Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II
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| Formato: | Preprint |
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2024
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| _version_ | 1866929297629904896 |
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| author | Takagi, Hiromichi |
| author_facet | Takagi, Hiromichi |
| contents | 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 $Υ_{\mathbb{A}}^{14}$ and a $15$-dimensional affine variety $Π_{\mathbb{A}}^{15}$ 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 $Υ_{\mathbb{A}}^{14}$ and $Π_{\mathbb{A}}^{15}$ 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II Takagi, Hiromichi Algebraic Geometry Rings and Algebras 14J45, 14E30, 17C10, 17C50 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 $Υ_{\mathbb{A}}^{14}$ and a $15$-dimensional affine variety $Π_{\mathbb{A}}^{15}$ 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 $Υ_{\mathbb{A}}^{14}$ and $Π_{\mathbb{A}}^{15}$ 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]. |
| title | Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II |
| topic | Algebraic Geometry Rings and Algebras 14J45, 14E30, 17C10, 17C50 |
| url | https://arxiv.org/abs/2403.15991 |