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arXiv 2003.10761math.AG
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非封育田块上Mukai品种的合理性

Rationality of Mukai varieties over non-closed fields

Alexander Kuznetsov, Yuri Prokhorov

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摘要：

我们讨论了Mukai簇的双有理性质，即，在零特征的任意域$\mathsf{k}$上的属$g \in \{7,8,9,10\}$的素数Fano三重的高维类似物。在维数$n \ge 4$的情况下，我们证明了这些簇是$\mathsf{k}$ -有理的当且仅当它们有一个$\mathsf{k}$ -点，除了在亏格$9$的情况下，我们假设$n \ge 5$ .进一步证明了亏格为$g \in \{7,8,9,10\}$且维数为$n \ge 5$的Mukai簇若有$\mathsf{k}$ -点则包含柱面.最后，证明了亏格$12$的素Fano三重的嵌入$X \hookrightarrow \mathrm{Gr}(3,7)$在任意域上都是正则定义的，并利用这一点给出了合理性判据的一个新的证明.

英文摘要：

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.

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