I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations', where he proves Gabriel's Theorem characterizing connected quivers with finitely many isoclasses of indecomposable representations. There he has defined, for a finite quiver $Q=(Q_0,Q_1)$ without oriented cycles and for a (fixed) dimension vector $\textbf{d}:= (d_i)_{i \in Q_0} \in \mathbb{Z}_{\geq 0}^n$, the space $E_\textbf{d}$ of representations $M$ of $Q$ having dimension vector equal to $\textbf{d}$, and the group $G_\textbf{d} := \prod_{i \in Q_0} GL_{d_i} (k)$, which acts on $E_\textbf{d}$ by conjugation: that is, for $g:=(g_i)_{i \in Q_0} \in G_\textbf{d}$ and $M := (M_i, \phi_\alpha)_{i \in Q_0, \alpha \in Q_1} \in E_\textbf{d}$, $$g \cdot M := (M_i, \hspace{1mm} g_{t(\alpha)} \hspace{0.5mm} \phi_\alpha \hspace{0.5mm} g_{s(\alpha)}^{-1})_{i \in Q_0, \alpha \in Q_1} \in E_\textbf{d}$$ where the source and target of arrow $\alpha \in Q_1$ are given by $s(\alpha)$ and $t(\alpha)$ respectively.
For a representation $M \in E_\textbf{d}$, we denote the orbit of $M$ under the above action by $\mathcal{O}_M$.
In Lemma 8.2, he views $\mathcal{O}_M, G_\textbf{d}$ and $Aut(M)$ as varieties, and gives a relation between their dimensions (as varieties), which arises essentially out of the standard bijection used to prove the Orbit-Stabilizer Theorem in elementary group theory. Unfortunately, the only definition of an algebraic variety that I am familiar with is that it is the set of common zeroes of a system of polynomial equations over the real or complex numbers. It is not clear to me how $\mathcal{O}_M, G_\textbf{d}$ and $Aut(M)$ may be viewed as varieties in this definition (or if there is an equivalent definition in which things are clearer), and I would really appreciate some help regarding the same.