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 | References | Similar Articles | Additional Information

Abstract: Associated to an -dimensional integral convex polytope is a toric variety and divisor , such that the integral points of represent . We study the free resolution of the homogeneous coordinate ring as a module over . 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 , satisfies Green's condition if contains at least lattice points.

**1.**Christina Birkenhake,*Linear systems on projective spaces*, Manuscripta Math.**88**(1995), no. 2, 177–184. MR**1354104**, 10.1007/BF02567815**2.**Winfried Bruns, Joseph Gubeladze, and Ngô Viêt Trung,*Normal polytopes, triangulations, and Koszul algebras*, J. Reine Angew. Math.**485**(1997), 123–160. MR**1442191****3.**Günter Ewald and Alexa Schmeinck,*Representation of the Hirzebruch-Kleinschmidt varieties by quadrics*, Beiträge Algebra Geom.**34**(1993), no. 2, 151–156. MR**1264282****4.**Günter Ewald and Uwe Wessels,*On the ampleness of invertible sheaves in complete projective toric varieties*, Results Math.**19**(1991), no. 3-4, 275–278. MR**1100674**, 10.1007/BF03323286**5.**William Fulton,*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037****6.**Francisco Javier Gallego and B. P. Purnaprajna,*Some results on rational surfaces and Fano varieties*, J. Reine Angew. Math.**538**(2001), 25–55. MR**1855753**, 10.1515/crll.2001.068**7.**Mark L. Green,*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785****8.**M. Hochster,*Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes*, Ann. of Math. (2)**96**(1972), 318–337. MR**0304376****9.**A. G. Khovanskiĭ,*The Newton polytope, the Hilbert polynomial and sums of finite sets*, Funktsional. Anal. i Prilozhen.**26**(1992), no. 4, 57–63, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl.**26**(1992), no. 4, 276–281 (1993). MR**1209944**, 10.1007/BF01075048**10.**Robert Jan Koelman,*A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics*, Beiträge Algebra Geom.**34**(1993), no. 1, 57–62. MR**1239278****11.**Giorgio Ottaviani and Raffaella Paoletti,*Syzygies of Veronese embeddings*, Compositio Math.**125**(2001), no. 1, 31–37. MR**1818055**, 10.1023/A:1002662809474**12.**J. M. Wills,*On an analog to Minkowski’s lattice point theorem*, The geometric vein, Springer, New York-Berlin, 1981, pp. 285–288. MR**661786**

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Additional Information

**Hal Schenck**

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

Email:
schenck@math.tamu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07523-9

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