Resolving quotient singularities II

One of the first tales of algebraic geometry that I was told was the story of Jung-Hirzebruch continued fractions and their role in resolving cyclic quotient singularities. The Jung-Hirzebruch continued fraction expansion of a rational number q is the expression

q=b_0-\frac{1}{b_1-\frac{1}{b_2-\frac{1}{\ddots}}}

which one often condenses to [b_0,b_1,\dots]. These are always finite expansions, at least for rational numbers as contemplated here. As promised, for the second instalment of this miniseries on quotient singularities, I will explain the application of these expansions to resolving quotient singularities of the form \frac{1}{r}(1,a) as well as why one is justified in just considering these singularities to represent all of those arising from quotients of smooth varieties by finite group actions.

Here are my notes, which are largely adapted from Kollar’s book on resolutions. Like many things, this can all be conveniently viewed via toric geometry where the continued fractions also make a ready appearance. It is also easier there to sight the final few unjustified claims in my notes, such as the fact that the continued fraction expansion knows the intersection pairing between exceptional curves.

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