k-Wahl chains and cyclic quotient singularities

TL;DR

This paper investigates cyclic quotient singularities defined by $k$-Wahl chains, linking their combinatorics to group representations and exploring deformation theory implications.

Contribution

It introduces the class of $k$-Wahl chains, generalizes classical Wahl singularities, and connects continued fraction combinatorics with group representations and deformation theory.

Findings

Continued fraction combinatorics are encoded in special group representations.

Zero continued fractions relate to dual singularity properties.

Existence of extremal P-resolutions for 1-Wahl chains is established.

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.