Exercise: Resolve the 5 forms of DuVal Singularities (Part 1/5: $A_n$ singularities).
The Minimal Model Program has specific conventions in the ways in which it classifies singularities on varieties. In order of weak (less severe) to strong (more severe) singularities, some of the classes singularities a variety can exhibit are terminal, canonical, log terminal, and log canonical singularities. Terminal singularities do not appear in 2 dimensions, hence the weakest surface singularities are the canonical singularities. These are also known as Du Val singularities (2 dimensional canonical singularities). Fix $X$ to be a normal surface over $k:=\Bbb C$. Since $X$ is normal, $X_{\text{sing}}$ is isolated. Recall that given 2 points in 2 varieties, $P\in X,Q\in Y$, we say $P$ and $Q$ are analytically isomorphic if their complete local rings are isomorphic as $k$-algebras, $\hat\OO_{X,P}\cong\hat\OO_{Y,Q}$. Up to analytic isomorphism, there are 5 types of Du Val singularities, known as ADE singularities. That is to say, a point $P\in X$ is a Du Val singularity if and only if $P$ is analytically isomorphic to the singular point on the surface $\{f=0\}\subset \A{k}{3}$, where $f$ is one of the following 5 polynomials：
$A_n:\quad x^2+y^2+z^{n+1}$
$D_n:\quad x^2+z(y^2+z^{n-2}),\:\:n\ge 4$
$E_6:\quad x^2+y^3+z^4$
$E_7:\quad x^2+y(y^2+z^3)$
$E_8:\quad x^2+y^3+z^5.$
The reason for naming these as $ADE$ singularities is because every Du Val singularity arises as the quotient of $\A{k}{2}$ by a finite group. This introduces another type of singularity called quotient singularities, where a variety is isomorphic to the quotient of a smooth variety by a finite group. These types of varieties have an especially nice description when they are toric. Given a lattice $N$ with a sub-lattice $N\subset N’$ of finite index the group $G=N/N’$ acts on the torus $T_{N’}$ and has $\Bbb C[M’]^G=\Bbb C[M]$. Fixing generators $e_1,…,e_n$ for $N$ and generators $m_1e_1,…,m_ne_n$ for $N’$, with $m_i\in \Z$, we have $\Bbb C[M’]=\C[U_1,…,U_n]$ and $\C[M]=\C[X_1,…,x_n]$ with $X_i=U_i^{m_i}$. An element $(a_1,…,a_n)\subset G\cong\bigoplus_i\Z/m_i\Z$ acts on a monomial $U^\alpha$ by $U^\alpha\mapsto e^{2\pi i\sum_i\frac{a_i\alpha_i}{m_i}}U^\alpha$. Given any full dimensional cone $\sigma\subset N_\R$ whose generators in $N_\R$ are linearly independent (also called an $n$-simplex), we can take $N’$ to be the sub-lattice generated by the minimal ray generators of $\sigma\cap N$, yielding a cone $\sigma ‘\subset N’$ with $\A{\C}{n}\cong U_{\sigma’}\ra U_\sigma$, where the morphism is induced by the identity map on $N$. Then $G$ acts on $U_{\sigma’}$, and $U_\sigma =U_{\sigma’}/G$ because $A_\sigma=A_{\sigma’}\cap\C[M]=A_{\sigma’}\cap\C[M’]^G=A_{\sigma’}^G$. It follows that if $\sigma$ is any simplex then $U_\sigma$ is a product of a quotient and a torus. Therefore, varieties of simplicial fans have at worst quotient singularities.
Now we will begin to give resolutions of the above canonical singularities. Let $X=\{x^2+y^2+z^{n+1}=0\}\subset\A{k}{3}$. I claim that $X\cong \A{k}{2}/\mu_n$, where $\mu_n$ is the cyclic group of $n$th roots of unity, acting on the closed points of $\A{k}{2}$ by $\zeta (x,y)\mapsto (\zeta x,\zeta^n y)$.
We will first recall some general facts about affine toric varieties. Every two dimensional singular affine toric variety is isomorphic to the affine variety of a cone $\sigma\subset \R^2$ with minimal ray generators $e_2,me_1-ke_2\in\Z^2$, where $0\le k<m$ and $(m,k)=1$. Call such a variety type $(m,k)$. In this situation $N’=m\Z\oplus\Z\subset \Z^2$ so $G=\Z/m\Z\cong\mu_m$ and the action of $G$ on $U_\sigma\cong\A{k}{2}$ described above is just $\zeta (x,y)\mapsto(\zeta x,\zeta^k y)$. If we take $\sigma = \op{cone}(e_2, (n+1)e_1-ne_2)$ with $n\ge 1$, then
$$A_\sigma= k[x,y]^{\mu_{n+1}}=k[x^{n+1},y^{n+1}, xy ]\cong k[x,y,z]/(xy-z^{n+1}).$$
After a transform $x\mapsto \frac{ix-y}{2},:y\mapsto\frac{ix+y}{2}$, we see that in this case $X=\spec{k[x,y,z]/(x^2+y^2+z^{n+1})}=\A{k}{2}/\mu_n$.
Every affine toric surface of type $(m,n)$ has a Hirzebruch-Jung resolution of singularities. The resolution is given as follows. Let $\sigma_0:=\sigma=\op{cone}(e_2,me_1-ke_2)$ be the initial singular cone. Insert the ray $\R_{\ge 0}e_2$ yielding a fan that consists of the smooth cone $\delta_1=\op{cone}(e_2,e_1)$ and the cone $\sigma_1=\op{cone}(e_1,me_1-ke_2)$. The lattice isomorphism that puts $\sigma_1$ in its standard form is first a rotation $R= \left(\begin{smallmatrix} 0 & -1 \ 1 & 0 \end{smallmatrix} \right)$ and then $ \left(\begin{smallmatrix} 1 & 0 \ a_1 & 1 \end{smallmatrix} \right)$, making $\sigma_1=\op{cone}(e_2,m_1e_1-k_1e_2)$ where $0\le k_1<m_1$ are coprime, $m_1=k$, and $k_1=a_1k-m$, for some integer $a_1\ge 2$. If $k_1=0$ then the cone $\sigma_1$ is smooth and we stop. Otherwise, $\frac{m}{k}=a_1-\frac{1}{\frac{k_1}{m}}$ and we repeat the process. Step $i$ is done as follows. We have a cone $\sigma_{i-1}=\op{cone}(e_2,m_{i-1}e_1-k_{i-1}e_2)$. Adding the ray $\R_{\ge 0}e_1$ yields a nonsingular cone $\delta_i$ and a new cone $\sigma_i=\op{cone}(e_1,m_{i-1}e_1-k_{i-1}e_2)$. The transformation of $\sigma_i$ into its standard form is given by $A_iR$ where $A_i= \left(\begin{smallmatrix} 1 & 0 \ a_i & 1 \end{smallmatrix} \right)$, which takes $(m_{i-1},-k_{i-1})\mapsto (m_i,-k_i)=(k_{i-1},-(a_ik_{i-1}-m_{i-1}))$ and $a_i\ge 2$ is an integer. In step $i\ge 2$, adding the ray $\R_{\ge 0}e_1$ to $\sigma_{i-1}$ corresponds to adding $(A_1R)^{-1}\cdots (A_{i-1}R)^{-1}e_1=:v_{i}$ to $\sigma_0$. We set $v_1:=e_1$ in step 1. The process ends after step $r$ after adding the ray $\R_{\ge 0}v_r$ to $\sigma_{r-1}$, when $k_{r}=0\implies \frac{m_{r-1}}{k_{r-1}}=a_r$. When this happens, denote $\sigma_r:=\delta_r$. This immediately gives the relations $v_{i+1}+v_{i-1}=a_iv_i$ for each $1\le i\le r$ where $v_0:=e_2,v_{r+1}=(m,-k)$:
$$v_{i+1}=(A_1R)^{-1}\cdots (A_{i}R)^{-1}e_1=(A_1R)^{-1}\cdots (A_{i-1}R)^{-1}(a_ie_1-e_2)=a_iv_i-v_{i-1}.$$
Note that this process does eventually end, because $k_i=a_ik_i-m_i< m_i=k_{i-1}$. Also,
$$ \frac{m}{k}=a_1-\frac{1}{a_2-\frac{1}{a_3-\frac{1}{\cdots -\frac{1}{a_r}}}}.$$
Call $a=(a_1,…,a_r)$ and $\Delta$ the resulting fan. The fan consists of ray generators $v_0,…,v_{r+1}$ in clockwise order such that $\delta_i=\op{cone}(v_{i-1},v_i)$ for all $1\le i\le r+1$, and $U_{\sigma_i}\cong\A{k}{2}$. Every ray $\tau_i=\R_{\ge 0}v_i$ corresponds to a torus invariant divisor $V(\tau_i)$ on $X_\Delta$. Let $E_i$ be the divisors on $X_\Delta$ corresponding to the rays $v_i$, $1\le i\le r$. Letting $\gamma_i =\op{cone}(v_{i-1},v_i),\gamma_i’=\op{cone}(v_i,v_{i+1})$, we have $E_i\subset U_{\gamma_i}\cup U_{\gamma_i’}$. Also $E_i\cap U_{\gamma_i}\cong \{Y=0\}\subset \spec{\C[X,Y]}$ and $E_i\cap U_{\gamma_i’}\cong\{YX^{a_i}=0\}\subset\spec{\C[X^{-1},YX^{a_i}]}$ are glued by $\spec{\C[X,X^{-1}]}$ so it follows that $E_i\cong \PP{k}{1}$ is a rational curve on $X_\Delta$. We see that $E_i\cap E_j=\varnothing$ if $j\notin{i-1,i,i+1}$, and $E_i\cap E_{i+1}\subset U_{\gamma_i’}$. We have $E_i\cap E_{i+1}\cong \spec{\C[X^{-1},YX^{a_i}]/(X^{-1},YX^{a_i})}\cong \spec{\C}$ is a reduced point and thus $E_i.E_{i+1}=1$ for every $1\le i\le r-1$. Lastly, we see that $E_i^2=-a_i$. Indeed, if $u\in M$ then $\op{div}\chi^u=\displaystyle\sum_{i=0}^{r+1} \op{ord}_{V(\tau_i)}(\chi^u) V(\tau_i)$. We have $\op{ord}_{V(\tau_i)}(\chi^u)=\op{proj}_{\tau_i}(u)=\langle u,v_i\rangle$. Choosing $u\in M$ with $\op{ord}{V(\tau_i)}(\chi^u)\ne 0$, it follows that
$$ 0=E_i.\op{div}\chi^u=\langle u,v_{i-1}\rangle+\langle u,v_i\rangle E_i^2+\langle u,v_{i+1}\rangle\implies E_i^2=-a_i.$$
Now returning to our surface $X$ with a type $A_n$ singularity, we see that the fraction $\frac{n+1}{n}$ has $a=(2,…,2)\in\N^n$, and therefore its Hirzebruch-Jung resolution is just $n$ weighted blowups. This yields a chain of $n$ copies of $\PP{k}{1}$ each with self intersection $-2$ and each intersecting its neighbor once.
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