Abstract
The purpose of this short note is to draw more attention to a very general finite generation problem arising in valutation theory with exciting links to both algebra and geometry. In particular, we propose a few problems with the aim of connecting finite generation in local versus global settings.
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References
D. Anderson, Okounkov bodies and toric degenerations. Math. Ann. 356(3), 1183–1202 (2013)
D. Anderson, A. Küronya, V. Lozovanu, Okounkov bodies of finitely generated divisors. Int. Math. Res. Not. IMRN 2014(9), 2343–2355 (2014)
C. Birkar, P. Cascini, C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)
S. Boucksom, A. Küronya, C. Maclean, T. Szemberg, Vanishing sequences and Okounkov bodies. Math. Ann. 361(3–4), 811–834 (2015)
P. Cascini, V. Lazić, New outlook on the minimal model program, I. Duke Math. J. 161(12), 2415–2467 (2012)
C. Ciliberto, M. Farnik, A. Küronya, V. Lozovanu, J. Roé, C. Shramov, Newton–Okounkov bodies sprouting on the valuative tree, arXiv:1602.02074v1 (to appear in the Rendiconti del Circ. Mat. Palermo)
F. Delgado, C. Galindo, A. Núñez, Generating sequences and Poincaré series for a finite set of plane divisorial valuations. Adv. Math. 219, 1632–1655 (2008)
T. de Fernex, L. Ein, M. Mustaţă, Vanishing theorems and singularities in birational geometry (preprint, 2014), pp. 401, http://homepages.math.uic.edu/~ein/DFEM.pdf
C. Galindo, F. Monserrat, The cone of curves associated to a plane configuration. Comment. Math. Helv. 80(1), 75–93 (2005)
C. Galindo, F. Monserrat, The cone of curves and the cox ring of rational surfaces given by divisorial valuations. Adv. Math. 290, 1040–1061 (2016)
M. Harada, K. Kaveh, Integrable systems, toric degenerations, and Okounkov bodies, integrable systems, toric degenerations and Okounkov bodies. Invent. Math. 202(3), 927–985 (2015)
S-Y. Jow, Fano varieties with finitely generated semigroups in the Okounkov body construction, arXiv:1511.01197
R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 48 (Springer, Berlin, 2004), p. xviii+387. ISBN: 3-540-22533-1
R. Lazarsfeld, M. Mustaţă, Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 783–835 (2009)
B. Teissier, Overweight deformations of affine toric varieties and local uniformization, Valuation Theory in Interaction. EMS Series of Congress Report (European Mathematical Society, Zürich, 2014), pp. 474–565
L. van Langenhoven, W. Veys, Semigroup and Poincaré series for a finite set of divisorial valuations. Rev. Mat. Complut. 28, 191–225 (2015)
O. Zariski, The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 76(2), 560–615 (1962)
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Küronya, A., Roé, J. (2018). Sufficient Conditions for the Finite Generation of Valuation Semigroups. In: Alberich-Carramiñana, M., Galindo, C., Küronya, A., Roé, J. (eds) Extended Abstracts February 2016. Trends in Mathematics(), vol 9. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-00027-1_2
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