Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

Kiumars Kaveh; Askold Georgievich Khovanskii

doi:http://doi.org/10.4007/annals.2012.176.2.5

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.

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[Anderson] D. Anderson, Okounkov bodies and toric degenerations.

Show bibtex

@misc{Anderson, author={Anderson, D.}, TITLE={Okounkov bodies and toric degenerations}, ARXIV={1001.4566}, }
[Arnold] $Go to document$ V. I. Arnold, "Higher-dimensional continued fractions," Regul. Chaotic Dyn., vol. 3, iss. 3, pp. 10-17, 1998.

Show bibtex

@article {Arnold, MRKEY = {1704965}, AUTHOR = {Arnold, V. I.}, TITLE = {Higher-dimensional continued fractions}, JOURNAL = {Regul. Chaotic Dyn.}, FJOURNAL = {Regular \& Chaotic Dynamics. Regulyarnaya \& Khaoticheskaya Dinamika}, VOLUME = {3}, YEAR = {1998}, NUMBER = {3}, PAGES = {10--17}, ISSN = {1560-3547}, MRCLASS = {11A55 (11J70 13A99 52B70)}, MRNUMBER = {1704965}, MRREVIEWER = {Thomas A. Schmidt}, DOI = {10.1070/rd1998v003n03ABEH000076}, ZBLNUMBER = {1044.11596}, }
[Bernstein] D. N. Bernstein, "The number of roots of a system of equations," Functional Anal. Appl., vol. 9, iss. 3, pp. 183-185, 1976.

Show bibtex

@article {Bernstein, MRKEY = {0435072}, AUTHOR = {Bernstein, D. N.}, TITLE = {The number of roots of a system of equations}, JOURNAL = {Functional Anal. Appl.}, FJOURNAL = {Akademija Nauk SSSR. Funkcional$'$nyi Analiz i ego Priloženija}, VOLUME = {9}, YEAR = {1976}, NUMBER = {3}, PAGES = {183--185}, ISSN = {0374-1990}, MRCLASS = {14B99 (12E99 58C25)}, MRNUMBER = {0435072}, MRREVIEWER = {A. H. Wallace}, ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER}, ZBLNUMBER = {0328.32001}, }
[BC] $Go to document$ S. Boucksom and H. Chen, "Okounkov bodies of filtered linear series," Compos. Math., vol. 147, iss. 4, pp. 1205-1229, 2011.

Show bibtex

@article {BC, MRKEY = {2822867}, AUTHOR = {Boucksom, S{é}bastien and Chen, Huayi}, TITLE = {Okounkov bodies of filtered linear series}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {147}, YEAR = {2011}, NUMBER = {4}, PAGES = {1205--1229}, ISSN = {0010-437X}, MRCLASS = {14G25 (11G50)}, MRNUMBER = {2822867}, DOI = {10.1112/S0010437X11005355}, ZBLNUMBER = {1231.14020}, }
[Burago-Zalgaller] Y. D. Burago and V. A. Zalgaller, Geometric Inequalities, New York: Springer-Verlag, 1988, vol. 285.

Show bibtex

@book {Burago-Zalgaller, MRKEY = {0936419}, AUTHOR = {Burago, Yu. D. and Zalgaller, V. A.}, TITLE = {Geometric Inequalities}, SERIES = {Grundleheren 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 = {xiv+331}, ISBN = {3-540-13615-0}, MRCLASS = {52A40 (53-02)}, MRNUMBER = {0936419}, ZBLNUMBER = {0633.53002}, }
[Askold-Chulkov] S. Chulkov and A. G. Khovanskii, Geometry of Semigroup $\mathbb{Z}^n_{\geq 0}$ and its Appearances in Combinatorics, Algebra and Differential Equations, Moscow: MCMME, 2006.

Show bibtex

@book{Askold-Chulkov, author={Chulkov, S. and Khovanskii, A. G.}, TITLE={Geometry of Semigroup {$\mathbb{Z}^n_{\geq 0}$} and its Appearances in Combinatorics, Algebra and Differential Equations}, PUBLISHER={MCMME}, ADDRESS={Moscow}, YEAR={2006}, }
[Eisenbud] D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, New York: Springer-Verlag, 1995, vol. 150.

Show bibtex

@book {Eisenbud, MRKEY = {1322960}, AUTHOR = {Eisenbud, David}, TITLE = {Commutative Algebra. With a View Toward Algebraic Geometry}, SERIES = {Grad. Texts in Math.}, VOLUME = {150}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1995}, PAGES = {xvi+785}, ISBN = {0-387-94268-8; 0-387-94269-6}, MRCLASS = {13-01 (14A05)}, MRNUMBER = {1322960}, MRREVIEWER = {Matthew Miller}, ZBLNUMBER = {0819.13001}, }
[Fujita] $Go to document$ T. Fujita, "Approximating Zariski decomposition of big line bundles," Kodai Math. J., vol. 17, iss. 1, pp. 1-3, 1994.

