Non-rationality of the symmetric sextic Fano threefold

Abstract

We prove that the symmetric sextic Fano threefold, defined by the equations $\sum X_i=\sum X_i^2=\sum X_i^3=0$ in $\mathbb{P}^6$, is not rational. In view of the work of Prokhorov [P], our result implies that the alternating group $\mathfrak{A}_7$ admits only one embedding into the Cremona group $\mathrm{Cr}_3$ up to conjugacy.