The functor of a smooth toric variety

In EGA I [3], projective space P n k is described as the variety representing the functor (1) Y ↦ → {line bundle quotients of O n+1 Y → L gives n + 1 sections of L which don’t. The goal of this paper is to generalize this representation to the case of an arbitrary smooth toric variety. We will work with schemes over an algebraically closed field k of characteristic zero, and we will fix a smooth n-dimensional toric variety X determined by the fan ∆ in NR = Rn. As usual, M denotes the dual lattice of N and ∆(1) denotes the 1-dimensional cones This is easy to prove since a surjection O n+1 Y vanish simultaneously and hence determine a map Y → Pn k of ∆. We will use ∑ ρ to mean ρ∈∆(1) , and similarly for ⊗ρ. Each ρ ∈ ∆(1) determines a divisor Dρ ⊂ X and a generator nρ of ρ ∩ N. Finally, let ∆max denote the maximal cones in ∆ (i.e., those which are not proper faces of cones in ∆). §1. ∆-Collections and Functors. If a fan ∆ determines a smooth toric variety X, then we can generalize the data in (1) as follows: Definition 1.1. Given a scheme Y over k, a ∆-collection on Y consists of line bundles Lρ and sections uρ ∈ H0 (Y, Lρ), indexed by ρ ∈ ∆(1), and isomorphisms cm: ⊗ρL ⊗〈m,nρ〉 ρ ≃ OY, indexed by m ∈ M, such that: (i) (Compatibility) cm ⊗ cm ′ = cm+m ′ for all m, m ′ ∈ M. (ii) (Nondegeneracy) uρ ∈ H0 (Y, Lρ) gives uρ: OY → Lρ, which induces u ∗ ρ: L−1 ρ → OY. Then the map ∑: → OY is a surjection. σ∈∆max ⊗ρ̸⊂σu ∗ ρ σ∈∆max