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Lattice polygons and Green's theorem

Author: Hal Schenck
Journal: Proc. Amer. Math. Soc. 132 (2004), 3509-3512
MSC (2000): Primary 14M25; Secondary 14J30, 52B35
Published electronically: May 21, 2004
MathSciNet review: 2084071
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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.

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Additional Information

Hal Schenck
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Toric variety, Green's theorem, free resolution, syzygy
Received by editor(s): April 10, 2002
Received by editor(s) in revised form: August 26, 2003
Published electronically: May 21, 2004
Additional Notes: The author was supported in part by NSA Grant #MDA904-03-1-0006
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society