# 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 $$\bar{1} \cdot (z,w) = (z e^{\frac{2\pi i}{m}}, w e^{\frac{-2\pi i}{m}}),$$ 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?

• 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. Dec 23 '17 at 16:27
• 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. Dec 23 '17 at 16:29
• Thank you for correcting the sign mistake. Now the equation is correct. 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 $\mathbb{P}^2$).