(Support function)
For a Cartier divisor \(D = \sum a_\rho D_\rho\) the support function \(\varphi_D : \abs{\Sigma} \to \RR\) is the unique function that is linear on each cone with \(\varphi_D(u_\rho) = -a_\rho\). On a maximal cone \(\sigma\) it is \(\varphi_D = \inp{m_\sigma}{\wait}\) for the Cartier datum \(m_\sigma \in M\). It is integral if \(\varphi_D(\abs{\Sigma} \intersect N) \subseteq \ZZ\), convex if \[ \varphi_D(t u + (1-t) v) \geq t \varphi_D(u) + (1-t)\varphi_D(v) , \] and strictly convex if in addition \(m_\sigma \neq m_{\sigma'}\) for distinct maximal cones.
(Positivity)
Let \(X_\Sigma\) be complete.
\(D\) is base point free, equivalently \(\OO(D)\) is globally generated, iff \(\varphi_D\) is convex, iff \(m_\sigma \in P_D\) for every \(\sigma \in \Sigma(n)\), iff \(P_D = \operatorname{Conv}\ts{m_\sigma \st \sigma \in \Sigma(n)}\).
\(D\) is ample iff \(\varphi_D\) is strictly convex, iff \(P_D\) is \(n\)-dimensional with the \(m_\sigma\) as its distinct vertices.
\(D\) is very ample iff it is ample and, for every vertex \(m_\sigma\), the semigroup \(S_\sigma\) is generated by \(\ts{m - m_\sigma \st m \in P_D \intersect M}\).
The last clause is a saturation condition on the vertex cone, and it is the only place the three notions can come apart. On a smooth complete toric variety ample implies very ample, so the distinction is invisible until a singular example is produced; the cone over the rational normal curve is where to look for one.
A convex piecewise-linear function bends upwards across every wall. That is the picture to draw: the graph of \(\varphi_D\) over the fan is a tent, and \(D\) is ample when every wall is a genuine crease.
Dependencies
D-TORQDThe polytope of a divisor, and its lattice points as sectionsuses
Backlinks
FE-FULAVAAn ample divisor that is not very ampleusesD-FULNORMPOLYNormal polytopes and very ample dilationsusesD-TORREFLReflexive polytopes, Gorenstein Fano toric varieties and the five toric del PezzosusesPR-FULCRITSmooth, complete, projective, Fano, read off the fanuses