On the ideals of general binary orbits

TL;DR

This paper calculates the minimal generators of the defining ideals of the orbit closures of general binary forms under SL_2 action for degrees 4 to 10, revealing unexpected 'invisible' generators and introducing the 'graded threshold character' concept.

Contribution

It introduces the 'graded threshold character' to compute equivariant minimal generators of orbit ideal closures for binary forms of degrees 4 to 10, highlighting the occurrence of 'invisible' generators.

Findings

Computed minimal generators for degrees 4 to 10

Discovered the occurrence of 'invisible' generators

Identified a dichotomy based on degree d

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 $d \geqslant 4$.