A type of the Lefschetz hyperplane section theorem on \({\mathbb{Q}}\)-Fano 3-folds with Picard number one and \({\frac{1}{2}(1,1,1)}\)-singularities (Q438958)

Let \(X\) be a projective \(3\)-fold with cyclic singularities of type \(\frac{1}{2}(1,1,1)\) and with Picard number \(1\). Assume further that the class group of \(X\) is generated by one element \(H\), that \({-}2K_X\) is very ample, and that the linear system \(|{-}K_X|\) has a member which is smooth on the smooth locus of \(X\), and has singularities of type \(\frac{1}{2}(1,1)\) elsewhere. If \(D\sim nH\) is a Weil divisor which has singularities of type \(\frac{1}{2}(1,1)\) along the singular locus of \(X\) and is otherwise smooth, then the main result of the paper under review is that the map \(\mathrm{Cl}(X)\to \mathrm{Cl}(D)/(\mathrm{torsion})\) is injective and has a torsion-free cokernel. Then the author gives a new example of a Calabi-Yau \(3\)-fold with \(\text{Pic}(X)=\mathbb{Z}H\), where \(H^3=8\), \(c_2(X)\cdot H=44\), and whose topological Euler characteristic is \(-88\).