论文标题
Mukai品种在非关闭领域的合理性
Rationality of Mukai varieties over non-closed fields
论文作者
论文摘要
我们讨论了Mukai品种的生育特性，即，在任意字段$ \ Mathsf {k k} $ Zero特征上，Prime Fano的三倍的fano fano $ g \属的高维类似物。在尺寸$ n \ ge 4 $的情况下，我们证明这些品种为$ \ mathsf {k} $ - 合理时，仅当它们具有$ \ mathsf {k} $ - 点$ 9 $，我们假设$ n \ ge 5 $。此外，我们证明了\ in \ in \ {7,8,9,10 \} $的Mukai品种，如果它们具有$ \ Mathsf {k} $ - 点。最后，我们证明，Prime Fano三倍的三倍$ 12 $的嵌入$ x \ hookrightArrow \ mathrm {gr}（3,7）$在任何字段上都定义了典型的定义，并使用它来提供新的合理性标准。
We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hookrightarrow \mathrm{Gr}(3,7)$ for prime Fano threefolds of genus $12$ is defined canonically over any field and use this to give a new proof of the criterion of rationality.
