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

Technical Summary: Actions of $(\mathbb{Z}/4)^4$ on Rationally Connected Threefolds

Problem Statement
The paper addresses the classification of finite abelian subgroups acting faithfully on rationally connected threefolds over the complex numbers $\mathbb{C}$. Specifically, it investigates the group $G = (\mathbb{Z}/4)^4$. While finite subgroups of the Cremona group $\text{Cr}_3(\mathbb{C})$ (birational automorphisms of $\mathbb{P}^3$) are not fully classified, the study of actions on the broader class of rationally connected varieties is motivated by the fact that such actions do not necessarily embed into $\text{Cr}_3(\mathbb{C})$ if the variety is not rational. Previous work has identified non-abelian simple groups (e.g., $\text{PSL}_2(\mathbb{F}_{11})$) and non-abelian groups (e.g., $S_7$) that act on rationally connected threefolds but do not embed into $\text{Cr}_3(\mathbb{C})$. The paper seeks to determine if the abelian group $(\mathbb{Z}/4)^4$ shares this property and to establish sharp bounds for the embedding of groups of the form $(\mathbb{Z}/m)^r$ into $\text{Cr}_3(\mathbb{C})$ versus $\text{Bir}(X)$ for rationally connected $X$.

Methodology
The authors employ the equivariant Minimal Model Program (MMP) and the theory of $G\mathbb{Q}$-factorial varieties. The strategy proceeds through several stages:

1. Reduction to Fano Threefolds: Using the equivariant MMP, any rationally connected threefold $X$ with a faithful $G$-action is shown to be $G$-birational to a terminal $G\mathbb{Q}$-Fano threefold $X'$. The paper rules out the case where $X'$ is a Mori fiber space over a positive-dimensional base, as this would imply $G$ is of "product type," which $(\mathbb{Z}/4)^4$ is not.
2. Analysis of the Anticanonical System: The authors distinguish between the Gorenstein and non-Gorenstein cases for the Fano threefold $X$.
   • Gorenstein Case: If $|-K_X| \neq \emptyset$, the existence of a $G$-invariant anticanonical divisor $S$ is established. $S$ is shown to be a smooth K3 surface. The action of $G$ on $S$ induces a sequence $1 \to C \to G \to H \to 1$, where $C$ is cyclic and $H$ acts faithfully on $S$. By analyzing invariant lattices of K3 surfaces and representation theory (specifically, restrictions on subgroups of $\text{PGL}_4(\mathbb{C})$ and $\text{Aut}(Q)$), the authors eliminate candidates such as $\mathbb{P}^3$, quartic double solids, and double covers of quadrics. This narrows $X$ to a smooth quartic hypersurface in $\mathbb{P}^4$ or a prime Fano threefold of genus 9. Further lattice arguments eliminate the genus 9 case, leaving only the smooth quartic.
   • Non-Gorenstein Case: If $|-K_X| = \emptyset$, the authors utilize Reid's orbifold Riemann-Roch formula and the Reid basket of singularities. By combining divisibility conditions on orbit lengths (derived from the structure of $(\mathbb{Z}/4)^4$) with an equivariant Euler-characteristic congruence on a resolution, they derive a contradiction, proving that no non-Gorenstein terminal $G\mathbb{Q}$-Fano threefold admits such an action.
3. Equation Identification: For the surviving case (smooth quartic), the authors analyze the linear system $|-K_X|$ and the induced representation on $H^0(X, -K_X)$. They demonstrate that the only smooth quartic admitting a faithful $(\mathbb{Z}/4)^4$ action is the Fermat quartic $x_0^4 + x_1^4 + x_2^4 + x_3^4 + x_4^4 = 0$, with the standard diagonal action.
4. Extension to $(\mathbb{Z}/m)^r$: The paper extends these results to general groups $(\mathbb{Z}/m)^r$ by analyzing primary components and using bounds on the number of generators for abelian $p$-subgroups in $\text{Bir}(X)$ (based on results by Kollár and Zhuang). A specific argument excludes $(\mathbb{Z}/6)^4$ by showing it would require a non-Gorenstein Fano threefold with impossible basket multiplicities.

Key Contributions and Results

• Main Theorem (Theorem 1.3): If $X$ is a rationally connected threefold with a faithful action of $G = (\mathbb{Z}/4)^4$, then $X$ is $G$-birational to the Fermat quartic threefold $X_4$. If $X$ is a terminal $G\mathbb{Q}$-Fano threefold, this birational map is an isomorphism.
• Non-Embeddability (Corollary 1.4): The group $(\mathbb{Z}/4)^4$ does not embed into the Cremona group $\text{Cr}_3(\mathbb{C})$. This is because the Fermat quartic is non-rational (a known result by Iskovskikh and Manin), and any embedding into $\text{Cr}_3(\mathbb{C})$ would imply birationality to $\mathbb{P}^3$.
• Maximal Automorphism Group: The paper confirms that the automorphism group of the Fermat quartic is $\Gamma = (\mathbb{Z}/4)^4 \rtimes S_5$, which is the maximal possible order for a smooth quartic threefold. Consequently, $\text{Bir}(X_4) = \text{Aut}(X_4) = \Gamma$.
• Classification of $(\mathbb{Z}/m)^r$ (Theorem 1.7): The paper provides a complete classification of pairs $(m, r)$ for which $(\mathbb{Z}/m)^r$ embeds into $\text{Cr}_3(\mathbb{C})$ and into $\text{Bir}(X)$ for rationally connected $X$:
  • In $\text{Cr}_3(\mathbb{C})$:
    • $m=2, r \le 6$
    • $m=3, r \le 4$
    • $m \ge 4, r \le 3$
  • In $\text{Bir}(X)$ (rationally connected):
    • $m=2, r \le 6$
    • $m \in \{3, 4\}, r \le 4$
    • $m \ge 5, r \le 3$
• Uniqueness of $(\mathbb{Z}/4)^4$: Among groups of the form $(\mathbb{Z}/m)^r$, the group $(\mathbb{Z}/4)^4$ is the unique example that embeds into $\text{Bir}(X)$ for some rationally connected threefold but fails to embed into $\text{Cr}_3(\mathbb{C})$.

Significance
The paper claims to provide a complete classification of the pairs $(m, r)$ for which the group $(\mathbb{Z}/m)^r$ embeds into the Cremona group $\text{Cr}_3(\mathbb{C})$ and into the birational automorphism group of a rationally connected threefold. By establishing $(\mathbb{Z}/4)^4$ as the unique abelian group of this form that acts faithfully on a rationally connected threefold without embedding into $\text{Cr}_3(\mathbb{C})$, the work fills a gap in the understanding of finite abelian subgroups in dimension three. It complements earlier classifications of simple groups and non-abelian groups, solidifying the trichotomy of finite abelian group actions (product type, K3 type, and the exceptional Fano case) proposed in prior literature. The result relies on the birational superrigidity of the Fermat quartic and the specific constraints imposed by the group structure on the geometry of Fano threefolds.