Lattice polygons and Green's theorem

Associated to an n-dimensional integral convex polytope P is a toric variety X and divisor D, such that the integral points of P represent H ⁰(O _X(D)). We study the free resolution of the homogeneous coordinate ring ⊕ _m∈ℤ H ⁰(mD) as a module over Sym(H ⁰(O _X(D))). It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface. In particular, for a planar polytope P = H ⁰(O _X(D)), D satisfies Green's condition N _p if ∂P contains at least p + 3 lattice points.