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

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--Möller 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 $α$-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.