Show bibtex

@article {Fujita, MRKEY = {1262949}, AUTHOR = {Fujita, Takao}, TITLE = {Approximating {Z}ariski decomposition of big line bundles}, JOURNAL = {Kodai Math. J.}, FJOURNAL = {Kodai Mathematical Journal}, VOLUME = {17}, YEAR = {1994}, NUMBER = {1}, PAGES = {1--3}, ISSN = {0386-5991}, MRCLASS = {14J60 (14C20)}, MRNUMBER = {1262949}, MRREVIEWER = {Yves Laszlo}, DOI = {10.2996/kmj/1138039894}, ZBLNUMBER = {0814.14006}, }
[Hartshorne] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.

Show bibtex

@book {Hartshorne, MRKEY = {0463157}, AUTHOR = {Hartshorne, Robin}, TITLE = {Algebraic Geometry}, SERIES = {Grad. Texts in Math.}, VOLUME={52}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {xvi+496}, ISBN = {0-387-90244-9}, MRCLASS = {14-01}, MRNUMBER = {0463157}, MRREVIEWER = {Robert Speiser}, ZBLNUMBER = {0367.14001}, }
[Jacobson] N. Jacobson, Basic Algebra. II, San Francisco, CA: W. H. Freeman and Co., 1980.

Show bibtex

@book {Jacobson, MRKEY = {0571884}, AUTHOR = {Jacobson, Nathan}, TITLE = {Basic Algebra. {II}}, PUBLISHER = {W. H. Freeman and Co.}, ADDRESS = {San Francisco, CA}, YEAR = {1980}, PAGES = {xix+666}, ISBN = {0-7167-1079-X}, MRCLASS = {00A05}, MRNUMBER = {0571884}, MRREVIEWER = {M. F. Smiley}, ZBLNUMBER = {0441.16001}, }
[Jow] $Go to document$ S. Jow, "Okounkov bodies and restricted volumes along very general curves," Adv. Math., vol. 223, iss. 4, pp. 1356-1371, 2010.

Show bibtex

@article {Jow, MRKEY = {2581374}, AUTHOR = {Jow, Shin-Yao}, TITLE = {Okounkov bodies and restricted volumes along very general curves}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {223}, YEAR = {2010}, NUMBER = {4}, PAGES = {1356--1371}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {14C20}, MRNUMBER = {2581374}, MRREVIEWER = {Antonio Lanteri}, DOI = {10.1016/j.aim.2009.09.015}, ZBLNUMBER = {1187.14012}, }
[Askold-Kiumars-arXiv-1] K. Kaveh and A. G. Khovanskii, Convex bodies and algebraic equations on affine varieties.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@article {Askold-Kiumars-arXiv-2, MRKEY = {2722802}, AUTHOR = {Kaveh, Kiumars and Khovanskii, A. G.}, TITLE = {Mixed volume and an extension of intersection theory of divisors}, JOURNAL = {Mosc. Math. J.}, FJOURNAL = {Moscow Mathematical Journal}, VOLUME = {10}, YEAR = {2010}, NUMBER = {2}, PAGES = {343--375}, ISSN = {1609-3321}, MRCLASS = {14C20 (14E05 52A39)}, MRNUMBER = {2722802}, MRREVIEWER = {Joaquim Ro{é}}, ZBLNUMBER = {05938487}, URL={http://www.ams.org/distribution/mmj/vol10-2-2010/cont10-2-2010.html}, }
[Askold-Kiumars-Kaz] $Go to document$ 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.

Show bibtex

@article {Askold-Kiumars-Kaz, MRKEY = {2729398}, AUTHOR = {Kaveh, Kiumars and Khovanskii, A. G.}, 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] $Go to document$ K. Kaveh and A. G. Khovanskii, "Newton polytopes for horospherical spaces," Mosc. Math. J., vol. 11, iss. 2, pp. 265-283, 407, 2011.

Show bibtex

@article {Askold-Kiumars-horo, MRKEY = {2859237}, AUTHOR = {Kaveh, Kiumars and Khovanskii, A. G.}, TITLE = {Newton polytopes for horospherical spaces}, JOURNAL = {Mosc. Math. J.}, FJOURNAL = {Moscow Mathematical Journal}, VOLUME = {11}, YEAR = {2011}, NUMBER = {2}, PAGES = {265--283, 407}, 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] $Go to document$ K. Kaveh and A. G. Khovanskii, "Convex bodies associated to actions of reductive groups," Mosc. Math. J., vol. 12, pp. 369-396, 2012.

