Computing graded Betti tables of toric surfaces
 Author
 Wouter Castryck (UGent) , Filip Cools, Jeroen Demeyer (UGent) and Alexander Lemmens
 Organization
 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)(P2) subset of P27 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)(P2).
 Keywords
 LATTICE POLYGONS, SYZYGIES, GEOMETRY, ALGEBRA
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU8619041
 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 authordate
 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 authordate (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)(P2) subset of P27 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)(P2).}, author = {Castryck, Wouter and Cools, Filip and Demeyer, Jeroen and Lemmens, Alexander}, issn = {00029947}, journal = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, keywords = {LATTICE POLYGONS,SYZYGIES,GEOMETRY,ALGEBRA}, language = {eng}, pages = {68696903}, 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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