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Classification of threefold enc cDV quotient singularities

Jingjun Han, Jihao Liu

TL;DR

This note provides a rough classification framework for enc cyclic cDV quotient singularities by translating the problem into combinatorial data of weights for a group action on $\mathbb{C}^4$. It distinguishes cA and non-cA (including Odd and cD-E) types of defining equations $f$ and leverages toric index bounds, the terminal lemma, and generalized non-canonical lemmas to prove that, for each fixed $k$, either the order $r$ or a primitive weight vector $\beta$ must lie in a finite set. This finiteness enables a finite, structured search over possible singularities, contributing toward ACC-type results and the termination analysis in related work. Overall, the paper reduces the local classification problem to a finite combinatorial problem on weights and group actions, clarifying the landscape of enc cyclic cDV quotient singularities and guiding further, complete classifications.

Abstract

We provide a rough classification of threefold exceptionally non-canonical cDV quotient singularities by studying their combinatorial behavior.

Classification of threefold enc cDV quotient singularities

TL;DR

This note provides a rough classification framework for enc cyclic cDV quotient singularities by translating the problem into combinatorial data of weights for a group action on . It distinguishes cA and non-cA (including Odd and cD-E) types of defining equations and leverages toric index bounds, the terminal lemma, and generalized non-canonical lemmas to prove that, for each fixed , either the order or a primitive weight vector must lie in a finite set. This finiteness enables a finite, structured search over possible singularities, contributing toward ACC-type results and the termination analysis in related work. Overall, the paper reduces the local classification problem to a finite combinatorial problem on weights and group actions, clarifying the landscape of enc cyclic cDV quotient singularities and guiding further, complete classifications.

Abstract

We provide a rough classification of threefold exceptionally non-canonical cDV quotient singularities by studying their combinatorial behavior.
Paper Structure (6 sections, 20 theorems, 101 equations)

This paper contains 6 sections, 20 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. cA type
  4. Non-cA type
  5. Odd type
  6. cD-E type

Key Result

Theorem 1.2

Notations and conditions as in Setting Setting: before terminal lem. Then either $r$ or $\beta\not=\bm{0}$ belongs to a finite set depending only on $k$.

Theorems & Definitions (42)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2: HL22
  • proof
  • Theorem 2.3: cf. Rei87, Jia21
  • Lemma 2.4
  • ...and 32 more