Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II

Abstract: Subsequent to the previous paper [Tak5], we are concerned with the classification of complex prime $\mathbb{Q}$-Fano $3$-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibrations. Such affine varieties or their appropriate weighted projectivizations (possibly allowing some coordinates have weights $0$) are called key varieties for prime $\mathbb{Q}$-Fano 3-folds. The purpose of this paper is to give new constructions of a $14$-dimensional affine variety $\Upsilon_{\mathbb{A}}^{14}$ and a $15$-dimensional affine variety $\Pi_{\mathbb{A}}^{15}$ related with $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibration, which were constructed and were shown to be key varieties in the papers [Tak3,Tak4] and [Tay]. It is well-known that the affine cone of the Segre embedded $\mathbb{P}^{2}\times\mathbb{P}^{2}$ is defined as the null loci of the so called $\sharp$-mapping of a 9-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Inspired with this fact, we construct $\Upsilon_{\mathbb{A}}^{14}$ and $\Pi_{\mathbb{A}}^{15}$ in the same way coordinatizing $J$ with 9 and 10 parameters, respectively. The coordinatization with $9$ parameters is derived by using a fixed primitive idempotent, and the associated Peirce decomposition. The coordinatization with $10$ parameters is derived from the construction of a quadratic Jordan subalgebra generated by $\sharp$-products of two elements due to Petersson [Pe1].