标题： Gorenstein奇点与$\mathbb{G}_m$作用及全纯微分的模空间 显示英文标题

标题： Gorenstein singularities with $\mathbb{G}_m$-action and moduli spaces of holomorphic differentials

摘要： 给定一个光滑曲线上的全纯微分，我们通过一个测试配置将其与一个具有$\mathbb{G}_m$-作用的Gorenstein奇点相关联。这个构造将全纯微分的层分解为这些奇点的负梯度完备形变空间，将Pinkham关于单项式奇点和Weierstrass半群之间的对应关系推广到Looijenga框架下具有良好$\mathbb{G}_m$-作用的多分支Gorenstein奇点的情况。此外，这个构造为出现在完备形变空间边界上的奇异曲线提供了自然描述，将对称半群和局部完全交的特殊情况推广到任意具有指定零点阶数的全纯除子的Gorenstein曲线。我们的构造提供了一种统一的方法来描述所得奇点及其不变量，如权重和特征，这些最初由Alper--Fedorchuk--Smyth研究过。作为应用，我们分类了Chen--Möller和Yu--Zuo工作中每个非变化层的具有$\mathbb{G}_m$-作用的唯一Gorenstein奇点，将每个非变化层与相应奇点的光滑形变的区域相联系，并研究这些非变化层何时可以通过加权射影空间进行紧化。此外，我们在Hassett--Keel对$\overline{\mathcal M}_g$的对数极小模型程序中，分类了具有有界$\alpha$-不变量的此类奇点。我们还研究了这些奇点的斜率，并利用它们来限制$\overline{\mathcal M}_g$中有效除子的斜率。最后，我们证明了具有固定半群的次典范点的区域具有平凡的塔图利克环，并提供了一个判断它们是否为仿射的准则。 显示英文摘要

摘要： Given a holomorphic differential on a smooth curve, we associate to it a Gorenstein singularity with $\mathbb{G}_m$-action via a test configuration. This construction decomposes the strata of holomorphic differentials into negatively graded versal deformation spaces of such singularities, refining Pinkham's correspondence between monomial singularities and Weierstrass semigroups to the case of Gorenstein singularities with multiple branches in the framework of Looijenga's deformations with good $\mathbb{G}_m$-action. Additionally, this construction provides a natural description for the singular curves that appear in the boundary of the versal deformation spaces, generalizing various special cases from symmetric semigroups and local complete intersections to arbitrary Gorenstein curves that admit canonical divisors with prescribed orders of zeros. Our construction provides a uniform approach to describe the resulting singularities and their invariants, such as weights and characters, initially studied by Alper--Fedorchuk--Smyth. As an application, we classify the unique Gorenstein singularity with $\mathbb{G}_m$-action for each nonvarying stratum of holomorphic differentials in the work of Chen--M\"oller and Yu--Zuo, identify each nonvarying stratum with the locus of smooth deformations of the corresponding singularity, and study when these nonvarying strata can be compactified by weighted projective spaces. Additionally, we classify such singularities with bounded $\alpha$-invariants in the Hassett--Keel log minimal model program for $\overline{\mathcal M}_g$. We also study the slopes of these singularities and utilize them to bound the slopes of effective divisors in $\overline{\mathcal M}_g$. Finally, we show that the loci of subcanonical points with fixed semigroups have trivial tautological rings and provide a criterion to determine whether they are affine.