywords: stable rationality, Chow group of zero-cycles, cyclic covers. 
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J.-L. Colliot-Thélène; A. Pirutka. Cyclic covers that are not stably rational. Izvestiya. Mathematics , Tome 80 (2016) no. 4, pp. 665-677. http://geodesic.mathdoc.fr/item/IM2_2016_80_4_a3/

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