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 $ab\mid n$. Therefore
$$
\operatorname{Pic}(\mathbb P(1,a,b))\cong\mathbb Z,
$$
and 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>.

Solution

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

For coprime positive integers $a,b$,
$$
\operatorname{Cl}(\mathbb P(1,a,b))\cong\mathbb Z,
\qquad
\operatorname{Pic}(\mathbb P(1,a,b))\cong\mathbb Z,
$$
and the natural map identifies the Picard group with $ab\mathbb Z$ inside the class group. The plane is smooth exactly when $a=b=1$.


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