AI 中文总结
本文研究了$A_r$稳定曲线模空间存在的障碍，并构造了一个开子栈证明其在特定条件下具有分离良好模空间。
AI 中文摘要
我们研究了$A_r$稳定曲线模栈$\mathcal{M}_{g,n}^r$的开子栈的分离良好模空间存在的障碍。我们的方法基于对覆盖$\Theta_R$和$\overline{\text{ST}}_R$的曲线族的分析，建立在之前关于$\mathcal{M}_{g,n}^r$局部几何的工作之上。我们证明$\mathcal{M}_{g,n}^r$既不是$\Theta$-完整也不是$\textsf{S}$-完整。然后我们构造了一个开子栈$\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$，并显示在$\mathcal{M}_{g,n}^r$中识别出的反例不发生在该子栈中。此外，我们证明$\mathcal{U}_{g,n}^r$不能严格包含在任何其他允许分离良好模空间的$\mathcal{M}_{g,n}^r$子栈中。进一步地，我们证明$\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$的包含关系既是$\Theta$-完整也是$\textsf{S}$-完整。这些结果将在即将发表的论文中用于证明当$r \leq 5$时，$\mathcal{U}_{g,n}^r$具有分离且确实为proper的良好模空间。
英文摘要
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 $Θ_R$ 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 $Θ$- 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 $Θ$- 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 $r \leq 5$.
Comments44 pages, 28 figures; comments are welcome
