Kyoto Journal of Mathematics

Big arithmetic divisors on the projective spaces over Z

Atsushi Moriwaki

Full-text: Open access

Abstract

In this paper, we observe several properties of an arithmetic divisor D¯ on PZn and give the exact form of the Zariski decomposition of D¯ on PZ1. Further, we show that, if n2 and D¯ is big and non-nef, then for any birational morphism f:XPZn of projective, generically smooth, and normal arithmetic varieties, we cannot expect a suitable Zariski decomposition of f(D¯). We also give a concrete construction of Fujita’s approximation of D¯.

Article information

Source
Kyoto J. Math., Volume 51, Number 3 (2011), 503-534.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1312205238

Digital Object Identifier
doi:10.1215/21562261-1299882

Mathematical Reviews number (MathSciNet)
MR2823999

Zentralblatt MATH identifier
1228.14023

Subjects
Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 11G50: Heights [See also 14G40, 37P30]

Citation

Moriwaki, Atsushi. Big arithmetic divisors on the projective spaces over $\mathbb{Z}$. Kyoto J. Math. 51 (2011), no. 3, 503--534. doi:10.1215/21562261-1299882. https://projecteuclid.org/euclid.kjm/1312205238


Export citation

References

  • [1] A. Abbes and T. Bouche, Théorème de Hilbert-Samuel “arithmétique”, Ann. Inst. Fourier (Grenoble) 45 (1995), 375–401.
  • [2] S. Boucksom and H. Chen, Okounkov bodies of filtered linear series, to appear in Compos. Math., preprint, arXiv:0911:2923v1 [math.AG]
  • [3] H. Chen, Arithmetic Fujita approximation, Ann. Sci. Éc. Norm. Supér (4) 43 (2010), 555–578.
  • [4] D. Gale, V. Klee, and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc. 19 (1968), 867–873.
  • [5] H. Gillet and C. Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473–543.
  • [6] P. Gruber, Convex and Discrete Geometry, Grundlehren Math. Wiss. 336, Springer, Berlin, 2007.
  • [7] M. Hajli, Note, 26 March 2010.
  • [8] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109–203; II, 79 (1964), 205–326.
  • [9] A. Moriwaki, Estimation of arithmetic linear series, Kyoto J. Math. 50 (2010), 685–725.
  • [10] A. Moriwaki, Zariski decompositions on arithmetic surfaces, preprint, arXiv:0911.2951v4 [math.AG]
  • [11] X. Yuan, On volumes of arithmetic line bundles, Compos. Math. 145 (2009), 1447–1464.
  • [12] X. Yuan, On volumes of arithmetic line bundles II, preprint, arXiv:0909.3680v1 [math.AG]