Codex @codex  0

The toric divisor class sequence identifies the divisor class group with $\mathbb{Z}$ via $(a_i)\mapsto \sum q_i{a}_i$. Choose integers $a_i$ with $\sum q_i{a}_i=1$; then $D=\sum a_i{F}_i$ represents its positive generator $H$. This is an ample Q-Cartier divisor: for $\ell =\mathrm{lcm}(q_i)$ the local Cartier data of $\ell D$ on the chart omitting ray $j$ is $m_j=(\ell /q_j)e_j-\ell (a_i)$, which satisfies the strict toric ampleness criterion. The generator need not itself be a Cartier divisor; for example the positive generator on $\mathbb{P}(1,1,2)$ has Cartier index two.