Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part I
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916988742270976 |
|---|---|
| author | Takagi, Hiromichi |
| author_facet | Takagi, Hiromichi |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part I Takagi, Hiromichi Algebraic Geometry Rings and Algebras 14J45, 14E30, 17C10, 17C50 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}. |
| title | Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part I |
| topic | Algebraic Geometry Rings and Algebras 14J45, 14E30, 17C10, 17C50 |
| url | https://arxiv.org/abs/2403.14991 |