= Divisorial section ring of a well-formed weighted projective space
{title2=$\bigoplus_{r\geq0}\mathcal L(rD)\cong k[X_0,\ldots,X_n],\quad\deg X_i=q_i$}
With $D=\sum a_iF_i$ and $\sum q_ia_i=1$, the dual <toric lattice> is $M=\{m\in\mathbb Z^{n+1}:\sum q_im_i=0\}$. A <torus character> $\chi^m$ is a section of $rD$ exactly when $b_i=m_i+ra_i\geq0$. The vector $b$ has weighted degree $\sum q_ib_i=r$, and the correspondence $\chi^m\mapsto X^b$ is bijective on bases in every degree and respects multiplication. This gives the displayed graded-ring isomorphism, even when $D$ is a non-Cartier <Weil divisor>.
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