Abstract
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice $\mathbb{Z}^n$ is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of the Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.
-
[Anderson] D. Anderson, Okounkov bodies and toric degenerations.
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author={Anderson, D.},
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} -
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[BC]
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AUTHOR = {Boucksom, S{é}bastien and Chen, Huayi},
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{\it Springer Series in Soviet Mathematics}},
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} -
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[Jow]
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[Askold-Kiumars-arXiv-1] K. Kaveh and A. G. Khovanskii, Convex bodies and algebraic equations on affine varieties.
@misc{Askold-Kiumars-arXiv-1,
author = {Kaveh, Kiumars and Khovanskii, A. G.},
TITLE = {Convex bodies and algebraic equations on affine varieties},
ARXIV={0804.4095},
SORTYEAR={2008},
} -
[Askold-Kiumars-arXiv-2]
K. Kaveh and A. G. Khovanskii, "Mixed volume and an extension of intersection theory of divisors," Mosc. Math. J., vol. 10, iss. 2, pp. 343-375, 2010.
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MRNUMBER = {2722802},
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} -
[Askold-Kiumars-Kaz]
K. Kaveh and A. G. Khovanskii, "Moment polytopes, semigroup of representations and Kazarnovskii’s theorem," J. Fixed Point Theory Appl., vol. 7, iss. 2, pp. 401-417, 2010.
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TITLE = {Moment polytopes, semigroup of representations and {K}azarnovskii's theorem},
JOURNAL = {J. Fixed Point Theory Appl.},
FJOURNAL = {Journal of Fixed Point Theory and Applications},
VOLUME = {7},
YEAR = {2010},
NUMBER = {2},
PAGES = {401--417},
ISSN = {1661-7738},
MRCLASS = {20G05 (05E10)},
MRNUMBER = {2729398},
MRREVIEWER = {Anthony Henderson},
DOI = {10.1007/s11784-010-0027-7},
ZBLNUMBER = {1205.14059},
} -
[Askold-Kiumars-horo]
K. Kaveh and A. G. Khovanskii, "Newton polytopes for horospherical spaces," Mosc. Math. J., vol. 11, iss. 2, pp. 265-283, 407, 2011.
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AUTHOR = {Kaveh, Kiumars and Khovanskii, A. G.},
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FJOURNAL = {Moscow Mathematical Journal},
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YEAR = {2011},
NUMBER = {2},
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ISSN = {1609-3321},
MRCLASS = {14M17 (14M25)},
MRNUMBER = {2859237},
URL={http://www.ams.org/distribution/mmj/vol11-2-2011/cont11-2-2011.html},
} -
[Askold-Kiumars-reductive]
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author={Kaveh, Kiumars and Khovanskii, A. G.},
TITLE = {Convex bodies associated to actions of reductive groups},
JOURNAL={Mosc. Math. J.},
VOLUME={12},
PAGES={369--396},
YEAR={2012},
URL={http://www.ams.org/distribution/mmj/vol12-2-2012/cont12-2-2012.html},
} -
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@incollection {Askold-BZ, MRKEY = {0936419},
AUTHOR = {Khovanskii, A. G.},
TITLE={Algebra and mixed volumes},
BOOKTITLE = {Geometric Inequalities},
SERIES = {Grundlehren Math. Wissen.},
VOLUME = {285},
NOTE = {translated from the Russian by A. B. Sosinski{\u\i},
{\it Springer Series in Soviet Mathematics}},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1988},
PAGES = {182--207},
ISBN = {3-540-13615-0},
MRCLASS = {52A40 (53-02)},
MRNUMBER = {0936419},
ZBLNUMBER = {0633.53002},
} -
[Askold-Hilbert-poly]
A. G. Khovanskii, "The Newton polytope, the Hilbert polynomial and sums of finite sets," Funktsional. Anal. i Prilozhen., vol. 26, iss. 4, pp. 57-63, 96, 1992.
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AUTHOR = {Khovanskii, A. G.},
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JOURNAL = {Funktsional. Anal. i Prilozhen.},
FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional$'$nyĭAnaliz i ego Prilozheniya},
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YEAR = {1992},
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ISSN = {0374-1990},
MRCLASS = {14M25 (20M14 52B20)},
MRNUMBER = {1209944},
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} -
[Askold-finite-sets]
A. G. Khovanskii, "Sums of finite sets, orbits of commutative semigroups and Hilbert functions," Funktsional. Anal. i Prilozhen., vol. 29, iss. 2, pp. 36-50, 95, 1995.
