(One table)
Let \(X_\Sigma\) be the toric variety of a fan \(\Sigma\) in \(N_\RR \cong \RR^n\).
| \(X_\Sigma\) has the property | exactly when \(\Sigma\) satisfies |
|---|---|
| normal | always: each \(S_\sigma\) is saturated |
| smooth | every cone is spanned by part of a \(\ZZ\)-basis of \(N\); in dimension \(n\), \(\det = \pm 1\) |
| simplicial, equivalently \(\QQ\)-factorial, equivalently finite quotient singularities only | every cone is spanned by \(\RR\)-independent vectors, that is \(\det \neq 0\) |
| complete, equivalently compact | \(\abs{\Sigma} = N_\RR\) |
| projective | \(\Sigma\) is the normal fan of a lattice polytope, equivalently some \(D\) has strictly convex support function |
| Fano | \(\Sigma\) is the fan of cones over the faces of a reflexive polytope |
For a divisor \(D = \sum a_\rho D_\rho\) with support function \(\varphi_D\) on a complete \(\Sigma\):
| \(D\) has the property | exactly when |
|---|---|
| base point free | \(\varphi_D\) is convex |
| ample | \(\varphi_D\) is strictly convex |
| very ample | ample, and each vertex semigroup \(\ts{m - m_\sigma \st m \in P_D \intersect M}\) is saturated in \(M\) |
(How the rows depend on each other)
Smooth implies simplicial, so a cone that is not simplicial is singular before any determinant is computed. Projective implies complete, and the converse fails: a complete fan admitting no strictly convex support function gives a proper variety that is not projective. Fano implies projective, since \(-K_X\) ample is itself a strictly convex support function.
Two rows collapse in low dimension. On a smooth complete toric variety ample and very ample agree, and on any complete toric surface they agree as well, so a counterexample needs dimension three and a singular point.
(Worked positivity on two families)
On \(\PP^n\) with \(D = \sum a_i D_i\), base point free means \(\sum a_i \geq 0\) and ample means \(\sum a_i > 0\).
On \(\FF_m\) with rays labelled as \(D_1, \ldots, D_4\) in cyclic order, \(\Pic(\FF_m) = \gens{D_1, D_4}\), and \(D = a D_1 + b D_4\) is ample exactly when \(a > 0\) and \(b > 0\).