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

For $m=(u,v)\in M\cong {\mathbb{Z}}^2$, membership in the dual cone is equivalent toThe Hilbert basis of a rational cone is $(0,1),(1,1),(2,1),(3,1)$. Therefore the coordinate ring of an affine toric variety is

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

Insert the ray generated by $w=(0,1)$ between $v_1=(1,0)$ and $v_2=(-1,3)$. Sincethe subdivided fan is smooth and gives a toric resolution of singularities. The unique exceptional divisor is $D_w$. Asthe self-intersection formula for a toric surface divisor gives $D_w^2=-3$.