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

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