ASSOCIATED FORMS OF BINARY QUARTICS AND TERNARY CUBICS

Let Qnd 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 Qnd the so-called associated form, which is an element of Qnd(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 Φ induces a rational equivariant involution on the projective space ℙ(Qnd), which is in fact the only nontrivial rational equivariant involution on ℙ(Qnd). 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.