Toric divisor

Every toric ray $\rho$ of a normal toric variety determines a torus-invariant prime Weil divisor $D_{\rho }$. An invariant toric toric divisor is an integer combination $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$.

Such a toric divisor is Cartier exactly when each toric cone $\sigma$ admits a algebraic torus character $m_{\sigma }\in M$ satisfyingThen the corresponding local algebraic torus character sections trivialize its divisor line bundle. Without this integral local condition, the toric divisor gives a divisorial rank-one reflexive sheaf and need not define a line bundle. The positive class generator of $\mathbb{P}(1,1,2)$, for example, has Cartier index two. Thus an arbitrary combination of invariant prime toric divisors must not be silently identified with invertible-sheaf data.^[ref]

For a character ${\chi }^m$ of the dense torus,

For a toric variety whose rays span $N_{\mathbb{R}}$, the divisor class group is computed bywhere $m\mapsto (⟨m,v_{\rho }⟩{)}_{\rho }$.

For a toric divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a normal split toric variety, define the rational polyhedronIts integral points index the torus character basis of global sections of the divisorial sheaf ${\mathcal{O}}_X(D)$. If $D$ is a Cartier divisor, this is an invertible sheaf. For a general Weil divisor it is a rank-one reflexive sheaf.

The polyhedron can be unbounded. For affine space ${\mathbb{A}}^1$ and $D=0$, its inequality is $m\ge 0$, and the integral points give the basis $1,t,t^2,\dots$ of $k[t]$. If the toric fan is complete, the recession toric cone is zero, so a nonempty $P_D$ is bounded. Even then its vertices need not be integral for a non-Cartier Weil divisor. ^[ref]