Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 3, page 717-753
  • ISSN: 0240-2963

Abstract

top
In this paper, we give a numerical characterization of nef arithmetic ℝ -Cartier divisors of C 0 -type on an arithmetic surface. Namely an arithmetic ℝ -Cartier divisor D ¯ of C 0 -type is nef if and only if D ¯ is pseudo-effective and deg ^ ( D ¯ 2 ) = vol ^ ( D ¯ ) .

How to cite

top

Moriwaki, Atsushi. "Numerical characterization of nef arithmetic divisors on arithmetic surfaces." Annales de la faculté des sciences de Toulouse Mathématiques 23.3 (2014): 717-753. <http://eudml.org/doc/275323>.

@article{Moriwaki2014,
abstract = {In this paper, we give a numerical characterization of nef arithmetic $\mathbb\{R\}$-Cartier divisors of $C^0$-type on an arithmetic surface. Namely an arithmetic $\mathbb\{R\}$-Cartier divisor $\overline\{D\}$ of $C^0$-type is nef if and only if $\overline\{D\}$ is pseudo-effective and $\widehat\{\mathrm\{deg\}\}(\overline\{D\}^2) = \widehat\{\mathrm\{vol\}\}(\overline\{D\})$.},
author = {Moriwaki, Atsushi},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Arakelov theory; arithmetic surfaces},
language = {eng},
number = {3},
pages = {717-753},
publisher = {Université Paul Sabatier, Toulouse},
title = {Numerical characterization of nef arithmetic divisors on arithmetic surfaces},
url = {http://eudml.org/doc/275323},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Moriwaki, Atsushi
TI - Numerical characterization of nef arithmetic divisors on arithmetic surfaces
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 3
SP - 717
EP - 753
AB - In this paper, we give a numerical characterization of nef arithmetic $\mathbb{R}$-Cartier divisors of $C^0$-type on an arithmetic surface. Namely an arithmetic $\mathbb{R}$-Cartier divisor $\overline{D}$ of $C^0$-type is nef if and only if $\overline{D}$ is pseudo-effective and $\widehat{\mathrm{deg}}(\overline{D}^2) = \widehat{\mathrm{vol}}(\overline{D})$.
LA - eng
KW - Arakelov theory; arithmetic surfaces
UR - http://eudml.org/doc/275323
ER -

References

top
  1. Abbes (A.) and Bouche (T.).— Théorème de Hilbert-Samuel “arithmétique”, Ann. Inst. Fourier(Grenoble) 45, 375-401 (1995). Zbl0818.14011MR1343555
  2. Autissier (P.).— Points entiers sur les surfaces arithmétiques, Journal für die reine und angewandte Mathematik 531, 201-235 (2001). Zbl1007.11041MR1810122
  3. Berman (R.) and Demailly (J.-P.).— Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in Analysis, Geometry and Topology: On the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics 296, 39-66. Zbl1258.32010MR2884031
  4. Boucksom (S.) and Chen (H.).— Okounkov bodies of filtered linear series, Compositio Math. 147, 1205-1229 (2011). Zbl1231.14020MR2822867
  5. Demailly (J.-P.).— Complex Analytic and Differential Geometry. 
  6. Faltings (G.).— Calculus on arithmetic surfaces, Ann. of Math. 119, 387-424 (1984). Zbl0559.14005MR740897
  7. Gillet (H.) and Soulé (C.).— An arithmetic Riemann-Roch theorem, Invent. Math. 110, 473-543 (1992). Zbl0777.14008MR1189489
  8. Hriljac (P.).— Height and Arakerov’s intersection theory, Amer. J. Math., 107, 23-38 (1985). Zbl0593.14004MR778087
  9. Ikoma (H.).— Boundedness of the successive minima on arithmetic varieties, to appear in J. Algebraic Geometry. Zbl1273.14048MR3019450
  10. Khovanskii (A.).— Newton polyhedron, Hilbert polynomial and sums of finite sets, Funct. Anal. Appl. 26, 276-281 (1993). Zbl0809.13012MR1209944
  11. Lipman (J.).— Desingularization of two-dimensional schemes, Ann. of Math., 107, 151-207 (1978). Zbl0349.14004MR491722
  12. Moriwaki (A.).— Continuity of volumes on arithmetic varieties, J. of Algebraic Geom. 18, 407-457 (2009). Zbl1167.14014MR2496453
  13. Moriwaki (A.).— Zariski decompositions on arithmetic surfaces, Publ. Res. Inst. Math. Sci. 48, p. 799-898 (2012). Zbl1281.14017MR2999543
  14. Moriwaki (A.).— Big arithmetic divisors on ℙ ℤ n , Kyoto J. Math. 51, p. 503-534 (2011). Zbl1228.14023MR2823999
  15. Moriwaki (A.).— Toward Dirichlet’s unit theorem on arithmetic varieties, to appear in Kyoto J. of Math. (Memorial issue of Professor Maruyama), see also (arXiv:1010.1599v4 [math.AG]). Zbl1270.14008MR3049312
  16. Moriwaki (A.).— Arithmetic linear series with base conditions, Math. Z., (DOI) 10.1007/s00209-012-0991-2. Zbl1260.14029MR2995173
  17. Randriambololona (H.).— Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faiseaux cohérents, J. Reine Angew. Math. 590, p. 67-88 (2006). Zbl1097.14020MR2208129
  18. Zhang (S.).— Small points and adelic metrics, Journal of Algebraic Geometry 4, p. 281-300 (1995). Zbl0861.14019MR1311351

NotesEmbed ?

top

You must be logged in to post comments.