The variety of flexes of plane cubics

Abstract

Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with an algebraic action of ${\rm PSL}_3$; (2) $X$ is ${\rm PSL}_3$-equivariantly birationally isomorphic to a homogeneous fiber space over ${\rm PSL}_3/K$ with fiber $\mathbb P^1$ for some subgroup $K$ isomorphic to the binary tetrahedral group ${\rm SL}_2(\mathbb F_3)$.