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Abstract

We construct a $13$-dimensional affine variety $\mathscr{H}_{\mathbb{A}}^{13}$ associated with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations of relative Picard number $1$. The construction is modelled on the fact that the affine cone over the Segre-embedded $\mathbb{P}^{2}\times\mathbb{P}^{2}$ is the null locus of the $\sharp$-map of the $9$-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Using three fixed complementary primitive idempotents and the Peirce decomposition, we coordinatize $J$ by $8$ parameters and thereby obtain $\mathscr{H}_{\mathbb{A}}^{13}$. We then produce complex prime $\mathbb{Q}$-Fano $3$-folds, anticanonically embedded of codimension $4$, as weighted complete intersections in suitable weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$, in its subvarieties, or in their weighted cones (allowing some coordinates of weight $0$). We refer to $\mathscr{H}_{\mathbb{A}}^{13}$ and these projectivizations as key varieties for prime $\mathbb{Q}$-Fano $3$-folds. As an application, we show that a prime $\mathbb{Q}$-Fano $3$-fold of genus $3$ with three $1/2(1,1,1)$-singularities of type No.,5.4 as in \cite{Tak1} arises as a linear section of a weighted projectivization of $\mathscr{H}_{\mathbb{A}}^{13}$ with all coordinates of positive weight, and conversely any such threefold is obtained in this way. Moreover, relating $\mathscr{H}_{\mathbb{A}}^{13}$ to the $C_{2}$-cluster variety of Coughlan--Ducat \cite{CD1}, we show that weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$ or of its subvarieties serve as key varieties for prime $\mathbb{Q}$-Fano $3$-folds belonging to $108$ classes in the online database \cite{GRDB}.