For a fan \(\Sigma\) in \(N_\RR\) with \(\dim N = n\), there is an inclusion-reversing bijection \[ \ts{\text{cones } \sigma \in \Sigma} \longleftrightarrow \ts{T\text{-orbits in } X_\Sigma} , \qquad \sigma \mapsto O(\sigma) = T \cdot x_\sigma , \] under which \(\dim O(\sigma) = n - \dim \sigma\), and \(\overline{O(\sigma)} = \bigcup_{\tau \supseteq \sigma} O(\tau)\).
The distinguished point \(x_\sigma\) is the semigroup homomorphism \(S_\sigma \to \ts{0,1}\) sending \(m\) to \(1\) exactly when \(m \in \sigma^\perp\), which is the limit of a one-parameter subgroup in the relative interior of \(\sigma\).
The two ends of the correspondence are the ones to say aloud: the zero cone gives the dense torus orbit, and a maximal cone of dimension \(n\) gives a fixed point. Rays give the \(T\)-invariant prime divisors, so \(\Sigma(1)\) is the set of boundary divisors, and this is the start of every divisor computation on a toric variety: \[ \ZZ^{\Sigma(1)} \to \Cl(X_\Sigma) \to 0 , \qquad \text{with kernel } M \text{ when } X_\Sigma \text{ has no torus factor} . \] The class group is a cokernel of an explicit integer matrix, which is why toric examples can realise prescribed class groups: the cone on \((0,1)\) and \((d,-1)\) has \(\Cl = \ZZ/d\).
Dependencies
D-Q7Q2NFrom a fan to a toric varietyuses
Backlinks
T-TORDIVDivisors, class group and Picard group from the raysusesPR-D2F15Smoothness and resolution, read off the fanusesPR-FULCOTANThe cotangent space at a fixed point is the Hilbert basis of the semigroupusesD-TORMOMENTThe moment map and the polytope as an orbit spaceusesD-FULSTARThe star of a cone, and orbit closures as toric varietiesusesPR-TORMORToric morphisms, completeness, and invariants read off the fanuses