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

Abstract

We construct a $13$-dimensional affine variety ${\mathcal{H}}_{\mathbb{A}}^{13}$ associated with ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations of relative Picard number $1$. The construction is modelled on the fact that the affine cone over the Segre-embedded ${\mathbb{P}}^2\times {\mathbb{P}}^2$ is the null locus of the $♯$-map of the $9$-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Using three fixed complementary primitive idempotents and the Peirce decomposition, we coordinatize $J$ by $8$ parameters and thereby obtain ${\mathcal{H}}_{\mathbb{A}}^{13}$. We then produce complex prime $\mathbb{Q}$-Fano $3$-folds, anticanonically embedded of codimension $4$, as weighted complete intersections in suitable weighted projectivizations of ${\mathcal{H}}_{\mathbb{A}}^{13}$, in its subvarieties, or in their weighted cones (allowing some coordinates of weight $0$). We refer to ${\mathcal{H}}_{\mathbb{A}}^{13}$ and these projectivizations as key varieties for prime $\mathbb{Q}$-Fano $3$-folds. As an application, we show that a prime $\mathbb{Q}$-Fano $3$-fold of genus $3$ with three $1/2(1,1,1)$-singularities of type No.,5.4 as in \cite{Tak1} arises as a linear section of a weighted projectivization of ${\mathcal{H}}_{\mathbb{A}}^{13}$ with all coordinates of positive weight, and conversely any such threefold is obtained in this way. Moreover, relating ${\mathcal{H}}_{\mathbb{A}}^{13}$ to the $C_2$-cluster variety of Coughlan--Ducat \cite{CD1}, we show that weighted projectivizations of ${\mathcal{H}}_{\mathbb{A}}^{13}$ or of its subvarieties serve as key varieties for prime $\mathbb{Q}$-Fano $3$-folds belonging to $108$ classes in the online database \cite{GRDB}.

40 pages. We include an explicit construction of the Sarkisov link in the proof of Theorem 1.3 (1); minor edits throughout