Boundedness of canonical \(\mathbb{Q}\)-Fano 3-folds (Q1587880)

Let \(X\) be a terminal weak \(\mathbb{Q}\)-Fano 3-fold over an algebraically closed field of characteristic zero. The Gorenstein index \(I(X)\) of \(X\) is the smallest positive integer \(I\) such that \(IK_X\) is a Cartier divisor. In this paper the authors give an effective bound for \(I(X)\) and for \((-K_X)^3\). Moreover they prove that the terminal \(\mathbb{Q}\)-Fano 3-folds are bounded (i.e. there is a morphism of scheme of finite type whose geometric fibers include all terminal \(\mathbb{Q}\)-Fano 3-folds). The main tools used in the proofs are a ``gluing technique'' for rational curves and a structure theorem of the cone of nef curves.