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For an ample invariant divisor $D$ and the blowup $\pi$ of a torus-fixed smooth point in dimension at least two, $c{\pi }^{\ast }D-E$ is ample for sufficiently large integers $c$. In the subdivided cone, its local characters are $c{m}_{\sigma }+u_i$, where $u_i$ is dual to the omitted ray generator. The new strict inequalities have gap one on the omitted ray and grow linearly with $c$ elsewhere, proving the claim by the toric ampleness criterion.

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.

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.

The dense algebraic torus has coordinate ring $k[M]$, a Laurent polynomial ring and hence a unique factorization domain. Its divisor class group is therefore zero. Restrict an arbitrary Weil divisor $D$ to the algebraic torus and choose a rational function on an algebraic variety $f$ whose divisor there is $D{|}_T$. Then $D-\mathrm{div}(f)$ is supported on the algebraic torus boundary. The prime divisors in that boundary are precisely the $F_j$ corresponding to the toric rays, by the torus orbit-cone correspondence. ConsequentlySince the variety is smooth, every Weil divisor is Cartier. This gives invariant representatives for all divisor classes, without assuming that the original divisor was invariant.

Now use such a representative. The associated vector space isThe principal divisor on a toric variety formula gives $\mathrm{div}({\chi }^m)={\sum }_j⟨m,e_j⟩F_j$. Define the lattice polytope of a toric divisorFor a maximal toric cone $\sigma$, its basic toric ray vectors are a basis of $N$, so there is a unique integral $m_{\sigma }$ with $⟨m_{\sigma },e_j⟩=-a_j$ on its toric rays. The algebraic torus character ${\chi }^{m_{\sigma }}$ is a local frame for $\mathcal{O}(D)$ on $U_{\sigma }$. Every global section restricts on the algebraic torus to a finite Laurent polynomial ${\sum }_m{c}_m{\chi }^m$. Relative to this frame, regularity on $U_{\sigma }$ means ${\sum }_m{c}_m{\chi }^{m-m_{\sigma }}\in k[{\sigma }^{\vee }\cap M]$. Since distinct algebraic torus characters are independent, this holds exactly when each exponent with nonzero coefficient satisfies the inequalities for the toric rays of $\sigma$. Taking all charts provesCompleteness makes $P_D$ bounded when nonempty: a nonzero recession vector would be nonnegative on every toric ray of a complete toric fan, and therefore on all of $N_{\mathbb{R}}$, which is impossible. The displayed space is consequently finite-dimensional.

At the torus-fixed point $p_{\sigma }$ in the smooth chart $U_{\sigma }\cong {\mathbb{A}}^n$, a section ${\chi }^m$ has local monomial ${\chi }^{m-m_{\sigma }}$. Its value is nonzero exactly when this monomial is constant, namely when $m=m_{\sigma }$. Thus if the complete linear system of a divisor has no base point, some section is nonzero at $p_{\sigma }$, forcing $m_{\sigma }\in P_D$. Conversely, if $m_{\sigma }\in P_D$ for every maximal toric cone, its algebraic torus character is a global section whose local expression on $U_{\sigma }$ is one. It therefore vanishes nowhere on that whole chart, and these charts cover $X$. This proves the toric basepoint-free criterion:The analogous toric ampleness criterion, stated without proof, replaces the inequalities off $\sigma$ by strict ones:Equivalently, the normal fan of a polytope $P_D$ is exactly $\Sigma$.

For the final fan subdivision, take $n\ge 2$ so that $\tilde{e}$ is a new toric ray, as the construction requires. The vectors $e_1,\dots ,e_n$ are a lattice basis, so their sum is primitive. Replacing any one basis vector by this sum gives another basis. Thus the star subdivision gives a smooth toric blowup at a torus-fixed point $\pi :X_{{\Sigma }^{\prime }}\to X_{\Sigma }$, with algebraic exceptional divisor $E=\tilde{F}$. On the old chart, a local equation of $D$ is ${\chi }^{-m_{\sigma }}$, whose order on the new toric ray is $-⟨m_{\sigma },\tilde{e}⟩={\sum }_{i=1}^n{a}_i=a$. All old toric ray coefficients are unchanged. ThereforeWe prove ampleness of $D_c=c{D}^{\prime }-E$ directly by inequalities, giving the ample divisor after blowing up a torus-fixed point result.

Let $u_1,\dots ,u_n$ be the basis of $M$ dual to $e_1,\dots ,e_n$. On the new maximal toric cone ${\sigma }_i$, which omits $e_i$, the required local algebraic torus character isIndeed its pairings on $e_j$, $j\ne i$, are $-c{a}_j$, while its pairing on $\tilde{e}$ is $-ca+1$, exactly the negative of the new coefficient $ca-1$. Its inequality on the omitted old toric ray has strict gap one:For any toric ray $e_j$ outside the original toric cone, put ${\delta }_j=⟨m_{\sigma },e_j⟩+a_j$. The original ampleness gives positive integers ${\delta }_j$, and the new strict gaps arefor all sufficiently large $c$.

For any unchanged maximal toric cone $\eta \ne \sigma$, the local algebraic torus character is $c{m}_{\eta }$. All strict inequalities on old toric rays follow from those for $D$. Its inequality on the new toric ray has gapEach summand is nonnegative, and at least one is positive because $\eta \ne \sigma$ cannot contain all its toric rays. Hence ${ϵ}_{\eta }$ is a positive integer, making this gap positive for $c\ge 2$. There are only finitely many remaining inequalities. For example, any integer satisfyingworks, with the inner maximum omitted when its index set is empty. The toric ampleness criterion now givesThe new-ray assumption is essential to the printed construction. In dimension one, $\tilde{e}=e_1$ and the fan subdivision is unchanged, so the two purported distinct divisors coincide. If the displayed definition of $D^{\prime }$ is applied literally on ${\mathbb{P}}^1$ with coefficients $a_1=-2$, $a_2=3$, then $D$ has degree one but $D^{\prime }$ has degree $-1$, contradicting the conclusion. The argument above applies to the intended blowup in dimension at least two.