The variety of flexes of plane cubics

TL;DR

This paper studies the geometric structure of the variety of flexes of plane cubics, showing it is an irreducible rational variety with a specific group action and describing its birational equivalence to a homogeneous fiber space.

Contribution

It establishes the irreducibility and rationality of the variety of flexes and describes its structure as a homogeneous fiber space with a specific fiber and group action.

Findings

$X$ is an irreducible rational algebraic variety.

$X$ admits a faithful algebraic action of ${ m PSL}_3$.

$X$ is birationally isomorphic to a homogeneous fiber space over ${ m PSL}_3/K$ with fiber $P^1$.

Abstract

Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with a faithful 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 ${\rm SL}_2(\mathbb F_3)$.