The functor of a smooth toric variety

Abstract

(1) Y 7→ {line bundle quotients of O Y } . This is easy to prove since a surjection O Y → L gives n + 1 sections of L which don’t vanish simultaneously and hence determine a map Y → Pnk . 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 = R. As usual, M denotes the dual lattice of N and ∆(1) denotes the 1-dimensional cones of ∆. We will use ∑