Solution
= Solution
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>
$$
X_\sigma\cong\mathbb A^2/\mu_3.
$$