Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 2

For the rays $v_1=(1,0)$, $v_2=(0,1)$, and $v_3=(-a,-b)$, the determinants of consecutive pairs have absolute valuesBy the smoothness criterion for a toric variety, $\mathbb{P}(1,a,b)$ is smooth exactly when $a=b=1$.

The toric divisor class sequence isThe two characters giveThusgenerated by $[D_3]$.

An invariant Weil divisor is Cartier precisely when its local linear support functions are integral on every maximal cone. Applying this to the three cones shows that a class $n[D_3]$ is Cartier exactly when both $a$ and $b$ divide $n$. Since $\gcd (a,b)=1$, this means $ab\mid n$. Thereforeand its map to the class group identifies it with $ab\mathbb{Z}\subseteq \mathbb{Z}$. This is the divisor class and Picard groups of a weighted projective plane.

The Cox ring has one variable for each ray and is $\mathbb{C}[x,y,z]$, graded byThe irrelevant locus is the common zero of all three variables. The quasitorus associated with the class group is ${\mathbb{C}}^{\ast }$ and acts byThe Cox construction therefore identifies the closed points of $\mathbb{P}(1,a,b)$ with

A toric morphism is induced by a lattice map carrying each source cone into a target cone. Compose a hypothetical fan map from the fan of ${\mathbb{P}}^2$ to the product fan with either coordinate projection to the fan of ${\mathbb{P}}^1$. If the three primitive source rays are $r_1,r_2,r_3$ with $r_1+r_2+r_3=0$, their scalar images must have each adjacent pair in one half-line. This is impossible for three numbers summing to zero unless all are zero. Both coordinate projections of the lattice map vanish, so the map itself is zero and the toric morphism is constant. This proves no nonconstant toric morphism from the projective plane to the product of projective lines.