Affine toric variety ($U_{\sigma }$)

For a rational strongly convex cone $\sigma \subseteq N_{\mathbb{R}}$, the affine toric variety is

For $r>1$ and $\gcd (r,q)=1$, the quotient of ${\mathbb{A}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{q}y)$ is a cyclic quotient surface singularity. Its toric model uses the first-quadrant cone in ${\mathbb{Z}}^2+\mathbb{Z}(1,q)/r$. Conversely, every singular full-dimensional affine toric surface over an algebraically closed field of characteristic zero has this form. Smooth cones and cones with a torus factor require separate treatment.

// Target: geometry-and-topology.bigb

The cyclic quotient surface singularity $\frac{1}{r}(q,1)$ is the only singular point of a complete four-ray toric surface exactly under the displayed condition. Normalize the three smooth cones to the consecutive rays $(a,-1),(1,0),(0,1),(-1,b)$. The singular cone has determinant $1-ab=r$, so $ab=1-r$. In the basis $(-1,b),(0,-1)$, its other generator is $-a(-1,b)+r(0,-1)$, giving quotient parameter $a\mathrm{mod}r$. Thus $a=\pm s$ for a positive divisor $s$ of $r-1$. Conversely these rays construct the surface. Exchanging the two axes replaces $q$ by $q^{-1}$; replacing a factor by its complementary factor preserves the criterion under this equivalence.

For a strongly convex rational cone $\sigma$, the group generated by $S={\sigma }^{\vee }\cap M$ is $M$. Inverting its finitely many monomial generators gives $k[M]$, so $\mathrm{Spec}k[M]$ is a dense open algebraic torus in $\mathrm{Spec}k[S]$. The map ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ extends torus multiplication to the entire variety.

// Target: geometry-and-topology.bigb

The coordinate ring of $U_{\sigma }$ is generated by the characters ${\chi }^m$ for lattice points $m\in {\sigma }^{\vee }\cap M$, with multiplication ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$.

The cone generated by $(1,0)$ and $(-1,3)$ has semigroup ringwhere ${\mu }_3$ acts diagonally with weight one. Inserting the ray through $(0,1)$ resolves it, with one exceptional curve of self-intersection $-3$.