Show bibtex

@article{Askold-Kiumars-reductive, 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}, }
[Askold-BZ] A. G. Khovanskii, "Algebra and mixed volumes," in Geometric Inequalities, New York: Springer-Verlag, 1988, vol. 285, pp. 182-207.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@article {Askold-Hilbert-poly, MRKEY = {1209944}, AUTHOR = {Khovanskii, A. G.}, TITLE = {The {N}ewton polytope, the {H}ilbert polynomial and sums of finite sets}, JOURNAL = {Funktsional. Anal. i Prilozhen.}, FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional$'$nyĭAnaliz i ego Prilozheniya}, VOLUME = {26}, YEAR = {1992}, NUMBER = {4}, PAGES = {57--63, 96}, ISSN = {0374-1990}, MRCLASS = {14M25 (20M14 52B20)}, MRNUMBER = {1209944}, MRREVIEWER = {Aleksandr G. Aleksandrov}, DOI = {10.1007/BF01075048}, NOTE={translated in {\it Funct. Anal. Appl.} {\bf 26} (1992), 276--281}, }
[Askold-finite-sets] $Go to document$ 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.

Show bibtex

@article {Askold-finite-sets, MRKEY = {1340302}, AUTHOR = {Khovanskii, A. G.}, TITLE = {Sums of finite sets, orbits of commutative semigroups and {H}ilbert functions}, JOURNAL = {Funktsional. Anal. i Prilozhen.}, FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional$'$nyĭAnaliz i ego Prilozheniya}, VOLUME = {29}, YEAR = {1995}, NUMBER = {2}, PAGES = {36--50, 95}, ISSN = {0374-1990}, MRCLASS = {20M14 (06F20 13D40)}, MRNUMBER = {1340302}, MRREVIEWER = {V. L. Popov}, DOI = {10.1007/BF01080008}, NOTE={translated in {\it Funct. Anal. Appl.} {\bf 29} (1995), 109--112}, }
[Askold-Hilbert-function] A. G. Khovanskii, "Intersection theory and Hilbert function," Funct. Anal. Appl., vol. 45, pp. 305-315, 2011.

Show bibtex

@article{Askold-Hilbert-function, author = {Khovanskii, A. G.}, TITLE = {Intersection theory and {H}ilbert function}, JOURNAL={Funct. Anal. Appl.}, VOLUME={45}, YEAR={2011}, PAGES={305--315}, NOTE={translation in {\it Funct. Anal. Appl.} {\bf 45} (2011), 305--315}, }
[Askold-convex-families] $Go to document$ A. G. Khovanskii, "Completion of convex families of convex bodies," Mat. Zametki, vol. 91, pp. 440-458, 2012.

Show bibtex

@article{Askold-convex-families, author = {Khovanskii, A. G.}, TITLE = {Completion of convex families of convex bodies}, JOURNAL={Mat. Zametki}, VOLUME={91}, PAGES={440--458}, YEAR={2012}, NOTE={translation in {\it Math. Notes} {\bf 91} (2012), 415--429}, DOI = {10.1134/S000143461203011X}, }
[Kuronya] $Go to document$ A. Kuronya, V. Lozovanu, and C. Maclean, "Convex bodies appearing as Okounkov bodies of divisors," Adv. Math., vol. 229, pp. 2622-2639, 2012.

Show bibtex

@article{Kuronya, author={Kuronya, A. and Lozovanu, V. and Maclean, C.}, TITLE={Convex bodies appearing as {O}kounkov bodies of divisors}, JOURNAL={Adv. Math.}, VOLUME={229}, YEAR={2012}, PAGES={2622--2639}, DOI = {10.1016/j.aim.2012.01.013}, }
[Kushnirenko] A. G. Kouchnirenko, "Polyèdres de Newton et nombres de Milnor," Invent. Math., vol. 32, iss. 1, pp. 1-31, 1976.

Show bibtex

@article {Kushnirenko, MRKEY = {0419433}, AUTHOR = {Kouchnirenko, A. G.}, TITLE = {Polyèdres de {N}ewton et nombres de {M}ilnor}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {32}, YEAR = {1976}, NUMBER = {1}, PAGES = {1--31}, ISSN = {0020-9910}, MRCLASS = {14B05}, MRNUMBER = {0419433}, MRREVIEWER = {H. B. Laufer}, }
[Lazarsfeld] $Go to document$ R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, New York: Springer-Verlag, 2004, vol. 48.

