Khovanskii Bases of Cox–Nagata Rings and Tropical Geometry

  • Martha Bernal Guillén
  • Daniel Corey
  • Maria Donten-BuryEmail author
  • Naoki Fujita
  • Georg Merz
Part of the Fields Institute Communications book series (FIC, volume 80)


The Cox ring of a del Pezzo surface of degree 3 has a distinguished set of 27 minimal generators. We investigate conditions under which the initial forms of these generators generate the initial algebra of this Cox ring. Sturmfels and Xu provide a classification in the case of degree 4 del Pezzo surfaces by subdividing the tropical Grassmannian \(\mathop{\mathrm{TGr}}\nolimits (2, \mathbb{Q}^{5})\). After providing the necessary background on Cox–Nagata rings and Khovanskii bases, we review the classification obtained by Sturmfels and Xu. We then describe our classification problem in the degree 3 case and its connections to tropical geometry. In particular, we show that two natural candidates, \(\mathop{\mathrm{TGr}}\nolimits (3, \mathbb{Q}^{6})\) and the Naruki fan, are insufficient to carry out the classification.

MSC 2010 codes:

14Q10 14T05 14D06 



This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. The authors are very grateful to Bernd Sturmfels for suggesting the problem, discussions and encouragement. Daniel Corey was supported by NSF CAREER DMS-1149054. Maria Donten-Bury was supported by a Polish National Science Center project 2013/11/D/ST1/02580. Naoki Fujita was supported by Grant-in-Aid for JSPS Fellows (No. 16J00420).


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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Martha Bernal Guillén
    • 1
  • Daniel Corey
    • 2
  • Maria Donten-Bury
    • 3
    Email author
  • Naoki Fujita
    • 4
  • Georg Merz
    • 5
  1. 1.Unidad Académica de MatemáticasUniversidad Autónoma de ZacatecasZacatecasMexico
  2. 2.Department of MathematicsYale UniversityNew HavenUSA
  3. 3.Institute of MathematicsUniversity of WarsawWarszawaPoland
  4. 4.Department of MathematicsTokyo Institute of TechnologyTokyoJapan
  5. 5.Mathematisches InstitutGeorg-August Universität GöttingenGöttingenGermany

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