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k-Wahl chains and cyclic quotient singularities

Yusuke Sato

Abstract

We study two-dimensional cyclic quotient singularities defined by $k$-Wahl chains, a class of Hirzebruch--Jung continued fractions obtained inductively starting from $[k+2]$. This class includes the classical Wahl singularities in the case $k=2$ and also contains cyclic quotient singularities arising from $k$-generalized Markov triples. For singularities defined by $k$-Wahl chains, we prove that the combinatorics of the continued fraction is encoded in the special representations of the associated cyclic group. We also study zero continued fractions on the dual side and obtain consequences for the deformation theory of these singularities, including the existence of extremal P-resolutions in the case of $1$-Wahl chains.

k-Wahl chains and cyclic quotient singularities

Abstract

We study two-dimensional cyclic quotient singularities defined by -Wahl chains, a class of Hirzebruch--Jung continued fractions obtained inductively starting from . This class includes the classical Wahl singularities in the case and also contains cyclic quotient singularities arising from -generalized Markov triples. For singularities defined by -Wahl chains, we prove that the combinatorics of the continued fraction is encoded in the special representations of the associated cyclic group. We also study zero continued fractions on the dual side and obtain consequences for the deformation theory of these singularities, including the existence of extremal P-resolutions in the case of -Wahl chains.

Paper Structure

This paper contains 6 sections, 14 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $k$-Wahl chains and cyclic quotient singularities
  3. Length encoding
  4. Special representations
  5. Length encoding of $k$-Wahl chain
  6. Zero continued fractions and P-resolutions

Key Result

Theorem 1.1

Let $G=\frac{1}{r}(1,a)$ be a cyclic quotient singularity with $\gcd(r,a)=1$, and suppose that the Hirzebruch--Jung continued fraction $\frac{r}{a}=[b_1,\dots,b_s]$ is a $k$-Wahl chain. Then the length encoding of $[b_1,\dots,b_s]$ coincides, up to reversal, with the length encoding of special repre

Figures (2)

  • Figure 1: Zero continued fractions and polygon triangulations
  • Figure 2: Adding a new vertex

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 2.4: GMS
  • Theorem 2.5: GM
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 25 more