There is one particular omitted detail in my previous post that I would like to address now. I gave a very classically-styled construction adapted from Corti and Heuberger’s paper of an orbifold del Pezzo surface with four singularities and degree 1/3, which turned out to be what was needed to move outside class TG. I stated that contracting the four (-3)-curves on the blown up surface produced four singularities without proof; I supply one here in slightly greater generality: that contracting a (-r)-curve produces a singularity.
This initial foray sets things up nicely to treat the sumptuous connection between Jung-Hirzebruch continued fractions and resolutions of surface quotient singularities. I’ve used this many times in my work to date but have never gone through the derivation in detail. As the watchmaker once said, it’s about time.