Equivariant deformation of minimally elliptic singularities

TL;DR

This work extends Wahl’s deformation theory to G-equivariant settings, focusing on minimally elliptic surface singularities and their equivariant smoothings and deformations. By developing quotient-stack techniques, lifting criteria for fundamental cycles, and ω*-constant deformation theory, it establishes that equivariant deformation components arising from lci minimally elliptic singularities admit a perfect obstruction theory and analyzes how partial resolutions and blow-downs behave under G-actions. The results include a G-equivariant Wahl-type theorem, smoothness of deformation maps in partial resolutions, and a framework for understanding the KSBA moduli of slc surfaces with such singularities. The findings have implications for the KSBA compactification and provide a robust toolkit for studying equivariant deformations, obstructions, and smoothing phenomena in cyclic quotients and related cusp/el-type singularities.

Abstract

We study certain equivariant deformation components of minimally elliptic surface singularities under finite group actions. Interesting examples include cyclic quotients of simple elliptic singularities and finite group quotients of cusp singularities, where the resulting quotients remain simple elliptic and cusp singularities, respectively. In cases where the minimally elliptic singularities are locally complete intersection (lci) singularities, we identify equivariant deformation components of general type surfaces containing such singularities that admit a perfect obstruction theory.

Figures (3)

• Figure 1: Node
• Figure 2: Tacnode
• Figure 3: Triangle

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• Definition 2.1
• Definition 2.2
• Remark 2.3