Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\rm II}_{2}$ projections
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914471517093888 |
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| author | Takagi, Hiromichi |
| author_facet | Takagi, Hiromichi |
| contents | In our paper [Tak6], we constructed eight families of quasi-smooth prime $\mathbb{Q}$-Fano threefolds, anticanonically embedded in codimension four, using weighted projectivizations of the $14$-dimensional affine variety $Π_{\mathbb{A}}^{14}$or its cone. Let $\widehat{f}\colon\widehat{X}\to X$ be the unique divisorial extraction at one specified singularity of maximal index. In this paper, we explicitly construct the Sarkisov link starting from $\widehat{f}$ for $X$ belonging to seven of these families. This is achieved by using the Sarkisov link associated with the weighted projectivization of $Π_{\mathbb{A}}^{14}$ or its cone corresponding to $X$. As a consequence, we show that the Sarkisov link ends with either a fibration whose general fiber is a del Pezzo surface of degree one or a divisorial contraction of type $(2,1)$ to weighted complete intersections of codimension at most two. We also provide more detailed descriptions of these Sarkisov links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\rm II}_{2}$ projections Takagi, Hiromichi Algebraic Geometry 14J45, 14E30 In our paper [Tak6], we constructed eight families of quasi-smooth prime $\mathbb{Q}$-Fano threefolds, anticanonically embedded in codimension four, using weighted projectivizations of the $14$-dimensional affine variety $Π_{\mathbb{A}}^{14}$or its cone. Let $\widehat{f}\colon\widehat{X}\to X$ be the unique divisorial extraction at one specified singularity of maximal index. In this paper, we explicitly construct the Sarkisov link starting from $\widehat{f}$ for $X$ belonging to seven of these families. This is achieved by using the Sarkisov link associated with the weighted projectivization of $Π_{\mathbb{A}}^{14}$ or its cone corresponding to $X$. As a consequence, we show that the Sarkisov link ends with either a fibration whose general fiber is a del Pezzo surface of degree one or a divisorial contraction of type $(2,1)$ to weighted complete intersections of codimension at most two. We also provide more detailed descriptions of these Sarkisov links. |
| title | Key variety construction of Sarkisov links for prime $\mathbb{Q}$-Fano threefolds of codimension four associated to Type ${\rm II}_{2}$ projections |
| topic | Algebraic Geometry 14J45, 14E30 |
| url | https://arxiv.org/abs/2604.12698 |