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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Computing graded Betti tables of toric surfaces
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by Wouter Castryck, Filip Cools, Jeroen Demeyer and Alexander Lemmens PDF
Trans. Amer. Math. Soc. 372 (2019), 6869-6903 Request permission

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
  • MR Author ID: 800098
  • Email: wouter.castryck@esat.kuleuven.be
  • Filip Cools
  • Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
  • MR Author ID: 771467
  • Email: f.cools@kuleuven.be
  • Jeroen Demeyer
  • Affiliation: Vakgroep Wiskunde, Universiteit Gent, Krijgslaan 281, Gent, Belgium
  • MR Author ID: 782465
  • Email: jeroen.demeyer@ugent.be
  • Alexander Lemmens
  • Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
  • MR Author ID: 1273465
  • Email: alexander.lemmens@kuleuven.be
  • 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)
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 6869-6903
  • MSC (2010): Primary 14M25, 13D02, 52B20
  • DOI: https://doi.org/10.1090/tran/7643
  • MathSciNet review: 4024541