Associated forms of binary quartics and ternary cubics

TL;DR

This paper studies a special involution on spaces of binary quartics and ternary cubics, linking it to projective duality and classical contravariants, revealing its unique role among rational equivariant involutions.

Contribution

It provides a new interpretation of the involution induced by the associated form map using projective duality and classical contravariants.

Findings

The involution is the only nontrivial rational equivariant involution on the space.

It can be interpreted through projective duality.

The involution is expressed via classical contravariants.

Abstract

Let ${\mathcal Q}_n^d$ be the vector space of forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $\Phi$, introduced in papers [EI], [AI1], that assigns every nondegenerate form in ${\mathcal Q}_n^d$ the so-called associated form, which is an element of ${\mathcal Q}_n^{n(d-2)*}$. We focus on two cases: those of binary quartics ($n=2$, $d=4$) and ternary cubics ($n=3$, $d=3$). In these situations the map $\Phi$ induces a rational equivariant involution on the projectivized space ${\mathbb P}({\mathcal Q}_n^d)$, which is in fact the only nontrivial rational equivariant involution on ${\mathbb P}({\mathcal Q}_n^d)$. In particular, there exists an equivariant involution on the space of elliptic curves with nonvanishing $j$-invariant. In the present paper, we give a simple interpretation of this involution in terms of projective duality. Furthermore, we express it via classical contravariants.