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} -
[Askold-Hilbert-function] A. G. Khovanskii, "Intersection theory and Hilbert function," Funct. Anal. Appl., vol. 45, pp. 305-315, 2011.
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VOLUME={45},
YEAR={2011},
PAGES={305--315},
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} -
[Askold-convex-families]
A. G. Khovanskii, "Completion of convex families of convex bodies," Mat. Zametki, vol. 91, pp. 440-458, 2012.
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PAGES={440--458},
YEAR={2012},
NOTE={translation in {\it Math. Notes} {\bf 91} (2012), 415--429},
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} -
[Kuronya]
A. Kuronya, V. Lozovanu, and C. Maclean, "Convex bodies appearing as Okounkov bodies of divisors," Adv. Math., vol. 229, pp. 2622-2639, 2012.
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author={Kuronya, A. and Lozovanu, V. and Maclean, C.},
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VOLUME={229},
YEAR={2012},
PAGES={2622--2639},
DOI = {10.1016/j.aim.2012.01.013},
} -
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YEAR = {1976},
NUMBER = {1},
PAGES = {1--31},
ISSN = {0020-9910},
MRCLASS = {14B05},
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} -
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VOLUME = {48},
PUBLISHER = {Springer-Verlag},
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PAGES = {xviii+387},
ISBN = {3-540-22533-1},
MRCLASS = {14-02 (14C20)},
MRNUMBER = {2095471},
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} -
[Nystrom] D. W. Nystrom, Transforming metrics on a line bundle to the Okounkov body.
@misc{Nystrom,
author={Nystrom, D. W.},
TITLE={Transforming metrics on a line bundle to the {O}kounkov body},
ARXIV={0903.5167},
} -
[Okounkov-Brunn-Minkowski]
A. Okounkov, "Brunn-Minkowski inequality for multiplicities," Invent. Math., vol. 125, iss. 3, pp. 405-411, 1996.
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JOURNAL = {Invent. Math.},
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VOLUME = {125},
YEAR = {1996},
NUMBER = {3},
PAGES = {405--411},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58F05 (14L30 52B11 58F06)},
MRNUMBER = {1400312},
DOI = {10.1007/s002220050081},
ZBLNUMBER = {0893.52004},
} -
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@incollection {Okounkov-log-concave, MRKEY = {1995384},
AUTHOR = {Okounkov, Andrei},
TITLE = {Why would multiplicities be log-concave?},
BOOKTITLE = {The Orbit Method in Geometry and Physics},
VENUE={{M}arseille, 2000},
SERIES = {Progr. Math.},
VOLUME = {213},
PAGES = {329--347},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2003},
MRCLASS = {20C35 (82B10)},
MRNUMBER = {1995384},
MRREVIEWER = {Nick Yu. Reshetikhin},
ZBLNUMBER = {1063.22024},
} -
[Parshin]
A. N. Parshin, "Chern classes, adèles and $L$-functions," J. Reine Angew. Math., vol. 341, pp. 174-192, 1983.
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JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {341},
YEAR = {1983},
PAGES = {174--192},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {14F20 (11G25 11G40 14J20)},
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[Petersen] L. Petersen, Okounkov bodies of complexity-one $T$-varieties.
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TITLE={Okounkov bodies of complexity-one {$T$}-varieties},
ARXIV={1108.0632},
} -
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TITLE = {Commutative Algebra. {V}ol. {II}},
SERIES = {The University Series in Higher Mathematics},
PUBLISHER = {D. Van Nostrand Co., Princeton, NJ},
YEAR = {1960},
PAGES = {x+414},
MRCLASS = {16.00 (14.00)},
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} -
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JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
FJOURNAL = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B},
VOLUME = {288},
YEAR = {1979},
NUMBER = {4},
PAGES = {A287--A289},
ISSN = {0151-0509},
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} -
[Yuan]
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DOI = {10.1112/S0010437X0900428X},
ZBLNUMBER = {1197.14023},
}