On classification of non-Gorenstein \(\mathbb{Q}\)-Fano 3-folds of Fano index 1 (Q1907765)

We consider the classification problem of non-Gorenstein \(\mathbb{Q}\)-Fano 3- folds of Fano index 1. Let \(I\) be the singularity index of a non- Gorenstein \(\mathbb{Q}\)-Fano 3-fold \(X\), then by definition \(- IK_X = IH\) in \(\text{Pic} X\), where \(H\) is a Cartier divisor. If \(|H |\) contains a smooth surface \(S\), then \(S\) is an Enriques surface. So classifying these Fano 3-folds also answers the classification problem of 3-folds having Enriques surfaces as hyperplane sections in the case that they have only terminal singularities. In this paper we classify non-Gorenstein \(\mathbb{Q}\)-Fano 3-folds of Fano index 1 with the assumption that they have only cyclic quotient singularities.