Update: fluther solved my problem

I wasn’t expecting it, but someone on fluther managed to answer my question (see last post). Now I’m in an interesting position: I have a hunch that whenever this certain algebraic equation has nontrivial solutions, that corresponds to the geometric objects I’m studying being the same as each other. But there doesn’t seem to be any clear reason this would be true. As always, I need to know more. There’s a paper I’m going to dive in to, which might have the answer, or point me on the right path.