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

Abstract

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 $\Theta_R$ 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 $\Theta$- nor $\textsf{S}$-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 $\Theta$- and $\textsf{S}$-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 $r \leq 5$.