For \(D = \sum_{\rho} a_\rho D_\rho\) on \(X_\Sigma\) put \[ P_D = \ts{ m \in M_\RR \st \inp{m}{u_\rho} \geq -a_\rho \text{ for all } \rho \in \Sigma(1) } . \]
(Sections)
\[ H^0\big( X_\Sigma, \OO(D) \big) = \bigoplus_{m \in P_D \intersect M} k \cdot \chi^m , \] so \(h^0(\OO(D))\) is a count of lattice points.
Starting from a full-dimensional lattice polytope \(P\) with facet presentation \(\inp{m}{u_F} \geq -a_F\), the divisor \(D_P = \sum_F a_F D_F\) on \(X_P\) is ample and satisfies \(P_{D_P} = P\). Starting from a basepoint-free Cartier divisor \(D\) on a complete toric variety \(X_\Sigma\), the fan \(\Sigma\) refines the normal fan of \(P_D\).
Since \(K_X = -\sum_\rho D_\rho\), the anticanonical divisor has \(a_\rho = 1\) for every \(\rho\) and \[ P_{-K_X} = \ts{ m \in M_\RR \st \inp{m}{u_\rho} \geq -1 } , \] which, for \(\Sigma\) complete, is the polar dual of the convex hull of the ray generators. For \(X = X_P\) with \(P\) reflexive this recovers \(P_{-K_{X_P}} = P\).
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Backlinks
PR-TORPOSBase-point freeness, ampleness and very ampleness through support functionsusesD-TORREFLReflexive polytopes, Gorenstein Fano toric varieties and the five toric del PezzosusesT-FULDEGThe degree of a projective toric variety is the normalised volume of its polytopeusesFE-TORP2The projective plane, computed entirely from its fanuses