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Computing graded Betti tables of toric surfaces

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Abstract
We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface nu(6)(P-2) subset of P-27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface nu(d)(P-2).
Keywords
LATTICE POLYGONS, SYZYGIES, GEOMETRY, ALGEBRA

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MLA
Castryck, Wouter, et al. “Computing Graded Betti Tables of Toric Surfaces.” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 372, 2019, pp. 6869–903.
APA
Castryck, W., Cools, F., Demeyer, J., & Lemmens, A. (2019). Computing graded Betti tables of toric surfaces. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 372, 6869–6903.
Chicago author-date
Castryck, Wouter, Filip Cools, Jeroen Demeyer, and Alexander Lemmens. 2019. “Computing Graded Betti Tables of Toric Surfaces.” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 372: 6869–6903.
Chicago author-date (all authors)
Castryck, Wouter, Filip Cools, Jeroen Demeyer, and Alexander Lemmens. 2019. “Computing Graded Betti Tables of Toric Surfaces.” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 372: 6869–6903.
Vancouver
1.
Castryck W, Cools F, Demeyer J, Lemmens A. Computing graded Betti tables of toric surfaces. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. 2019;372:6869–903.
IEEE
[1]
W. Castryck, F. Cools, J. Demeyer, and A. Lemmens, “Computing graded Betti tables of toric surfaces,” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 372, pp. 6869–6903, 2019.
@article{8619041,
  abstract     = {We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface nu(6)(P-2) subset of P-27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface nu(d)(P-2).},
  author       = {Castryck, Wouter and Cools, Filip and Demeyer, Jeroen and Lemmens, Alexander},
  issn         = {0002-9947},
  journal      = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY},
  keywords     = {LATTICE POLYGONS,SYZYGIES,GEOMETRY,ALGEBRA},
  language     = {eng},
  pages        = {6869--6903},
  title        = {Computing graded Betti tables of toric surfaces},
  url          = {http://dx.doi.org/10.1090/tran/7643},
  volume       = {372},
  year         = {2019},
}

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