AI 中文总结
该研究通过结合斜率不等式与退化定理，揭示α-稳定性与Hassett-Keel纲领模紧化的关联，得到光滑曲线的新α-稳定性结果，恢复特定区间的Deligne-Mumford稳定性。
AI 中文摘要
我们研究来自模空间的超越GIT方法的曲线的α-稳定性这一内在概念。证明当9/11<α≤1时，该概念可恢复Deligne-Mumford稳定性；更一般地，当α>2/3-ε时，α-半稳定轨迹包含在Hassett-Keel纲领的已知模紧化中。作为应用，我们还得到光滑曲线的新α-稳定性结果。我们的方法结合了斜率不等式与一个退化定理，该定理表明每个具有非结点奇点的Gorenstein曲线都可等退化为具有G_m作用的Gorenstein曲线。
英文摘要
We study the intrinsic notion of $α$-stability for curves arising from the Beyond GIT approach to moduli spaces. We show that it recovers Deligne-Mumford stability for $9/11<α\le1$, and more generally that the $α$-semistable locus is contained in the known modular compactification of the Hassett-Keel proram for $α>2/3-\varepsilon$. As applications, we also obtain new $α$-stability results for smooth curves. Our approach combines slope inequalities with a degeneration theorem showing that every Gorenstein curve with a non-nodal singularity can be isotrivially degenerated to a Gorenstein curve with a $\mathbb G_m$-action.
Comments33 pages, 5 figures
