The diagonal action of $\mu_3$ on $\mathbb A^2_{x,y}$ fixes precisely the monomials whose total degree is divisible by three. Its invariant ring is generated by
$$
x^3,\quad x^2y,\quad xy^2,\quad y^3.
$$
The assignment
$$
x^{3-j}y^j\longmapsto t_1^jt_2,
\qquad0\leq j\leq3,
$$
identifies this invariant ring with the ring in part i. Taking spectra gives the <diagonal cyclic quotient singularity of order three>
$$
X_\sigma\cong\mathbb A^2/\mu_3.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 1 / a / ii / Solution

{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 $-3$.


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 Incoming links: Diagonal cyclic quotient singularity of order three