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

Abstract: We construct a $15$-dimensional affine variety $\Pi_{\mathbb{A}}^{15}$ with a ${\rm GL}{2}$- and $(\mathbb{C}^{*})^{4}$-actions. We denote by $\Pi{\mathbb{A}}^{14}$ the affine variety obtained from $\Pi_{\mathbb{A}}^{15}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). Let $\Pi_{\mathbb{P}}^{13}$ be several weighted projectivizations of $\Pi_{\mathbb{A}}^{14}$, and $\Pi_{\mathbb{P}}^{14}$ the weighted cone over $\Pi_{\mathbb{P}}^{13}$ with a weight one coordinate added. We show that $\Pi_{\mathbb{P}}^{13}$ or $\Pi_{\mathbb{P}}^{14}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to the eight classes No.308, 501, 512, 550, 577, 872, 878, and 1766 of the graded ring database. The construction of $\Pi_{\mathbb{A}}^{15}$ is based on a certain type of unprojection and is inspired by R.Taylor's thesis submitted to University of Warwick. We also show that a partial projectivization of $\Pi_{\mathbb{A}}^{15}$ has a $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibration over the affine space $\mathbb{A}^{10}$. To show this, we introduce another $13$-dimensional affine variety $H_{\mathbb{A}}^{13}$ whose product with an open subset of $\mathbb{A}^{2}$ is isomorphic to a sextic cover of an open subset of $\Pi_{\mathbb{A}}^{15}$.