Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 152 / 1

Let $v_1,\dots ,v_r$ be the primitive lattice generators of the rays of the strongly convex rational cone $\sigma \subseteq N_{\mathbb{R}}$. The smoothness criterion for a toric variety says that the affine toric variety $U_{\sigma }$ is smooth exactly when $v_1,\dots ,v_r$ form part of a $\mathbb{Z}$-basis of the cocharacter lattice of an algebraic torus $N$. Equivalently, there is a basis $e_1,\dots ,e_n$ of $N$ such thatIn particular, the cone must be a simplicial polyhedral cone and its ray generators must be primitive; for a full-dimensional cone the criterion says that those generators form a lattice basis.

The orbit-cone correspondence is an inclusion-reversing bijectionIf $n=\mathrm{rank}N$, thenThe affine chart and orbit closure in a toric variety areIn particular, the zero cone corresponds to the dense algebraic torus, while maximal cones correspond to torus-fixed points.

For every $t\in T_N$, the torus action mapis an automorphism of $X_{\Sigma }$. It induces isomorphisms of local rings, so it carries smooth points to smooth points and singular points to singular points. The singular locus is consequently invariant under $T_N$. If it contains one point $x$, it contains the entire orbit $T_N\cdot x$; hence the torus-invariance of the singular locus of a toric variety proves that $X_{\Sigma }^{\mathrm{sing}}$ is a union of the orbits in the orbit-cone correspondence.

Write $m=a{e}_1^{\ast }+b{e}_2^{\ast }\in M$. The inequalities defining the dual cone areThusIts Hilbert basis of a rational cone isIndeed, after subtracting copies of $(2,3)$ one reduces to $b=0,1,$ or $2$, and the remaining point is generated by $(1,0)$ and $(1,1)$. The coordinate ring of an affine toric variety is thereforePutting $A=x$, $B=xy$, and $C=x^2{y}^3$ gives the alternative presentation

The primitive ray generators of $\sigma$ areand $|\det (v_0,v_3)|=3$. They do not form a lattice basis, so the smoothness criterion for a toric variety shows that $U_{\sigma }$ is singular. The presentation from part (i) gives the same conclusion by the Jacobian criterion: all derivatives of $B^3-AC$ vanish at $A=B=C=0$.

Insert the primitive raysLet $\Sigma$ consist of the conesand all their faces. Sinceevery cone of $\Sigma$ is smooth. Its support is $\sigma$, so this fan subdivision induces a proper birational toric morphismThe source is smooth, and $p$ is an isomorphism over the dense algebraic torus; it is therefore a toric resolution of singularities.

Let $D_1$ and $D_2$ be the toric divisors corresponding respectively to the primitive ray generators $v_1=(3,-2)$ and $v_2=(0,1)$ of the original cone. The principal divisor on a toric variety formula applied to the characters $x={\chi }^{e_1^{\ast }}$ and $y={\chi }^{e_2^{\ast }}$ givesThe toric divisor class sequence therefore presents the divisor class group asThe class of $D_1$ is a generator and $[D_2]=2[D_1]$.