AI 中文总结
研究$(\mathbb{Z}/4)^4$对有理连通三维流形的作用，证明相关双有理结论，得出该群不嵌入$\operatorname{Cr}_3(\mathbb{C})$并结合早期结果，实现对$(\mathbb{Z}/m)^r$嵌入情况的完整分类。
AI 中文摘要
设$G = (\mathbb{Z}/4)^4$。我们证明，若$X$是具有$G$忠实作用的有理连通三维流形，则$X$与费马四次三维流形同$G$双有理。若$X$是终端$G\mathbb{Q}$-法诺三维流形，此双有理等价是双正则的。结果表明，群$G$忠实作用于有理连通三维流形但不嵌入$\operatorname{Cr}_3(\mathbb{C})$。结合早期结果，得到了$(\mathbb{Z}/m)^r$嵌入$\operatorname{Cr}_3(\mathbb{C})$以及嵌入有理连通三维流形$X$的$\operatorname{Bir}(X)$的$(m,r)$对的完整分类。
英文摘要
Let $G=(\mathbb{Z}/4)^4$. We prove that if $X$ is a rationally connected threefold with a faithful action of $G$, then $X$ is $G$-birational to the Fermat quartic threefold. If $X$ is a terminal $G\mathbb{Q}$-Fano threefold, this birational equivalence is biregular. Consequently, the group $G$ acts faithfully on a rationally connected threefold but does not embed into $\operatorname{Cr}_3(\mathbb{C})$. Combined with earlier results, this yields a complete classification of the pairs $(m,r)$ for which $(\mathbb{Z}/m)^r$ embeds into $\operatorname{Cr}_3(\mathbb{C})$, and of those for which it embeds into $\operatorname{Bir}(X)$ for a rationally connected threefold $X$.
