arXiv 2103.110862021-03math.AG

与$\mathbb{Q}$ -纤维化相关的三倍。部件I

Key Varieties for Prime $\mathbb{Q}$-Fano Threefolds related with $\mathbb{P}^2\times \mathbb{P}^2$-Fibrations. Part I

Hiromichi Takagi

备注：: 47 pages, largely revised, especially, explanations in the introduction and the section 7 are updated by taking account of R.Taylor's thesis, University of Warwick

摘要：

我们构造了一个具有$\rm{GL}_3$ -和$(\mathbb{C}^*)^6$ -作用的$14$ -维仿射簇$Σ^{14}_{\mathbb{A}}$.我们用$Σ^{13}_{\mathbb{A}}$来表示从$Σ^{14}_{\mathbb{A}}$通过将一个指定变量设置为$1$而获得的仿射簇（我们将精确定义参考本文的定义1.1）。我们发现，$Σ^{13}_{\mathbb{A}}$和$Σ^{14}_{\mathbb{A}}$的几个加权投影产生，作为加权完全交叉，素$\mathbb{Q}$ -Fano三倍余维四属于$24$类的分级环数据库的例子。除数据库中的No.360外，这些主要的$\mathbb{Q}$ -Fano三倍体具有I型Tom投影。此外，它们不是由Coughlan和Ducat引入的$C_2$型簇簇的加权完全交叉。我们还证明了$Σ^{14}_{\mathbb{A}}$的部分投影化在仿射空间$\mathbb{A}^9$上有一个$\mathbb{P}^2\times \mathbb{P}^2$ -纤维化。

英文摘要：

We construct a $14$-dimensional affine variety $Σ^{14}_{\mathbb{A}}$ with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by $Σ^{13}_{\mathbb{A}}$ the affine variety obtained from $Σ^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we refer the precise definition to Definition 1.1 of the paper). We show that several weighted projectivizations of $Σ^{13}_{\mathbb{A}}$ and $Σ^{14}_{\mathbb{A}}$ produce, as weighted complete intersections, examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to $24$ classes of the graded ring database. Except No.360 in the database, these prime $\mathbb{Q}$-Fano threefolds have a Type I Tom projection. Moreover, they are not weighted complete intersections of the cluster variety of type $C_2$ introduced by Coughlan and Ducat. We also show that a partial projectivization of $Σ^{14}_{\mathbb{A}}$ has a $\mathbb{P}^2\times \mathbb{P}^2$-fibration over the affine space $\mathbb{A}^9$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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