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Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow

However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
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resolution for the du Val's $(A_3)$-singularity

For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following $$ \bar{1} \cdot (z,w) = (z e^{\...