发表机构
East China Normal University; Embry-Riddle Aeronautical University（华东师范大学; 埃姆布里-里德尔航空大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究曲线模空间基本群的相对完备化，证明特定分离除子可给出相同完备化，并给出所需除子数量的下界。
AI 中文摘要
我们研究了紧致型曲线模栈及光滑模栈部分紧化中基本群的相对完备化。对于$g\geq3$，紧致型完备化具有交换幂单根，其李代数为$\Lambda^3_0H\oplus H^{\oplus n}$。固定$1\leq h\leq g-1$，我们证明了加入那些属$h$侧至多带有两个标记点的分离除子的一节点轨迹，可得到相同的相对完备化。证明使用了权重$-2$中的显式分离扭转计算、尖点Johnson同态以及灯笼关系。函子性特化给出了在正特征（对于不同于特征的奇数$\ell$）下的相应结果。作为推论，代数辛局部系统的迭代扩张到紧致型的延拓可仅在这些选定的除子上进行检验。我们还证明了任何给出相同相对完备化的分离除子集合至少包含$\binom n2$个除子。
英文摘要
We study relative completions of fundamental groups of moduli stacks of curves of compact type and of partial compactifications of the smooth moduli stack. For $g\geq3$, the compact-type completion has commutative prounipotent radical with Lie algebra $Λ^3_0H\oplus H^{\oplus n}$. Fix $1\leq h\leq g-1$. We show that adjoining the one-node loci of the separating divisors whose genus-$h$ side carries at most two markings gives the same relative completion. The proof uses explicit separating-twist calculations in weight $-2$, pointed Johnson homomorphisms, and the lantern relation. Functorial specialization gives the corresponding results in positive characteristic for odd $\ell$ different from the characteristic. As a consequence, extension to compact type of iterated extensions of algebraic symplectic local systems can be tested on these selected divisors. We also prove that any collection of separating divisors giving the same relative completion contains at least $\binom n2$ divisors.
Comments52 pages
