A criterion for a variety to be a cone (Q1095982)

Let \(X\subset {\mathbb{P}}_{{\mathbb{C}}}\) be a normal Cohen-Macaulay variety and \(L=O_{{\mathbb{P}}_{{\mathbb{C}}}}(1)|_ X\). Assume that \(L^ k=K_ X^{-1}\) for some \(k>0\) and let Irr(X) be the locus of non-rational singularities of X. The authors show that, if Irr(X)\(\neq \emptyset\), then dim(Irr(X))\(\geq k-1\) with equality if and only if Irr(X) is a linear \({\mathbb{P}}_{{\mathbb{C}}}^{k-1}\) and X is a cone with Irr(X) as vertex. As a consequence the authors settle affirmatively a conjecture of \textit{J. P. Murre} and the reviewer about 3-dimensional Gorenstein varieties with very ample anti-canonical bundle.