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

The diagonal action of ${\mu }_3$ on ${\mathbb{A}}_{x,y}^2$ fixes precisely the monomials whose total degree is divisible by three. Its invariant ring is generated byThe assignmentidentifies this invariant ring with the ring in part i. Taking spectra gives the diagonal cyclic quotient singularity of order three