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

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 $D_w^2=-3$.

Let $w$ be the primitive generator of the new ray $\tau$, and let $u,v$ be its adjacent primitive rays in the smooth subdivided fan. Smoothness gives a unique integer $a$ withBecause $\tau$ is new, it lies in the interior of an original two-dimensional cone. Its adjacent refined rays lie on opposite sides of it inside that cone, which forces $a>0$. Therefore $D_{\tau }^2<0$.

Let $v_1=(1,0)$ and $v_2=(1,m)$ generate the distinguished cone. If a complete fan had exactly one further ray with primitive generator $w=(a,b)$ and were smooth away from $x_{\sigma }$, the other two cones would be smooth. Completeness puts $w$ on the other side of the two boundary rays, so after choosing cyclic orientation $b=-1$. Smoothness would requirehence $ma=-2$. This has no integer solution for $m\ge 3$. Thus no proper toric variety can satisfy the three conditions from the PDF.