By using the method of Arakelov geometry in the function field setting, we associate, to each graded linear series which is birational and of sub-finite type, a convex body whose Lebesgue measure identifies with the volume of the graded linear series. Compared to other constructions in the literature, less non-intrinsic parameters of the projective variety are involved in this new approach. Moreover, this method does not require the existence of a regular rational point in the projective variety, which was assumed for example in the construction of Lazarsfeld and Mustaţǎ.
En utilisant les méthodes de la géométrie d’Arakelov dans le cadre de corps de fonctions, on associe, à chaque système linéaire gradué birationnel et de type sous-fini, un corps convexe dont la mesure de Lebesgue s’identifie au volume du système linéaire gradué. Comparé à d’autres approches dans la littérature, cette nouvelle approche demande moins de paramètres non intrinsèques de la variété projective. En outre, cette méthode n’exige pas l’existence d’un point rationnel régulier sur la variété projective, ce qui est supposé, par exemple, dans la construction de Lazarsfeld et Mustaţǎ.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1051
Keywords: Okounkov bodies, function field arithmetic
@article{JTNB_2018__30_3_829_0,
author = {Chen, Huayi},
title = {Newton{\textendash}Okounkov bodies: an approach of function field arithmetic},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {829--845},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {30},
number = {3},
year = {2018},
doi = {10.5802/jtnb.1051},
mrnumber = {3938628},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jtnb.1051/}
}
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Chen, Huayi. Newton–Okounkov bodies: an approach of function field arithmetic. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 829-845. doi : 10.5802/jtnb.1051. http://archive.numdam.org/articles/10.5802/jtnb.1051/
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