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

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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      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.
    @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},
      }

Authors

Kiumars Kaveh

Department of Mathematics
University of Pittsburgh
301 Thackeray Hall
Pittsburgh, PA 15260

A. G. Khovanskii

Department of Mathematics
University of Toronto
Bahen Centre
40 St. George St.
Toronto, Ontario
Canada M5S 2E4

and

Moscow Independent University
Institute for Systems Analysis
Russian Academy of Sciences