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

Dawei Chen, Fei Yu

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 $K(π,1)$-property for non-hypersurface complete intersection singularities of type $U_7$, $U_8$, $U_9$, and $S_{k}$. We also study singularities with bounded $α$-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.

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

Abstract

Given a holomorphic differential on a smooth complex algebraic curve, we associate to it a Gorenstein curve singularity with -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 -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 singularities, we establish the -property for non-hypersurface complete intersection singularities of type , , , and . We also study singularities with bounded -invariants in the log minimal model program for and utilize them to bound the slopes of effective divisors in . 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.

Paper Structure

This paper contains 55 sections, 19 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Gorenstein singularities with $\mathbb G_m$-action and test configurations
  3. Uniqueness of singularities for the nonvarying strata
  4. Adding ordinary marked points and the $K(\pi,1)$-property
  5. Weights and characters of singularities
  6. Singularities with bounded $\alpha$-invariants
  7. Slopes of singularities and effective divisors in $\overline{\mathcal{M}}_g$
  8. Affine and tautological geometry of subcanonical points
  9. Further directions
  10. Outline of the paper
  11. Singularities and test configurations
  12. Deformations of singularities with $\mathbb G_m$-action
  13. From differentials to singularities
  14. Properties of the construction
  15. The boundary of $\mathbb P{\rm Def}^{-} (Y, q)$
  16. ...and 40 more sections

Key Result

Theorem 1.1

In the above construction, $Y$ admits an isolated Gorenstein singularity at $q$ with $\mathbb G_m$-action. Canonical divisors in $\mathbb P\mathcal{H} (\mu)$ that produce the same isomorphism class of $(Y, q)$ form an open subset $\mathbb P{\rm Def}^{-}_s(Y, q)$ of smooth deformations in the negativ

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 2.1: Deformations of the ordinary cusp
  • Example 2.2: Deformations with repulsive $\mathbb G_m$-action
  • Remark 2.3
  • ...and 41 more