On the ideals of general binary orbits

Let $E$ denote a general complex binary form of order $d$ (seen as a point in ${\mathrm{\P }}^d$), and let ${\mathrm{\Omega }}_E\subseteq {\mathrm{\P }}^d$ denote the closure of its $S{L}_2$-orbit. In this note, we calculate the equivariant minimal generators of its defining ideal $I_E\subseteq \text{complex}[a_0,\mathrm{.}\mathrm{.}\mathrm{.},a_d]$ 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 $d⩾4$.