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 [Chr91] and [Ste91].
Constructors
cyclic_quotient_singularity — Methodcyclic_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)
falseAttributes
continued_fraction_hirzebruch_jung — Methodcontinued_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]-1//(cf[2]-QQFieldElem(1, cf[3]))
7//5dual_continued_fraction_hirzebruch_jung — Methoddual_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[1] - QQFieldElem(1, dcf[2])
7//2Auxiliary Methods
continued_fraction_hirzebruch_jung_to_rational — Methodcontinued_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//5rational_to_continued_fraction_hirzebruch_jung — Methodrational_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