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Equivariant smoothing of cusp singularities

Yunfeng Jiang

TL;DR

This work extends Looijenga's cusp smoothing theory to an equivariant framework, showing that a cusp admitting a one-parameter smoothing can be realized as the quotient of a smoothing of an lci cusp under a finite group action. The authors develop equivariant Type III canonical degenerations by leveraging Inoue-Hirzebruch surfaces and equivariant Looijenga pairs, establishing a robust mechanism to translate deformations through $G$-actions. They prove that a hyperbolic $G$-action on a Looijenga pair yields a $G$-equivariant smoothing whose quotient provides the smoothing of the dual cusp, and they demonstrate this with concrete examples. Additionally, the paper connects these results to moduli theories, providing evidence for the moduli stack of $ ext{lci}$ covers over semi-log-canonical surfaces and clarifying how equivariant smoothings inform KSBA-type compactifications and lci-cover constructions.

Abstract

We generalize Looijenga's conjecture for smoothing surface cusp singularities to the equivariant setting. Moreover, we prove that for any cusp singularity which admits a one-parameter smoothing, the smoothing can always be induced by smoothing of locally complete intersection cusps. The result provides evidence for the existence of the moduli stack of $\lci$ covers over semi-log-canonical surfaces.

Equivariant smoothing of cusp singularities

TL;DR

This work extends Looijenga's cusp smoothing theory to an equivariant framework, showing that a cusp admitting a one-parameter smoothing can be realized as the quotient of a smoothing of an lci cusp under a finite group action. The authors develop equivariant Type III canonical degenerations by leveraging Inoue-Hirzebruch surfaces and equivariant Looijenga pairs, establishing a robust mechanism to translate deformations through -actions. They prove that a hyperbolic -action on a Looijenga pair yields a -equivariant smoothing whose quotient provides the smoothing of the dual cusp, and they demonstrate this with concrete examples. Additionally, the paper connects these results to moduli theories, providing evidence for the moduli stack of covers over semi-log-canonical surfaces and clarifying how equivariant smoothings inform KSBA-type compactifications and lci-cover constructions.

Abstract

We generalize Looijenga's conjecture for smoothing surface cusp singularities to the equivariant setting. Moreover, we prove that for any cusp singularity which admits a one-parameter smoothing, the smoothing can always be induced by smoothing of locally complete intersection cusps. The result provides evidence for the existence of the moduli stack of covers over semi-log-canonical surfaces.
Paper Structure (30 sections, 23 theorems, 120 equations, 4 figures)

This paper contains 30 sections, 23 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Inoue-Hirzebruch surfaces with a finite group action
  4. Inoue-Hirzebruch surfaces
  5. Hirzebruch-Inoue modular surface
  6. Finite group action on Inoue-Hirzebruch surfaces
  7. Equivariant anticanonical pairs
  8. Looijenga pairs
  9. Corner blow-ups and internal blow-ups
  10. The minimal model of Looijenga pairs with a finite group action
  11. Finite group action on Looijenga pairs
  12. Toric model of $G$-Looijenga pairs
  13. Equivariant Type III canonical degenerations
  14. Universal deformation of Inoue-Hirzebruch surfaces
  15. Type III equivariant degeneration
  16. ...and 15 more sections

Key Result

Theorem 1.1

(Looijenga, GHK15,Engel) The cusp singularity $(\overline{V}, p^\prime)$ admits a smoothing if and only if the dual cycle $D$ of the dual cusp $p$ is the anti-canonical divisor of a Looijenga pair $(Y,D)$.

Figures (4)

  • Figure 1: cusp $(5,2)$ and its dual
  • Figure 2: resolution $(-4,-3,-3,-1)$
  • Figure 3: cusp $(3,2,2,2,2,2)$
  • Figure 4: blow-down process

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Example 1
  • Definition 3.1
  • Definition 3.2
  • ...and 41 more