Obstructions to the existence of good moduli spaces of $A_r$-stable curves

We study obstructions to the existence of separated good moduli spaces for open substacks of the moduli stack ${\mathcal{M}}_{g,n}^r$ of $A_r$-stable curves. Our approach is based on an analysis of families of curves over ${\mathrm{\Theta }}_R$ and ${\overline{\text{ST}}}_R$, building on prior work on the local geometry of ${\mathcal{M}}_{g,n}^r$. We prove that ${\mathcal{M}}_{g,n}^r$ is neither $\mathrm{\Theta }$- nor $\mathsf{\text{S}}$-complete. We then construct an open substack ${\mathcal{U}}_{g,n}^r\subset {\mathcal{M}}_{g,n}^r$ and show that the counterexamples identified in ${\mathcal{M}}_{g,n}^r$ do not occur within this substack. Moreover, we prove that ${\mathcal{U}}_{g,n}^r$ cannot be strictly contained in any other substack of ${\mathcal{M}}_{g,n}^r$ that admit a separated good moduli space. Furthermore, we show that the inclusion ${\mathcal{U}}_{g,n}^r\subset {\mathcal{M}}_{g,n}^r$ is both $\mathrm{\Theta }$- and $\mathsf{\text{S}}$-complete. These results will be used in a forthcoming paper to prove that ${\mathcal{U}}_{g,n}^r$ admits a separated, and indeed proper, good moduli space for $r\le 5$.