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Smoothing of surface singularities via equivariant smoothing of lci covers

Yunfeng Jiang

TL;DR

This work develops a framework to study smoothing of surface singularities via equivariant smoothing of $ ext{lci}$ covers, building on Looijenga-Wahl’s quadratic and discriminant structures to classify which simple elliptic and cusp singularities (and their cyclic quotients) admit $ ext{lci}$ smoothing liftings. By constructing the moduli stack of $ ext{lci}$ covers with a perfect obstruction theory, the authors connect smoothing liftings to KSBA moduli of slc surfaces and derive criteria for the existence of liftings in key cases (notably degrees $d=1,2,3,4,8,9$ for simple elliptic and corresponding cusp scenarios). They provide explicit descriptions of permissible isotropic subgroups and covering data, yielding concrete classifications: for simple elliptic degrees $8$ and $9$ there exist nontrivial $ ext{lci}$ cyclic covers (orders $2$ and $3$, respectively) that lift the smoothing; and for cusps, lifting via $ ext{lci}$ covers is achieved under suitable Milnor-fiber group conditions. The results connect topological invariants of smoothings to algebraic and moduli-theoretic frameworks, offering a structured approach to lifting singularities into $ ext{lci}$ covers and informing the KSBA compactification program.

Abstract

We provide some results of the smoothing of surface singularities by Looijenga-Wahl and study smoothing of isolated surface singularities induced by equivariant smoothing of locally complete intersection ($\lci$) singularities. We classify the situation where the smoothing of a simple elliptic singularity, a cusp singularity or its cyclic quotient is induced by the equivariant smoothing of the $\lci$ covers.

Smoothing of surface singularities via equivariant smoothing of lci covers

TL;DR

This work develops a framework to study smoothing of surface singularities via equivariant smoothing of covers, building on Looijenga-Wahl’s quadratic and discriminant structures to classify which simple elliptic and cusp singularities (and their cyclic quotients) admit smoothing liftings. By constructing the moduli stack of covers with a perfect obstruction theory, the authors connect smoothing liftings to KSBA moduli of slc surfaces and derive criteria for the existence of liftings in key cases (notably degrees for simple elliptic and corresponding cusp scenarios). They provide explicit descriptions of permissible isotropic subgroups and covering data, yielding concrete classifications: for simple elliptic degrees and there exist nontrivial cyclic covers (orders and , respectively) that lift the smoothing; and for cusps, lifting via covers is achieved under suitable Milnor-fiber group conditions. The results connect topological invariants of smoothings to algebraic and moduli-theoretic frameworks, offering a structured approach to lifting singularities into covers and informing the KSBA compactification program.

Abstract

We provide some results of the smoothing of surface singularities by Looijenga-Wahl and study smoothing of isolated surface singularities induced by equivariant smoothing of locally complete intersection () singularities. We classify the situation where the smoothing of a simple elliptic singularity, a cusp singularity or its cyclic quotient is induced by the equivariant smoothing of the covers.
Paper Structure (22 sections, 11 theorems, 75 equations)

This paper contains 22 sections, 11 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Results of Looijenga-Wahl
  3. Smoothing component
  4. Equivariant smoothing
  5. Applications-simple elliptic singularities
  6. Cusp singularities
  7. Moduli stack of lci covers
  8. Outline
  9. Quadratic functions
  10. Topological invariants of smoothing surface singularities
  11. Surface singularity
  12. Smoothing
  13. Gorenstein surface smoothing
  14. Equivariant smoothing
  15. Cover of smoothings
  16. ...and 7 more sections

Key Result

Theorem 1.1

(Looijenga-Wahl) Suppose that $M$ is the Milnor fiber of a smoothing of an isolated surface singularity $(X,0)$ with link $L_X$. Then the quadratic function $Q_M$ on $H_2(M)$ makes it an ordinary quadratic lattice with the associated non-degenerate lattice $(\overline{H}_2(M), Q_M)$. The correspondi where $P$ is a quotient of $H_1(L_X)/H_1(L_X)_{\mathop{\rm tor}\nolimits}\cong H_1(\widetilde{X})$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Proposition 2.1
  • Theorem 3.1
  • Proposition 3.2
  • ...and 5 more