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

TL;DR

This paper links holomorphic differentials on complex curves to Gorenstein singularities with $ ext{G}_m$-action, classifies these singularities, and explores their applications in moduli space compactifications, singularity theory, and algebraic geometry.

Contribution

It introduces a novel construction associating Gorenstein singularities to holomorphic differentials, classifies these singularities, and applies the results to moduli spaces and singularity classifications.

Findings

Classified Gorenstein singularities with $ ext{G}_m$-action for each nonvarying stratum.

Studied compactifications of nonvarying strata by weighted projective spaces.

Established $K( ext{pi},1)$-property for certain singularities and analyzed their role in the moduli space.

Abstract

Given a holomorphic differential on a smooth complex algebraic curve, we associate to it a Gorenstein curve singularity with $\mathbb G_m$-action via a test configuration. This construction decomposes the strata of holomorphic differentials with prescribed orders of zeros into negatively graded miniversal deformation spaces of such singularities. Additionally, it provides a natural description for the singular curves that appear in the boundary of the miniversal deformation spaces. Our approach leads to a number of applications. We classify the unique Gorenstein singularity with $\mathbb G_m$-action for each nonvarying stratum of holomorphic differentials and study when these nonvarying strata can be compactified by weighted projective spaces. Moreover, extending the classical results about $ADE$ singularities, we establish the $K(\pi,1)$-property for non-hypersurface complete intersection singularities of type $U_7$, $U_8$, $U_9$, and $S_{k}$. We also study singularities with bounded $\alpha$-invariants in the log minimal model program for $\overline{\mathcal M}_g$ 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 varieties.