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 
$$
\bar{1} \cdot (z,w) = (z e^{\frac{2\pi i}{m}}, w e^{\frac{-2\pi i}{m}}),
$$
where $\bar{1} \in \mathbb{Z}\_m$.
For $m=2$, in other words $(z,w) \sim (-z,-w)$, there is a resolution $\mathcal{O}(-2)$ for $\mathbb{C}^2/\mathbb{Z}\_2$ with the holomorphic map $\pi:\mathcal{O}(-2) \rightarrow \mathbb{C}^2/\mathbb{Z}\_2$ defined as 
$$
\pi:(z, \xi) \mapsto [z\sqrt{\xi}, \sqrt{\xi}]
$$ 
on $\mathcal{O}(-2)|\_{U\_1}$ where $U\_1 = \{[z,w] \in \mathbb{CP}^1|w\not = 0\}$,
and 
$$
\pi:(w, \eta) \mapsto [\sqrt{\eta}, w\sqrt{\eta}]
$$
on $U\_2 = \{[z,w] \in \mathbb{CP}^1|z\not = 0\}$ similarily.


However, for $m=3$ I have no idea to write down the similar holomorphic map and the resolution. Is there any reference point out the resolution of $A\_3$-singularity?



resolution for the du Val's $(A_3)$-singularity
The quotient $\mathbb{C}^2/\mathbb{Z}\_3$ is the hypersurface
$$
xz = y^3
$$
in $\mathbb{A}^3$. To resolve it it is enough to blow up the origin. The resulting variety is a hypersurface in the blowup of $\mathbb{A}^3$ at the origin (this blowup is isomorphic to the total space of $\mathcal{O}(-1)$ on $\mathbb{P}^2$).


