Newton–Okounkov bodies sprouting on the valuative tree

  • Ciro Ciliberto
  • Michal Farnik
  • Alex Küronya
  • Victor Lozovanu
  • Joaquim Roé
  • Constantin Shramov


Given a smooth projective algebraic surface X, a point \(O\in X\) and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (Ep) which is infinitely near O, in the sense that there is a sequence of blowups \(X' \rightarrow X\), mapping the smooth, irreducible rational curve \(E\subset X'\) to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (Ep) varies, focusing on the case \(X=\mathbb {P}^2\).


Newton-Okounkov body Valuative tree Valuation Linear system Algebraic geometry 

Mathematics Subject Classification

14C20 14N05 13A18 



This research was started during the workshop “Recent advances in Linear series and Newton–Okounkov bodies”, which took place in Padova, Italy, February 9–14, 2015. The authors enjoyed the lively and stimulating atmosphere of that event. Michal Farnik was partially supported by the Polish National Science Centre, grant number 2015/17/B/ST1/02637. Joaquim Roé was partially supported by the Spanish Mineco grant MTM2013-40680-P. Constantin Shramov was partially supported by the Russian Academic Excellence Project 5-100’, by RFBR grants 15-01-02164, 15-01-02158, 14-01-00160, and by Dynasty foundation.


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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly
  2. 2.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.Goethe-Universität Frankfurt am MainInstitut für MathematikFrankfurt am MainGermany
  4. 4.Université de Caen Normandie, Laboratoire de Mathématiques ”N. Oresme“Campus Côte de NacreCaenFrance
  5. 5.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra (Cerdanyola del Vallès)Spain
  6. 6.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  7. 7.National Research University Higher School of EconomicsMoscowRussia

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