Examples
Surface: ℂP²
Fan rays: 3
Max cones: 3
Polytope vertices: 3
Euler characteristic χ: 3
Toric variety: T = (ℂ*)ⁿ acts on X with dense orbit. Data = fan Σ in ℝⁿ (lattice N), or equivalently a lattice polytope (moment map image).
Fan: collection of cones. Each ray (1-cone) = torus-invariant divisor. Each 2-cone = torus-fixed point (T²-orbit closes).
Moment map μ: X → Δ (polytope). Polytope Δ ↔ fan Σ by normal fan duality.
Smooth iff each max cone is generated by a ℤ-basis (primitive, unimodular).