Newton–Okounkov bodies sprouting on the valuative tree

Abstract

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 \(X=\mathbb {P}^2\).

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Acknowledgements

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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Correspondence to Joaquim Roé.

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Ciliberto, C., Farnik, M., Küronya, A. et al. Newton–Okounkov bodies sprouting on the valuative tree. Rend. Circ. Mat. Palermo, II. Ser 66, 161–194 (2017). https://doi.org/10.1007/s12215-016-0285-3

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Keywords

  • Newton-Okounkov body
  • Valuative tree
  • Valuation
  • Linear system
  • Algebraic geometry

Mathematics Subject Classification

  • 14C20
  • 14N05
  • 13A18