On classification of $\mathbb{Q}$-Fano $3$-folds of Gorenstein index 2. II.

Abstract

In the previous paper, we obtained a list of numerical possibilities of $\mathbb{Q}$-Fano $3$-folds $X$ with Pic $X = \mathbb{Z} (-2K_{X})$ and $h^{0} (-K_{X}) \geq 4$ containing index $2$ points $P$ such that $(X, P) \simeq (\{ xy+z^{2}+u^{a} = 0 \} / \mathbb{Z}_{2}(1, 1, 1, 0), o)$ for some $a \in \mathbb{N}$. Moreover we showed that such an $X$ is birational to a simpler Mori fiber space. In this paper, we prove their existence except for a few cases by constructing a Mori fiber space with desired properties and reconstructing $X$ from it.