Show bibtex

@book {Lazarsfeld, MRKEY = {2095471}, AUTHOR = {Lazarsfeld, Robert}, TITLE = {Positivity in Algebraic Geometry. {I}. Classical Setting: Line Bundles and Linear Series}, SERIES = {Ergeb. Math. Grenzgeb.}, VOLUME = {48}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2004}, PAGES = {xviii+387}, ISBN = {3-540-22533-1}, MRCLASS = {14-02 (14C20)}, MRNUMBER = {2095471}, MRREVIEWER = {Mihnea Popa}, DOI = {10.1007/978-3-642-18808-4}, ZBLNUMBER = {1093.14501}, }
[Lazarsfeld-Mustata] R. Lazarsfeld and M. Mustata, "Convex bodies associated to linear series," Ann. Sci. Éc. Norm. Supér., vol. 42, iss. 5, pp. 783-835, 2009.

Show bibtex

@article {Lazarsfeld-Mustata, MRKEY = {2571958}, AUTHOR = {Lazarsfeld, Robert and Mustata, Mircea}, TITLE = {Convex bodies associated to linear series}, JOURNAL = {Ann. Sci. Éc. Norm. Supér.}, FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série}, VOLUME = {42}, YEAR = {2009}, NUMBER = {5}, PAGES = {783--835}, ISSN = {0012-9593}, MRCLASS = {14C20 (14E05)}, MRNUMBER = {2571958}, MRREVIEWER = {Zach Teitler}, ZBLNUMBER = {1182.14004}, }
[Nystrom] D. W. Nystrom, Transforming metrics on a line bundle to the Okounkov body.

Show bibtex

@misc{Nystrom, author={Nystrom, D. W.}, TITLE={Transforming metrics on a line bundle to the {O}kounkov body}, ARXIV={0903.5167}, }
[Okounkov-Brunn-Minkowski] $Go to document$ A. Okounkov, "Brunn-Minkowski inequality for multiplicities," Invent. Math., vol. 125, iss. 3, pp. 405-411, 1996.

Show bibtex

@article {Okounkov-Brunn-Minkowski, MRKEY = {1400312}, AUTHOR = {Okounkov, Andrei}, TITLE = {Brunn-{M}inkowski inequality for multiplicities}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, 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}, }
[Okounkov-log-concave] A. Okounkov, "Why would multiplicities be log-concave?," in The Orbit Method in Geometry and Physics, Boston, MA: Birkhäuser, 2003, vol. 213, pp. 329-347.

Show bibtex

@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] $Go to document$ A. N. Parshin, "Chern classes, adèles and $L$-functions," J. Reine Angew. Math., vol. 341, pp. 174-192, 1983.

Show bibtex

@article {Parshin, MRKEY = {0697316}, AUTHOR = {Parshin, A. N.}, TITLE = {Chern classes, adèles and {$L$}-functions}, 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)}, MRNUMBER = {0697316}, MRREVIEWER = {Gerd Faltings}, DOI = {10.1515/crll.1983.341.174}, ZBLNUMBER = {0518.14013}, }
[Petersen] L. Petersen, Okounkov bodies of complexity-one $T$-varieties.

Show bibtex

@misc{Petersen, author={Petersen, L.}, TITLE={Okounkov bodies of complexity-one {$T$}-varieties}, ARXIV={1108.0632}, }
[Zariski] O. Zariski and P. Samuel, Commutative Algebra. Vol. II, D. Van Nostrand Co., Princeton, NJ, 1960.

Show bibtex

@book {Zariski, MRKEY = {0120249}, AUTHOR = {Zariski, Oscar and Samuel, Pierre}, 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)}, MRNUMBER = {0120249}, MRREVIEWER = {H. T. Muhly}, ZBLNUMBER = {0121.27801}, }
[Teissier] B. Teissier, "Du théorème de l’index de Hodge aux inégalités isopérimétriques," C. R. Acad. Sci. Paris Sér. A-B, vol. 288, iss. 4, p. a287-a289, 1979.

Show bibtex

@article {Teissier, MRKEY = {0524795}, AUTHOR = {Teissier, Bernard}, TITLE = {Du théorème de l'index de {H}odge aux inégalités isopérimétriques}, 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}, CODEN = {CHASAP}, MRCLASS = {14C17 (14M99 28A75 52A40)}, MRNUMBER = {0524795}, MRREVIEWER = {I. Dolgachev}, ZBLNUMBER = {0406.14011}, }
[Yuan] $Go to document$ X. Yuan, "On volumes of arithmetic line bundles," Compos. Math., vol. 145, iss. 6, pp. 1447-1464, 2009.

Show bibtex

@article {Yuan, MRKEY = {2575090}, AUTHOR = {Yuan, Xinyi}, TITLE = {On volumes of arithmetic line bundles}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {145}, YEAR = {2009}, NUMBER = {6}, PAGES = {1447--1464}, ISSN = {0010-437X}, MRCLASS = {14G40 (11G35 11G50)}, MRNUMBER = {2575090}, MRREVIEWER = {Yuri Tschinkel}, DOI = {10.1112/S0010437X0900428X}, ZBLNUMBER = {1197.14023}, }

Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

Abstract

Authors