Maybe we can start with an easy example. I want to understand what is the moduli space of say $1\times 1$ matrices under conjugation. So if such a matrix is $A$, the conjugation means $A = G^{-1}AG$ where say $G\in GL_n$. What is the moduli space then? And what is the corresponding moduli space for $2\times 2$ such matrices? Let's say that $k = \mathbb{C}$. What is the moduli space for $n \times n$ such matrices? And do singularities appear?
This is the first question a speaker asked at a seminar last week. And I really want to understand not only what the moduli space of such matrices is but also when and why singularities appear.