Actions of $(\mathbb{Z}/4)^4$ on rationally connected threefolds

Abstract

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 $X$.

Disclosure

“–Roch formula. In Section 5 we show that the group (Z/6)4 cannot act faithfully on a rationally connected threefold, which is needed to prove Theorem 1.7. Finally, in Section 6 we prove the main results. Acknowledgements. Version 5.6 of OpenAI’s ChatGPT was used to explore proof ideas, test preliminary arguments, and identify possible references. The author independently checked every source and argument, wrote the final text, and assumes full responsibility for its content.”

PDF page 3

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:06:44.839659Z

Structural counts