Key Varieties for Prime $\mathbb{Q}$-Fano Threefolds related with $\mathbb{P}^2\times \mathbb{P}^2$-Fibrations. Part I

Abstract: We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$ with a $\rm{GL}3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $\Sigma^{13}{\mathbb{A}}$ the affine variety obtained from $\Sigma^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). We show that several weighted projectivizations of $\Sigma^{13}_{\mathbb{A}}$ and $\Sigma^{14}_{\mathbb{A}}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to $24$ classes of the graded ring database. Except No.360 in the database, these prime $\mathbb{Q}$-Fano threefolds have a Type I Tom projection. Moreover, they are not weighted complete intersections of the cluster variety of type $C_2$ introduced by Coughlan and Ducat. We also show that a partial projectivization of $\Sigma^{14}_{\mathbb{A}}$ has a $\mathbb{P}^2\times \mathbb{P}^2$-fibration over the affine space $\mathbb{A}^9$.