Symmetries of Fano varieties

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 $k\le m(n)$. We prove that $m(n)>n+\sqrt{2n}$ for every $n$. On the other hand, we show that ${\lim }_{n\to \infty }m(n)/(n+1{)}^2\le 1$. 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 $n\ge 4$. 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.

32 pages, 3 tables