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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Computing graded Betti tables of toric surfaces


Authors: Wouter Castryck, Filip Cools, Jeroen Demeyer and Alexander Lemmens
Journal: Trans. Amer. Math. Soc. 372 (2019), 6869-6903
MSC (2010): Primary 14M25, 13D02, 52B20
DOI: https://doi.org/10.1090/tran/7643
Published electronically: August 20, 2019
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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(\mathbb{P}^2) \subseteq \mathbb{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(\mathbb{P}^2)$.


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Additional Information

Wouter Castryck
Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium; Departement ESAT, KU Leuven and imec, Kasteelpark Arenberg 10/2452, Leuven, Belgium; Vakgroep Wiskunde, Universiteit Gent, Krijgslaan 281, Gent, Belgium
Email: wouter.castryck@esat.kuleuven.be

Filip Cools
Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
Email: f.cools@kuleuven.be

Jeroen Demeyer
Affiliation: Vakgroep Wiskunde, Universiteit Gent, Krijgslaan 281, Gent, Belgium
Email: jeroen.demeyer@ugent.be

Alexander Lemmens
Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
Email: alexander.lemmens@kuleuven.be

DOI: https://doi.org/10.1090/tran/7643
Received by editor(s): December 2, 2016
Received by editor(s) in revised form: June 4, 2018, and June 13, 2018
Published electronically: August 20, 2019
Additional Notes: This research was partially supported by the research project G093913N of the Research Foundation Flanders (FWO), by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement No. 615722 MOTMELSUM, and by the Labex CEMPI (ANR-11-LABX-0007-01)
The fourth author was supported by a Ph.D. fellowship of the Research Foundation Flanders (FWO)
Article copyright: © Copyright 2019 American Mathematical Society