(Toric morphism)
A lattice map \(\phi : N_1 \to N_2\) is compatible with fans \(\Sigma_1, \Sigma_2\) if every \(\sigma_1 \in \Sigma_1\) has some \(\sigma_2 \in \Sigma_2\) with \(\phi_\RR(\sigma_1) \subseteq \sigma_2\). Such a \(\phi\) induces \(X_{\Sigma_1} \to X_{\Sigma_2}\) carrying \(T_1\) into \(T_2\) by a group homomorphism, and every equivariant morphism arises this way. On affine pieces this is a semigroup map \(S_2 \to S_1\) inducing \(k[S_2] \to k[S_1]\).
(Completeness)
\(X_\Sigma\) is complete iff \(\abs{\Sigma} = N_\RR\). More generally \(\phi\) is proper iff \(\inverseof{\phi_\RR}(\abs{\Sigma_2}) = \abs{\Sigma_1}\).
(Why, in one picture)
Each \(u \in N\) gives a one-parameter subgroup \(\lambda^u : \GG_m \to T\), and the valuative criterion asks whether \(\lim_{t \to 0} \lambda^u(t)\) exists in \(X_\Sigma\). It does exactly when \(u\) lies in some cone of \(\Sigma\), and the limit is the distinguished point \(x_\sigma\) of the smallest cone containing \(u\). So full support means every one-parameter subgroup converges, which is completeness.
For \(\PP^2\) with \(T = \ts{(1 : s : t)}\) and \(\lambda^u(t) = (1 : t^{u_1} : t^{u_2})\), the three maximal cones of the fan are precisely the three regions of \(u\) sending the limit to each of the three coordinate points.
(Two cheap invariants)
\[ \pi_1(X_\Sigma) \cong N / N', \qquad N' = \gens{ \sigma \intersect N \st \sigma \in \Sigma } , \] so a fan whose rays generate \(N\) gives a simply connected variety. For \(X_\Sigma\) smooth and complete, the topological Euler characteristic is the number of maximal cones, \[ \chi(X_\Sigma) = \size \Sigma(n) , \] because the torus orbits give a cell decomposition and only the fixed points contribute.
Dependencies
PR-O8V3IThe orbit-cone correspondenceuses
Backlinks
T-FULBETTIBetti numbers of a smooth complete toric variety from the cone countsusesFE-TORBLOWBlowup as star subdivision, and P^2 blown up at a fixed pointusesPR-FULCRITSmooth, complete, projective, Fano, read off the fanusesFE-FULHIRZBUNThe Hirzebruch surface as a projective bundle over the lineuses