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Let $X_{\Sigma }$ be a normal complex toric variety whose toric rays span $N_{\mathbb{R}}$, so it has no torus factor. Form the Cox ring $S=\mathbb{C}[x_{\rho }]$, its toric irrelevant ideal $B_{\Sigma }$, and the diagonalizable groupThe divisor class group grading defines the action of $G$. The Cox construction recovers $X_{\Sigma }$ as the categorical quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by $G$. It is a geometric quotient exactly when the toric fan is simplicial.

For the projective line, $S=\mathbb{C}[x_0,x_1]$, $B=(x_0,x_1)$ and $G={\mathbb{C}}^{\ast }$ acts by simultaneous scaling, giving $({\mathbb{A}}^2\setminus \{0\})/{\mathbb{G}}_m={\mathbb{P}}^1$. The ray-span condition matters: a positive-dimensional algebraic torus has a zero toric fan and no toric ray variables, so its construction needs the omitted torus factor as well.^[ref]

First suppose $\Delta$ has full dimension in $M_{\mathbb{R}}$. Form the toric cone over it in $M_{\mathbb{R}}\oplus \mathbb{R}$ and the graded semigroup algebrawhere degree zero contains only $k$. The lattice semigroup is finitely generated by the Gordan lemma. The projective toric variety of a lattice polytope is ${\mathbb{P}}_{\Delta }=\mathrm{Proj}{R}_{\Delta }$. Using all lattice points in all dilates avoids any assumption that the degree-one lattice points generate the ring.

For each vertex $v$ of $\Delta$, the degree-zero part after inverting ${\chi }^{v}t$ isIndeed its exponents are $m-lv$ with $m\in l\Delta$. Conversely, every lattice point of the tangent toric cone has this form for sufficiently large integral $l$. These charts cover the Proj construction: each point of a dilate is a rational convex combination of vertices, so some power of its monomial is a product of vertex monomials. Their dual cones are the vertex toric cones of the inward normal fan of a polytope. The normal toric fan is complete because every linear functional on $\Delta$ attains its minimum on a convex face. This describes ${\mathbb{P}}_{\Delta }$ as a complete toric variety. Translating $\Delta$ by $m_0\in M$ gives the graded isomorphism $(m,l)\mapsto (m+l{m}_0,l)$, so containing the origin is unnecessary. If $\Delta$ is not full-dimensional, translate a lattice vertex to zero and use $M\cap {\mathrm{span}}_{\mathbb{R}}(\Delta -\Delta )$; the construction is then complete of dimension $\dim \Delta$.

For the simplex in question put $Q={\prod }_i{q}_i$. Its inequalities are $x_i\ge 0$ and ${\sum }_i{q}_i{x}_i\le Q$, so its inward normal toric rays are ${\mathbb{R}}_{+}{e}_i$ and ${\mathbb{R}}_{+}(-q_1,\dots ,-q_n)$. Here the $e_i$ in the dual space denote the standard dual basis. Apply the real linear map $A(e_i)=e_i/q_i$. It carries these toric rays and their convex face incidences to the standard simplex fan, with toric rays $e_1,\dots ,e_n,-{\sum }_i{e}_i$. The transported lattice isWith $\gcd (q_1,\dots ,q_n)=1$, the primitive vectors of this toric fan in $N^{\prime }$ are $v_i=e_i/q_i$ and $v_0=-{\sum }_i{e}_i$. The latter is primitive because $\lambda {\sum }_i{e}_i\in N^{\prime }$ requires each $\lambda q_i$ to be integral, whose smallest positive solution is $\lambda =1$. Their unique primitive relation isThus mutual coprimality beyond the displayed common gcd condition is not needed.

For a general complete toric variety, introduce one variable $z_{\rho }$ for each toric ray. Its toric homogeneous coordinate ring, or Cox ring, isThe toric divisor class sequence is $0\to M\to {\mathbb{Z}}^{\Sigma (1)}\to \mathrm{Cl}(X_{\Sigma })\to 0$, where the first map is $m\mapsto (⟨m,v_{\rho }⟩{)}_{\rho }$. Thus the graded piece $S_{\alpha }$ has basis the monomials $\prod z_{\rho }^{b_{\rho }}$ with $b_{\rho }\ge 0$ and $\sum b_{\rho }[F_{\rho }]=\alpha$. For an invariant divisor $D=\sum a_{\rho }{F}_{\rho }$, monomials in degree $[D]$ correspond to algebraic torus characters ${\chi }^m$ through $b_{\rho }=a_{\rho }+⟨m,v_{\rho }⟩\ge 0$. This also identifies that graded piece with the global sections of $\mathcal{O}(D)$.

In our example $M^{\prime }=(N^{\prime }{)}^{\ast }={⨁}_i{q}_i\mathbb{Z}{e}_i^{\ast }$, and its map to ${\mathbb{Z}}^{n+1}$ sends ${\sum }_i{q}_i{b}_i{e}_i^{\ast }$ to $(-{\sum }_i{q}_i{b}_i,b_1,\dots ,b_n)$. The quotient is therefore $\mathbb{Z}$, via $(c_0,\dots ,c_n)\mapsto c_0+{\sum }_i{q}_i{c}_i$. In particular $\mathrm{deg}{z}_0=1$ and $\mathrm{deg}{z}_i=q_i$, giving the explicit graded isomorphismThe target has the grading inherited from ordinary total degree in $k[Y_0,\dots ,Y_n]$. Its displayed generators are algebraically independent, which proves injectivity as well as surjectivity onto that subring.

To obtain the quotient, the toric irrelevant ideal is generated by ${\prod }_{\rho \notin \sigma (1)}{z}_{\rho }$. Here each maximal toric cone omits exactly one toric ray, so it is $(z_0,\dots ,z_n)$, and the open set is ${\mathbb{A}}^{n+1}\setminus \{0\}$. The diagonalizable algebraic group with algebraic torus character group $\mathrm{Cl}(X)=\mathbb{Z}$ is $k^{\ast }$. Its action on the Cox coordinates is dictated by their degrees, namelyHere the $y_i$ name the independent Cox coordinates; the graded-subring realization above does not change their weights. More concretely, the quotient chart where $y_i\ne 0$ has coordinate ring $(S[y_i^{-1}]{)}_0$, the invariant ring for this action. These charts glue to $\mathrm{Proj}S$ and to the specified toric fan charts. Positive weights make each orbit closed inside the punctured affine space, and in characteristic zero the finite stabilizers cause no failure of this geometric quotient. This is the weighted projective space construction.

For the polynomial Cox ring $S=k[z_{\rho }]$ of a normal split toric variety whose toric rays span the ambient real toric lattice space, the toric irrelevant ideal isThe Cox construction uses the open set $\mathrm{Spec}S\setminus V(B_{\Sigma })$. For the projective line, the two maximal toric cones give generators $z_0,z_1$, so the excluded locus is the origin. For affine space, one maximal toric cone contains every toric ray; the empty product is one, hence $B=(1)$ and nothing is excluded.

The ray-span/no-torus-factor hypothesis belongs to this polynomial presentation. A pure algebraic torus has no toric rays and requires its algebraic torus coordinates in addition to the displayed polynomial data.^[ref]

// Target: geometry-and-topology.bigb