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

Abstract

The main aim of this paper is to construct certain complex prime $\mathbb{Q}$-Fano $3$-folds anticanonically embedded of codimension $4$. We show that such $3$-folds can be obtained as weighted complete intersections of suitable weighted projectivizations of a $13$-dimensional affine variety ${\mathcal{H}}_{\mathbb{A}}^{13}$ and its related varieties, which we call key varieties. More precisely, we prove that a prime $\mathbb{Q}$-Fano $3$-fold of genus $3$ with three $1/2(1,1,1)$-singularities in the family No. 5.4 of Takagi (2002) arises as a linear section of a weighted projectivization of ${\mathcal{H}}_{\mathbb{A}}^{13}$, and conversely, any such $3$-fold can be obtained in this way. Furthermore, by relating ${\mathcal{H}}_{\mathbb{A}}^{13}$ to the $C_2$-cluster variety constructed by Coughlan and Ducat (2020), we recover their constructions of prime $\mathbb{Q}$-Fano $3$-folds belonging to $108$ families in the Graded Ring Database. The variety ${\mathcal{H}}_{\mathbb{A}}^{13}$ is constructed by adapting the description of the affine cone over the Segre variety ${\mathbb{P}}^2\times {\mathbb{P}}^2$, defined as the null locus of the $♯$-mapping of a $9$-dimensional quadratic Jordan algebra $J$ of a cubic form, to a coordinatization of $J$ with $8$ parameters derived via a Peirce decomposition with respect to three fixed complementary primitive idempotents.