# Cyclic quotient singularities

[2018-06-02 Sat]

I have been reading the paper "Flipping surfaces" by Hacking, Tevelev
and UrzĂșa, which is a detailed study of a certain class of 3-fold
flips. There is some background material about cyclic quotient surface
singularities which I keep having to figure out from first principles
whenever I return to this stuff and it would be helpful to have it
written down somewhere. To that end, here is the first in a series of
posts about this, covering the toric model for cyclic quotient
singularities and the minimal resolution.

## Toric picture of cyclic quotient singularities

Given coprime positive integers \(P,Q\), the cyclic quotient singularity of type \(\frac{1}{P}(1,Q)\) is defined to be the quotient \(\mathbf{C}^2/\Gamma\) where \(\Gamma\) is the action of the group of \(P^{th}\) roots of unity on \(\mathbf{C}^2\) given by \(\mu\cdot(x,y)=(\mu x,\mu^Q y)\).

The singularity \(\mathbf{C}^2/\Gamma\) admits a Hamiltonian torus
action generated by a moment map
\(\mu\colon\mathbf{C}^2/\Gamma\to\mathbf{R}^2\) whose image is the
wedge \[\pi(P,Q):=\{(x,y)\in\mathbf{R}^2\ :\ x\geq 0,\ Py\geq
Qx\}.\]

The Hamiltonian functions \(H_1(x,y):=\frac{1}{2}|x|^2\) and
\(H_2(x,y):=\frac{1}{2}|y|^2\) on \(\mathbf{C}^2/\Gamma_{P,Q}\)
Poisson-commute and generate an \(\mathbf{R}^2\)-action whose period
lattice is the set of points
\[(\phi_1,\phi_2)=\left(2\pi\left(\frac{k}{P}+\ell\right),
2\pi\left(\frac{kQ}{P}+m\right)\right),\quad
k,\ell,m\in\mathbf{Z}.\] (The

*period lattice*for an \(\mathbf{R}^n\)-action is the set of points in \(\mathbf{R}^n\) which act as the identity.) The Hamiltonians \(\frac{1}{P}(H_1+QH_2)\) and \(H_2\) therefore give us the standard period lattice \(2\pi\mathbf{Z}\oplus 2\pi\mathbf{Z}\), therefore define an effective Hamiltonian torus action. The image of \(\mathbf{C}^2/\Gamma_{P,Q}\) under \(\mu=\left(H_2,\frac{1}{P}(H_1+QH_2)\right)\) is precisely the polygon \(\pi(P,Q)\).

## Minimal resolution

The minimal resolution \(\pi\colon\tilde{X}\to X\) of a cyclic
quotient singularity \(X\) of type \(\frac{1}{P}(1,Q)\) has the
following exceptional locus: it is a *Hirzebruch-Jung* chain of
embedded spheres \(C_1,\ldots,C_r\):

- The first truncation should be made horizontally: this is the obvious way to ensure that the left-most vertex is Delzant.
- The new right-hand vertex now has outgoing edge vectors \((-1,0)\) and \((P,Q)\); if we write \(P=b_1Q-R_1\) for some \(0\leq R_1\leq Q-1\) then the matrix \(\left(\begin{array}{cc}0&1\\ -1&b_1\end{array}\right)\) applied to these edge vectors sends them to \((0,1)\) and \((Q,R_1)\), so the new right-hand vertex corresponds to a cyclic quotient singularity of type \(\frac{1}{Q}(1,R_1)\). If \(R_1=0\) then this vertex is smooth, so we stop truncating.
- Otherwise, having put this right-hand vertex into this position, we again make a horizontal truncation to make this vertex Delzant. We introduce a new right-hand vertex, and proceed in the same manner.
- Eventually, we get to the point where the remainder \(R_r\) vanishes
so the process terminates (this is essentially the Euclidean
algorithm). At this point, the final outgoing edge is pointing in
the \((1,0)\)-direction.