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

We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1\mathrm{/}2(1,1,1)$-singularities and with $h^0(-K_X)\ge 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1\mathrm{/}2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov link in higher dimensions.