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In various sources (e.g. here, Theorem 1.1 and here, Theorem 2.1 (3)), a certain notation which uses a fraction followed by a tuple is used to describe surface singularities. For example, the first source describes a particular singularities as $\frac{1}{3}(1,2)$ and $\frac{1}{7}(1,3)$, while the second discusses $\frac{1}{2}(1,1,0)$, $\frac{1}{3}(1,1,1)$, $\frac{1}{4}(2,1,1)$, and $\frac{1}{5}(3,2,1)$. I'm not familiar with this notation, and I can't find a reference for it. Thus, I'm wondering:

What does this notation mean?

The second source indicates that this is related to a newton polygon. How does one obtain a newton polygon from a singularity?

How are these related to other ways of describing singularities? Can one pick out by looking at the numbers whether the singular point is du Val? Canonical? Rational?

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    $\begingroup$ These are cyclic quotient singularities, in particular rational singularities. You can find a lot of information by googling these words. For $1/3(1, \, 2)$ and $1/7(1, \, 3)$, another relevant word to google is Hirzebruch-Jung strings. The others are not surface singularities, but $3$-fold singularities. $\endgroup$
    – Francesco Polizzi
    Commented Jul 21, 2021 at 18:15
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    $\begingroup$ For instance, a Du Val singularity of type $A_n$ is a cyclic quotient singularity of type $1/n(n, \, n-1)$. $\endgroup$
    – Francesco Polizzi
    Commented Jul 21, 2021 at 18:16
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    $\begingroup$ More precisely: if $\sigma $ is the automorphism of $\mathbb{C}^n$ given by the diagonal matrix with entries $(\zeta ^{a_1},\ldots ,\zeta ^{a_n})$, with $\zeta $ a primitive r-th root of 1, the quotient $\mathbb{C}^n/\langle\sigma \rangle$ has a singularity of type $\frac{1}{r}(a_1,\ldots ,a_n )$. $\endgroup$
    – abx
    Commented Jul 22, 2021 at 5:05

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Just to expand on Francesco Polizzi's comments, the notation $1/n(a,b)$ means that you take the quotient of $\mathbb{A}^2$ by the action of the group of $n$-th roots of unity where a root $\mu$ acts by sending $(x,y)$ to $(\mu^a x,\mu^b y)$. This generalises to higher dimensions in the obvious way. These are called cyclic quotient singularities.

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Just address the question as to whether one can read off the fact that $\tfrac1r(a_1,\ldots,a_n)$ is canonical from the numbers $r$ and $a_1,\ldots,a_n$: the answer is yes. It is determined by the Reid--Shepherd-Barron--Tai criterion. Let $0\leq \overline{x} < r$ denote the least nonnegative residue of $x$ modulo $r$. Then $\tfrac1r(a_1,\ldots,a_n)$ is canonical if $$ \frac1r \sum_{i=1}^n \overline{a_ik} \geq 1 $$ for all $k=1,\ldots,r-1$ (and terminal if we take $>1$ instead of $\geq1$ for all such $k$).

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