ASSOCIATED FORMS OF BINARY QUARTICS AND TERNARY CUBICS

Let \( {\mathcal{Q}}_n^d \) be the vector space of forms of degree d ≥ 3 on ℂn, with n ≥ 2. The object of our study is the map Φ, introduced in earlier articles by M. Eastwood and the first two authors, that assigns every nondegenerate form in \( {\mathcal{Q}}_n^d \) the so-called associated form, which is an element of \( {{\mathcal{Q}}_n^d}^{\left(d-2\right)*} \). 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 Φ induces a rational equivariant involution on the projective space ℙ\( \left({\mathcal{Q}}_n^d\right) \), which is in fact the only nontrivial rational equivariant involution on ℙ\( \left({\mathcal{Q}}_n^d\right) \). 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.