Abstract
Using methods developed by Kollár, Voisin, ourselves and Totaro, we prove that a cyclic cover of , , of prime degree , ramified along a very general hypersurface of degree , is not stably rational if . In dimension 3 we recover double covers of ramified along a very general surface of degree 4 (Voisin) and double covers of ramified along a very general surface of degree 6 (Beauville). We also find double covers of ramified along a very general hypersurface of degree 6. This method also enables us to produce examples over a number field.
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