I have a vague question:
Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells us when the stable locus $X^{s}$ will be non-empty?
Any kind of reference, addressing the above question, will be useful.
I'm particularly interested in the case of bound quivers (King's stability conditions) as follows:
Let $\alpha$ be a dimension vector and $A$ be a finite-dimensional algebra over an algebraically closed field of characteristic $0$. We take an irreducible component $C$ of $\operatorname{mod}_A(\alpha)$. Then, we have the action of $\operatorname{GL}_\alpha$ on $C$ and we take a stability parameter (or weight) $\theta$ and construct the G.I.T. quotient and hence we get the moduli space $\mathcal{M}(\alpha,\theta)$ and I would like to know if there are some conditions that allow us to say that $\mathcal{M}^s(\alpha,\theta)$ won't be empty. So, in particular, I'm asking if there some special conditions or some special algebras such that there would exist a $\theta$ stable module, for some stability parameter $\theta$.
