Associated forms of binary quartics and ternary cubics

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 ${\mathcal{Q}}_n^{n(d-2)\ast }$. 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.