On the ideals of general binary orbits

Abstract

Let $E$ denote a general complex binary form of order $d$ (seen as a point in $\P^d$), and let $\Omega_E \subseteq \P^d$ denote the closure of its $SL_2$-orbit. In this note, we calculate the equivariant minimal generators of its defining ideal $I_E \subseteq \complex[a_0,...,a_d]$ for $4 \leqslant d \leqslant 10$. In order to effect the calculation, we introduce a notion called the `graded threshold character' of $d$. One unexpected feature of the problem is the (rare) occurrence of the so-called `invisible' generators in the ideal, and the resulting dichotomy on the set of integers $d \geqslant 4$.