On the ideals of general binary orbits

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Let E denote a general complex binary form of order d (seen as a point in ¶ᵈ), and let Ωₑ ¶ᵈ denote the closure of its SL₂-orbit. In this note, we calculate the equivariant minimal generators of its defining ideal Iₑ [a₀,...,ad] for 4 d 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.