Abstract
We prove that a very general complex hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) containing an r-plane with multiplicity m is not stably rational under some mild assumptions for \(n \geqslant 3\), \(m, r > 0\). We also prove the failure of stable rationality of a very general hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) admitting several isolated ordinary double points.
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