Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with $\mathbb{P}^2\times \mathbb{P}^2$-fibrations

Abstract: In the previous papers [Tak8,Tak6], we construct the affine varieties $\Sigma_{\mathbb{A}}^{13}$ and $\Pi_{\mathbb{A}}^{14}$ whose partial projectivizations have $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations. In this paper, we produce prime $\mathbb{Q}$-Fano 3-folds of anticanonical codimension 4 belonging to 23 (resp. 8) classes of Graded Ring Database [GRDB] as weighted complete intersections of weighted projectivizations of $\Sigma_{\mathbb{A}}^{13}$ (resp. $\Pi_{\mathbb{A}}^{14}$ or the cone over it). Together with the results of [CD] and [Tak5], prime $\mathbb{Q}$-Fano 3-folds of anticanonical codimension 4 are constructed for 141 classes among the 143 classes in [GRDB].