# Cyclic Quotient Singularities

## Introduction

Cyclic quotient singularities are quotients of $\mathbb{C}^2$ by the action of $\mathbb{Z}/n\mathbb{Z}$ acting via $\left(\begin{array}{cc}\xi & 0\\0 & \xi^q\end{array}\right)$, where $\xi$ is a $n$-th root of unity, and $q$ and $n$ are integers, such that $q$ is coprime with $n$, and $0<q<n$.

For the notation we rely on Jan Arthur Christophersen (1991) and Jan Stevens (1991).

Warning

Note that Jan Arthur Christophersen (1991) and Jan Stevens (1991) use Hirzebruch-Jung continued fraction, which differ from the commonly known continued fraction from literature and used in the rest of OSCAR.

## Constructors

cyclic_quotient_singularityMethod
cyclic_quotient_singularity(n::ZZRingElem, q::ZZRingElem)

Return the cyclic quotient singularity for the parameters $n$ and $q$, with $0<q<n$ and $q, n$ coprime.

Examples

julia> cqs = cyclic_quotient_singularity(7, 5)
Cyclic quotient singularity Y(7, 5)

julia> is_affine(cqs)
true

julia> is_smooth(cqs)
false
source

## Attributes

continued_fraction_hirzebruch_jungMethod
continued_fraction_hirzebruch_jung(cqs::CyclicQuotientSingularity)

Return the Hirzebruch-Jung continued fraction associated with the cyclic quotient singularity, i.e. the Hirzebruch-Jung continued fraction corresponding to $n/q$.

The rational number corresponding to a Hirzebruch-Jung continued fraction $[c_1, c_2,\ldots, c_n]$ is $r([c_1, c_2,\ldots, c_n])\ =\ c_1-\frac{1}{r([c_2,\ldots, c_n])}$ where $r([c_n]) = c_n$. Note that this is differs in sign from what is commonly known as continued fraction.

Examples

julia> cqs = cyclic_quotient_singularity(7, 5)
Cyclic quotient singularity Y(7, 5)

julia> cf = continued_fraction_hirzebruch_jung(cqs)
3-element Vector{ZZRingElem}:
2
2
3

julia> ecf = cf-1//(cf-QQFieldElem(1, cf))
7//5
source
dual_continued_fraction_hirzebruch_jungMethod
dual_continued_fraction_hirzebruch_jung(cqs::CyclicQuotientSingularity)

Return the dual Hirzebruch-Jung continued fraction associated with the cyclic quotient singularity, i.e. the Hirzebruch-Jung continued fraction corresponding to $q/(n-q)$.

The rational number corresponding to a Hirzebruch-Jung continued fraction $[c_1, c_2,\ldots, c_n]$ is $r([c_1, c_2,\ldots, c_n])\ =\ c_1-\frac{1}{r([c_2,\ldots, c_n])}$ where $r([c_n]) = c_n$. Note that this is differs in sign from what is commonly known as continued fraction.

Examples

julia> cqs = cyclic_quotient_singularity(7, 5)
Cyclic quotient singularity Y(7, 5)

julia> dcf = dual_continued_fraction_hirzebruch_jung(cqs)
2-element Vector{ZZRingElem}:
4
2

julia> edcf = dcf - QQFieldElem(1, dcf)
7//2
source

## Auxiliary Methods

continued_fraction_hirzebruch_jung_to_rationalMethod
continued_fraction_hirzebruch_jung_to_rational(v::Vector{ZZRingElem})

Return the rational number corresponding to a Hirzebruch-Jung continued fraction given as a vector of (positive) integers.

The rational number corresponding to a Hirzebruch-Jung continued fraction $[c_1, c_2,\ldots, c_n]$ is $r([c_1, c_2,\ldots, c_n])\ =\ c_1-\frac{1}{r([c_2,\ldots, c_n])}$ where $r([c_n]) = c_n$. Note that this is differs in sign from what is commonly known as continued fraction.

Examples

julia> cqs = cyclic_quotient_singularity(7, 5)
Cyclic quotient singularity Y(7, 5)

julia> v = continued_fraction_hirzebruch_jung(cqs)
3-element Vector{ZZRingElem}:
2
2
3

julia> continued_fraction_hirzebruch_jung_to_rational(v)
7//5
source
rational_to_continued_fraction_hirzebruch_jungMethod
rational_to_continued_fraction_hirzebruch_jung(r::QQFieldElem)

Encode a (positive) rational number as a Hirzebruch-Jung continued fraction, i.e. find the Hirzebruch-Jung continued fraction corresponding to the given rational number.

The rational number corresponding to a Hirzebruch-Jung continued fraction $[c_1, c_2,\ldots, c_n]$ is $r([c_1, c_2,\ldots, c_n])\ =\ c_1-\frac{1}{r([c_2,\ldots, c_n])}$ where $r([c_n]) = c_n$. Note that this is differs in sign from what is commonly known as continued fraction.

Examples

julia> r = QQFieldElem(2464144958, 145732115)
2464144958//145732115

julia> cf = rational_to_continued_fraction_hirzebruch_jung(r)
7-element Vector{ZZRingElem}:
17
11
23
46
18
19
37

julia> continued_fraction_hirzebruch_jung_to_rational(cf)
2464144958//145732115

julia> r == continued_fraction_hirzebruch_jung_to_rational(cf)
true
source