Symmetries of Fano varieties
Louis Esser, Lena Ji, Joaquín Moraga
TL;DR
The paper investigates how large symmetric groups can act faithfully on $n$-dimensional Fano varieties, with analogous questions for Calabi--Yau varieties and klt singularities. It combines representation theory of $S_k$ (including Schur covers) with geometric techniques (Cox rings for toric varieties, boundedness up to degeneration) to derive sharp and asymptotic bounds on $k$ for classes like toric varieties and quasismooth weighted complete intersections. Notably, it establishes a universal quadratic bound, sharp bounds in toric and WCI cases, and a boundedness result for $S_8$-equivariant Fano $4$-folds, while showing $S_7$-equivariant Fano $4$-folds are unbounded. It also characterizes maximally symmetric Fano WCIs, linking them to Fano–Fermat constructions and clarifying how symmetry interacts with boundedness and potential moduli in birational geometry.
Abstract
We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded.