Diagonal cyclic quotient singularity of order three
= Diagonal cyclic quotient singularity of order three
{title2=$\mathbb A^2/\mu_3$}
The cone generated by $(1,0)$ and $(-1,3)$ has semigroup ring
$$
\mathbb C[t_2,t_1t_2,t_1^2t_2,t_1^3t_2]
\cong\mathbb C[x,y]^{\mu_3},
$$
where $\mu_3$ acts diagonally with weight one. Inserting the ray through $(0,1)$ resolves it, with one exceptional curve of self-intersection $-3$.