Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. C. Birkenhake, Linear systems on projective spaces, Manuscripta Math. 88 (1995), 177-184. MR 96h:14003
  • 2. W. Bruns, J. Gubeladze, N. Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123-160.MR 99c:52016
  • 3. G. Ewald, A. Schmeinck, Representation of the Hirzebruch-Kleinschmidt varieties by quadrics, Beiträge Algebra Geom. 34 (1993), 151-156. MR 95b:14036
  • 4. G. Ewald, U. Wessels, On the ampleness of invertible sheaves in complete projective toric varieties, Results Math. 19 (1991), 275-278. MR 92b:14028
  • 5. W. Fulton, Introduction to Toric Varieties, Princeton University Press, Princeton N.J., 1993. MR 94g:14028
  • 6. F. Gallego, B. Purnaprajna, Some results on rational surfaces and Fano varieties, J. Reine Angew. Math. 538 (2001), 25-55. MR 2002f:14024
  • 7. M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geometry 19 (1984), 125-171. MR 85e:14022
  • 8. M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972), 318-337. MR 46:3511
  • 9. A. Khovanskii, The Newton polytope, the Hilbert polynomial and sums of finite sets, Funct. Anal. Appl. 26 (1992), 276-281. MR 94e:14068
  • 10. R. Koelman, A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics, Beiträge Algebra Geom. 34 (1993), 57-62. MR 94h:14051
  • 11. G. Ottaviani, R. Paoletti, Syzygies of Veronese embeddings, Compositio Math. 125 (2001), 31-37. MR 2002g:13023
  • 12. J. Wills, On an analog to Minkowski's lattice point theorem. The geometric vein, Springer, New York-Berlin (1981), 285-288. MR 84b:52014

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14M25, 14J30, 52B35

Retrieve articles in all journals with MSC (2000): 14M25, 14J30, 52B35

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

American Mathematical Society