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Lattice polygons and Green's theorem

Authors:	Hal Schenck

Institution:	Department of Mathematics, Texas A&M University, College Station, Texas 77843

Abstract:	Associated to an -dimensional integral convex polytope is a toric variety and divisor , such that the integral points of 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 $P=H^0({\mathcal O}_X(D))$ , satisfies Green's condition if $\partial P$ contains at least lattice points.

Keywords:	Toric variety Green's theorem free resolution syzygy

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