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 $\Pi_{ \mathbb{ A}}^{ 15}$.
Categories: math.AG