The Versal Deformation of Cyclic Quotient Singularities

We describe the versal deformation of two-dimensional cyclic quotient singularities in terms of equations, following Arndt, Brohme and Hamm. For the reduced components the equations are determined by certain systems of dots in a triangle. The equations of the versal deformation itself are governed by a different combinatorial structure, involving rooted trees. One of the goals of singularity theory is to understand the versal deformations of singularities. In general the base space itself is a highly singular and complicated object. Computations for a whole class of singularities are only possible in the presence of many symmetries. A natural class of surface singularities to consider consists of the affine toric singularities. These are just the cyclic quotient singularities. Their infinitesimal deformations were determined by Riemenschneider [8]. Explicit equations for the versal deformation are the result of a series of PhD-theses. Arndt [1] gave a recipe to find equations of the base space. This was further studied by Brohme [3], who proposed explicit formulas. Their correctness was finally proved by Hamm [6]. One of the objectives of this paper is to describe these equations. Unfortunately it is difficult to find the structure of the base space from the equations. What one can do is to study the situation for low embedding dimension e. On the basis of such computations Arndt [1] conjectured that the number of irreducible components should not exceed the Catalan number Ce−3 = 1 e−2 (2(e−3) e−3 ) . This conjecture was proved in [11] using Kollár and Shepherd-Barron’s description [7] of smoothing components as deformation spaces of certain partial resolutions. It was observed by Jan Christophersen that the components are related to special ways of writing the equations of the singularity. In terms of his continued fractions, representing zero, these equations are given in [4, §2], and in terms of subdivisions of polygons in [11, Sect. 6]. A more direct way of operating with the equations was found by Riemenschneider [2]. We use it, and the combinatorics behind it, in this paper to describe the components. From the toric picture one finds immediately some equations, by looking at the Newton boundary in the lattice of monomials: zε−1zε+1 = z aε ε , 2 ≤ ε ≤ e− 1 . These form the bottom line of a pyramid of equations zδ−1zε+1 = pδ,ε. In computing these higher equations choices have to be made. We derive pδ,ε from pδ,ε−1 and pδ+1,ε. As zδ−1zε+1 = (zδ−1zε)(zδzε+1)/(zδzε), we have two natural choices for pδ,ε: pδ,ε−1pδ+1,ε pδ+1,ε−1 or pδ,ε−1pδ+1,ε zδzε . We encode the choice by putting a white or black dot at place (δ, ε) in a triangle of dots. Only for certain systems of choices we can write down (in an easy way) enough deformations to fill a whole component. We call the corresponding triangles of dots sparse coloured triangles.