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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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References

  1. Anderson, D., Küronya, A., Lozovanu, V.: Okounkov bodies of finitely generated divisors. Int. Math. Res. Note 132(5), 1205–1221 (2013)

    MATH  Google Scholar 

  2. Bădescu, L.: Algebraic Surfaces. Universitext. Springer, New York (2001)

    Book  MATH  Google Scholar 

  3. Bauer, T., Küronya, A., Szemberg, T.: Zariski chambers, volumes, and stable base loci. Journal für die reine und angewandte Mathematik 576, 209–233 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Mathematical Surveys and Monographs, 33. American Mathematical Society, Providence, RI, pp. x+169, ISBN:0-8218-1534-2 (1990)

  5. Boucksom, S.: Corps d’Okounkov [d’après Okounkov, Lazarsfeld–Mustaţă et Kaveh-Khovanskii]. Séminaire Bourbaki 1059, 38 (2012)

    MATH  Google Scholar 

  6. Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. RIMS Kyoto Univ 44, 449–494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boucksom, S., Küronya, A., MacLean, C., Szemberg, T.: Vanishing sequences and Okounkov bodies. Math. Ann. 361(3–4), 811–834 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cámara, A., Giné, I., Temkin, M., Thuillier, A., Ulirsch, M., Urbinati, S., Werner, A., Xarles, X.: Nonarchimedean analyitification and Newton–Okounkov bodies. Discussions held at the CRM Workshop Positivity and Valuations, Barcelona 22–26 (2016)

  9. Casas-Alvero, E.: Singularities of Plane Curves. London Math. Soc. Lecture Note Ser., vol. 276, Cambridge University Press, Cambridge (2000)

  10. Cutkosky, S.D., Srinivas, V.: On a problem of Zariski on dimensions of linear systems. Ann. Math. (2) 137(3), 531–559 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dumnicki, M., Harbourne, B., Küronya, A., Roé, J., Szemberg, T.: Very general monomial valuations of \({\mathbb{P}}^{2}\) and a Nagata type conjecture, Communications in Analysis and Geometry, to appear (2013). arXiv:1312.5549 [math.AG]

  12. Favre, C., Jonsson, M.: The Valuative Tree. Lecture Notes in Mathematics, vol. 1853. Springer, Berlin (2004)

    MATH  Google Scholar 

  13. Foster, T., Ranganathan, D.: Hahn analytification and connectivity of higher rank tropical varieties. Manuscr. Math. (2016). doi:10.1007/s00229-016-0841-3

  14. Fujita, T.: On Zariski problem. Proc. Jpn. Acad. Ser. A Math. Sci. 55(3), 106–110 (1979)

  15. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Math, vol. 52. Springer, New York (1977)

    MATH  Google Scholar 

  16. Huber, R.: Étale Cohomology of Rigid Analytic Varieties and Adic Spaces. Aspects of Mathematics, vol. 30. Friedrich Vieweg & Sohn, Braunschweig (1996)

    MATH  Google Scholar 

  17. Jow, S.-Y.: Okounkov bodies and restricted volumes along very general curves. Adv. Math. 223(4), 1356–1371 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaveh, K., Khovanskii, A.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176, 925–978 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Küronya, A., Lozovanu, V.: Local positivity of linear series, (2014). arXiv:1411.6205v1 [math.AG]

  20. Küronya, A., Lozovanu, V.: Positivity of line bundles and Newton–Okounkov bodies (2015). arXiv:1506.06525v1 [math.AG]

  21. Küronya, A., Lozovanu, V.: Infinitesimal Newton–Okounkov bodies and jet separation Duke Math. J. (2015). arXiv:1507.04339v1 [math.AG] (to appear)

  22. Küronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors. Adv. Math. 229(5), 2622–2639 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lazarsfeld, R.: Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 49. Springer, Berlin, pp. xviii+385, ISBN: 3-540-22534-X (2004)

  24. Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 783–835 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Okounkov, A.: Brunn–Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Orevkov, S.Y.: On rational cuspidal curves. I. Sharp estimate for degree via multiplicities. Math. Ann. 324(4), 657–673 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Roé, J.: Local positivity in terms of Newton–Okounkov bodies. Adv. Math. preprint arXiv:1505.02051 (to appear)

  28. Urabe, T.: Resolution of singularities of germs in characteristic positive associated with valuation rings of iterated divisor type, MPI (1999). arXiv:math/9901048v3 [math.AG]

  29. Vaquié, M.: Valuations and local uniformization. Singularity theory and its applications. Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo, pp. 477–527 (2006)

  30. Zariski, O., Samuel, P.: Commutative Algebra. Vol. II. Graduate Texts in Mathematics, 1960th edn. Springer, New York (1975)

    MATH  Google Scholar 

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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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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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