Toric Varieties and Lattice Polytopes

We begin with a lattice N isomorphic to Z. The dual lattice M of N is given by Hom(N,Z); it is also isomorphic to Z. (The alphabet may appear to be going backwards; but this notation is standard in the literature.) We write the pairing of v ∈ N and w ∈M as 〈v, w〉. A cone in N is a subset of the real vector space NR = N ⊗R generated by nonnegative R-linear combinations of a set of vectors {v1, . . . , vn} ⊆ N . We assume that cones are strongly convex, that is, they contain no line through the origin. Note that each face of a cone is a cone. (Strictly speaking, our “cones in N” are “strongly convex rational polyhedral cones”.) Any cone σ in N has a dual cone σ† in M given by {w ∈MR | 〈v, w〉 ≥ 0∀ v ∈ σ}. A fan consists of a finite collection of cones such that each face of a cone in the fan is also in the fan, and any pair of cones in the fan intersects in a common face. Note the analogy to simplicial complexes. A (convex) polytope in a finite-dimensional vector space is the convex hull of a finite set of points. We are interested in lattice polytopes, for which this finite set of points– the polytope’s vertices– are contained in our integer lattice. Given a lattice polytope in N containing 0, we may construct a fan by taking cones over each face of the polytope. Given a lattice polytope K in N , we define its polar polytope K to be K = {w ∈M | 〈v, w〉 ≥ −1∀ v ∈ K}. If K is also a lattice polytope, we say that K is a reflexive polytope and that K and K are a mirror pair.