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Let $v_i$ be the primitive lattice generator of the ray ${\tau }_i$. In the toric divisor class sequence, the first map sends a torus character exponent $m\in M$ to its principal divisor on a toric variety:The second map sends an invariant Weil divisor to its class modulo principal divisors. Completeness makes the rays span $N_{\mathbb{R}}$, so the first map is injective. These are the maps in the given sequence; its exactness need not be reproved.

For $D={\sum }_i{r}_i{F}_i$, letThe torus characters ${\chi }^m$ with $m\in P_D\cap M$ form a basis of $\mathcal{L}(D)=\{f\in \mathbb{C}(X):\mathrm{div}(f)+D\ge 0\}\cup \{0\}$. This works for Weil divisors, not only for Cartier divisors. It is the character basis associated to the lattice polytope of a toric divisor.

For a Cartier divisor, the toric ampleness criterion is as follows. Each maximal cone $\sigma$ has local Cartier data $m_{\sigma }\in M$ satisfying $⟨m_{\sigma },v_i⟩=-r_i$ on its rays. The divisor is ample exactly whenEquivalently the support function is strictly concave, with these linear pieces. For a Q-Cartier Weil divisor use rational $m_{\sigma }$ and clear their denominators; the criterion means that a positive multiple is ample Cartier. An arbitrary Weil divisor need not be Q-Cartier, so this qualification cannot be omitted.

Take positive weights $q=(q_0,\dots ,q_n)$, reduced so their common greatest common divisor is one. A toric definition of weighted projective space usesIf the weights are well-formed, meaning $\gcd (q_0,\dots ,\widehat{q_i},\dots ,q_n)=1$ for every $i$, the $v_i$ are primitive: the quotient $N/\mathbb{Z}{v}_i$ has torsion of order that omitted-weight greatest common divisor. The lattice $N$ itself is free because $q$ is primitive. The positive relation ${\sum }_i{q}_i{v}_i=0$ implies that proper subsets generate pointed cones and that they meet along common faces. Completeness can be seen directly: represent a vector by real coordinates $a_i$ and subtract ${\min }_i(a_i/q_i)$ times $q$. The resulting coordinates are nonnegative and at least one is zero, placing the vector in one of the displayed cones. Their common-face property follows from this unique normalization of each class modulo $\mathbb{R}q$. This gives the fan of the well-formed weighted projective space.

The dual lattice isIts map in the divisor sequence is simply inclusion in ${\mathbb{Z}}^{n+1}$, so the homomorphism $a\mapsto q\cdot a$ identifies the cokernel with $\mathbb{Z}$. It is onto by coprimality of all the weights. ThusChoose integers $a_i$ by the Bezout identity ${\sum }_i{q}_i{a}_i=1$. Then $D={\sum }_i{a}_i{F}_i$ explicitly represents the positive generator $H$.

To verify ampleness in the Weil/Q-Cartier sense, put $\ell =\mathrm{lcm}(q_0,\dots ,q_n)$. On the maximal cone omitting ray $j$, defineThis is integral and has $q\cdot m_j=\ell -\ell =0$. Its pairings with the included rays are $-\ell a_i$, while on the omitted ray they equal $-\ell a_j+\ell /q_j>-\ell a_j$. Consequently $\ell D$ is Cartier and ample by the numerical criterion. This is an explicit class-group generator of a well-formed weighted projective space.

The generator is generally ample Weil, not Cartier. If the source's word “ample” is interpreted as requiring a Cartier divisor, its generator request is false in general: on $\mathbb{P}(1,1,2)$, take $D=F_0$. On the chart omitting the weight-two ray, the Cartier equations force $m=(-1,0,1/2)$, which is not in the integral dual lattice. Twice this datum is integral, so the class-one divisor has Cartier index two and no Cartier divisor generates its class group. With the usual ample-Q-Cartier-Weil interpretation, the construction above proves exactly the requested result. This distinction also explains the use of the divisorial spaces $\mathcal{L}(rD)$ rather than assuming an invertible sheaf $\mathcal{O}(D)$.

Finally let an ample divisor represent the class-group generator. Its class is the positive generator; the negative generator cannot be ample because a positive Cartier multiple has negative degree on the curves where the positive generator has positive degree. After replacing by a linearly equivalent invariant divisor, write it as $D=\sum a_i{F}_i$ with $\sum q_i{a}_i=1$. A character basis element of $\mathcal{L}(rD)$ has exponent $m\in M$ with $m_i+r{a}_i\ge 0$. Set $b_i=m_i+r{a}_i$. Then $b\in {\mathbb{Z}}_{\ge 0}^{n+1}$ and $\sum q_i{b}_i=r$. Conversely every nonnegative exponent vector of weighted degree $r$ gives the unique $m=b-ra\in M$. Thereforeis a bijection of bases in each degree. Adding $m$ and $r$ under multiplication adds these nonnegative exponent vectors, so it preserves products and the unit. A different linearly equivalent representative changes the basis by the corresponding rational-function power and gives the same graded-ring isomorphism. Hence the divisorial section ring is the weighted polynomial ring:The character construction is defined over the ground field $k$, so this is an isomorphism of graded $k$-algebras. It proves the divisorial section ring of a well-formed weighted projective space assertion even when its ample generator is not Cartier.