Duke Mathematical Journal

Effective bound of linear series on arithmetic surfaces

Xinyi Yuan and Tong Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove effective upper bounds on the number of effective sections of a Hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert–Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.

Article information

Source
Duke Math. J., Volume 162, Number 10 (2013), 1723-1770.

Dates
First available in Project Euclid: 11 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1373546603

Digital Object Identifier
doi:10.1215/00127094-2322779

Mathematical Reviews number (MathSciNet)
MR3079259

Zentralblatt MATH identifier
1281.14019

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

Citation

Yuan, Xinyi; Zhang, Tong. Effective bound of linear series on arithmetic surfaces. Duke Math. J. 162 (2013), no. 10, 1723--1770. doi:10.1215/00127094-2322779. https://projecteuclid.org/euclid.dmj/1373546603


Export citation

References

  • [1] S. J. Arakelov, An intersection theory for divisors on an arithmetic surface (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179–1192.
  • [2] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004.
  • [3] J. B. Bost, Semi-stability and heights of cycles, Invent. Math. 118 (1994), 223–253.
  • [4] H. Chen, Positive degree and arithmetic bigness, preprint, arXiv:0803.2583v3 [math.AG].
  • [5] M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 21 (1988), 455–475.
  • [6] R. de Jong, Local invariants attached to Weierstrass points, Manuscripta Math. 129 (2009), 273–292.
  • [7] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), 387–424.
  • [8] H. Gillet and C. Soulé, On the number of lattice points in convex symmetric bodies and their duals, Israel J. Math. 74 (1991), 347–357.
  • [9] H. Gillet and C. Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473–543.
  • [10] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [11] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), 151–207.
  • [12] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques, Invent. Math. 98 (1989), 491–498.
  • [13] L. Moret-Bailly, “Hauteurs et classes de Chern sur les surfaces arithmétiques” in Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988), Astérisque 183, 1990, 37–58.
  • [14] A. Moriwaki, Lower bound of self-intersection of dualizing sheaves on arithmetic surfaces with reducible fibres, Math. Ann. 305 (1996), 183–190.
  • [15] A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101–142.
  • [16] A. Moriwaki, Continuity of volumes on arithmetic varieties, J. Algebraic Geom. 18 (2009), 407–457.
  • [17] A. Moriwaki, Continuous extension of arithmetic volumes, Int. Math. Res. Not. IMRN 2009, no. 19, 3598–3638.
  • [18] A. Moriwaki, Estimation of arithmetic linear series, Kyoto J. Math. 50 (2010), 685–725.
  • [19] A. Moriwaki, Zariski decompositions on arithmetic surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), 799–898.
  • [20] A. N. Parshin, “The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications” in Séminaire de Théorie des Nombres (Paris 1986–1987), Progr. Math. 75, Birkhäuser, Boston, 1988, 299–312.
  • [21] D.-K. Shin, Noether inequality for a nef and big divisor on a surface, Commun. Korean Math. Soc. 23 (2008), 11–18.
  • [22] G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449–466.
  • [23] X. Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), 603–649.
  • [24] X. Yuan, On volumes of arithmetic line bundles, Compos. Math. 145 (2009), 1447–1464.
  • [25] S.-W. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), 569–587.
  • [26] S.-W. Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), 171–193.
  • [27] S.-W. Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2010), 1–73.