On the ideals of general binary orbits: The low order cases

Let E denote a general complex binary form of order d (seen as a point in Pd), and let ΩE⊆Pd denote the closure of its SL2-orbit. In this note, we calculate the equivariant minimal generators of its defining ideal IE⊆C[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.