The variety of flexes of plane cubics

TL;DR

The paper studies the variety $X$ of flexes of plane cubics, establishing a faithful $PG={\rm PSL}_3(\mathbb C)$-action and three main results: (i) $X$ is irreducible and rational; (ii) $X$ is $PG$-equivariantly birational to the homogeneous fiber space $PG\times^{ {\rm Hes}_{i,j}} \ell$ with fiber $\ell\simeq{\mathbb P}^1$ and stabilizers ${\rm Hes}_{i,j}\cong{\rm SL}_2(\mathbb F_3)$; (iii) $X$ is rational. The key approach combines a detailed analysis of $G$-orbits on the cubic space, the Hesse pencil, and the theory of homogeneous fiber spaces and relative sections to produce a concrete birational model. The rationality is deduced by reducing to a product of a rational base $PG/{\rm Hes}_{i,j}$ and the rational line $\ell$, with careful treatment of the finite Hessian stabilizers. These results connect invariant theory, projective geometry, and classical pencils of cubics to give an explicit, computable description of the geometry of flexes.

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 ${\rm SL}_2(\mathbb F_3)$.

Figures (1)

• Figure :

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1: Poi, Kra
• Corollary 1
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 2