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

TL;DR

The paper builds a new 13-dimensional key affine variety $\mathscr{H}_{\mathbb{A}}^{13}$ from a quadratic Jordan algebra of a cubic form and coordinatizes it by 8 parameters to realize a family of complex prime $\mathbb{Q}$-Fano $3$-folds of codimension 4 as weighted complete intersections. It develops a rich geometric framework: a $(\mathrm{GL}_{2})^{3}\rtimes\mathfrak{S}_{3}$-symmetry, Cayley’s hyperdeterminant structure, and explicit charts, establishing Gorensteinness, normality, and factoriality of $\mathscr{H}_{\mathbb{A}}^{13}$ and its projectivizations. The work demonstrates that linear sections of a weighted projectivization $\mathscr{H}_{\mathbb{P}}^{12}$ yield No.$5.4$-type Fano $3$-folds, and constructs a toric Sarkisov link connecting to their birational models. It also relates the key variety to the $C_{2}$-cluster variety, and introduces relatives $\mathscr{M}_{\mathbb{A}}^{8}$, $\mathscr{S}_{\mathbb{A}}^{6}$, showing that several Fano classes arise from weighted projectivizations of these spaces. Altogether, the paper provides a cohesive bridge between Jordan-algebraic coordinate systems, cluster varieties, and the explicit birational geometry of a broad family of prime $\mathbb{Q}$-Fano $3$-folds, including 108 GRDB classes via concrete embeddings and linear sections.

Abstract

We construct a $13$-dimensional affine variety $\mathscr{H}_{\mathbb{A}}^{13}$ 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 $\sharp$-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 $\mathscr{H}_{\mathbb{A}}^{13}$. We then produce complex prime $\mathbb{Q}$-Fano $3$-folds, anticanonically embedded of codimension $4$, as weighted complete intersections in suitable weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$, in its subvarieties, or in their weighted cones (allowing some coordinates of weight $0$). We refer to $\mathscr{H}_{\mathbb{A}}^{13}$ 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 $\mathscr{H}_{\mathbb{A}}^{13}$ with all coordinates of positive weight, and conversely any such threefold is obtained in this way. Moreover, relating $\mathscr{H}_{\mathbb{A}}^{13}$ to the $C_{2}$-cluster variety of Coughlan--Ducat \cite{CD1}, we show that weighted projectivizations of $\mathscr{H}_{\mathbb{A}}^{13}$ 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}.

Theorems & Definitions (58)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 3.1
• Proposition 3.2