Nonrational weighted hypersurfaces

Abstract

The aim of this paper is to construct (i) infinitely many families of nonrational $\mathbb{Q}$-Fano varieties of arbitrary dimension $\ge 4$ with at most quotient singularities, and (ii) twelve families of nonrational $\mathbb{Q}$-Fano threefolds with at most terminal singularities among which two are new and the remaining ten give an alternate proof of nonrationality to known examples. These are constructed as weighted hypersurfaces with the reduction mod $p$ method introduced by Kollár [10].