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?

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    $\begingroup$ You have the wrong equation for the group action. Except for $m=2$, the group action you wrote has a non-Gorenstein quotient space. Every $A_m$-singularity is Gorenstein. $\endgroup$ Dec 23 '17 at 16:27
  • $\begingroup$ For the minimal resolution $X\to \mathbb{C}/\mathbb{Z}_m$, the fiber over the singular point is a chain of rational curves with $m-1$ curves, all self-intersections are $-2$. So, for $m=2$, you have only one rational curve, hence you could write the resolution you wrote. May be you should look at the original paper of Artin on such resolutions. $\endgroup$
    – Mohan
    Dec 23 '17 at 16:29
  • $\begingroup$ Thank you for correcting the sign mistake. Now the equation is correct. $\endgroup$ Dec 23 '17 at 16:47

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 $\mathbb{P}^2$).


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