Lattice polygons and Green’s theorem

Abstract

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 $H^0({\mathcal O}_X(D))$. We study the free resolution of the homogeneous coordinate ring $\bigoplus _{m \in \mathbb Z}H^0(mD)$ as a module over $Sym(H^0({\mathcal O}_X(D)))$. 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 $P=H^0({\mathcal O}_X(D))$, $D$ satisfies Green’s condition $N_p$ if $\partial P$ contains at least $p+3$ lattice points.