Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1\mathrm{/}2(1,1,1)$-singularites via key varieties

We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1\mathrm{/}2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six $1\mathrm{/}2(1,1,1)$-singularities, each equipped with a prescribed Sarkisov link. Our method involves constructing certain higher-dimensional $\mathbb{Q}$-Fano varieties $\mathrm{\Sigma }$, referred to as key varieties, by extending the Sarkisov links to higher dimensions. We prove that each such 3-fold $X$ arises as a linear section of the corresponding key variety $\mathrm{\Sigma }$, and conversely, any general linear section of $\mathrm{\Sigma }$ yields such an $X$. Various geometric properties of the key varieties $\mathrm{\Sigma }$ are also investigated and clarified.