[Le polygone caractéristique d'Hironaka et la résolution effective des surfaces]
Le concept, introduit par Hironaka, du polyèdre caractéristique d'une singularité a été une des idées les plus puissantes et profitables des dernières décennies dans la théorie des singularités. En fait, depuis son apparition les combinatoires sont devenues un outil central pour plusieurs résultats importants dans ce domaine. Pourtant, ce concept séminal n'est pas encore suffisant pour gérer quelques problèmes effectifs : par exemple, trouver une borne supérieure pour le nombre d'éclatements qu'on peut appliquer à une surface sans faire descendre sa multiplicité. Dans cette brève Note on montre pourquoi l'obtention d'une telle borne n'est pas possible, au moins avec les définitions originales.
Hironaka's concept of a characteristic polyhedron of a singularity has been one of the most powerful and fruitful ideas of the last decades in singularity theory. In fact, since then, combinatorics have become a major tool in many important results. However, this seminal concept is still not enough to cope with some effective problems: for instance, giving a bound on the maximum number of blowing-ups to be performed on a surface before its multiplicity decreases. This short Note shows why such a bounding is not possible, with the original definitions.
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@article{CRMATH_2007__344_5_309_0,
author = {Piedra, Ramon and Tornero, Jos\'e M.},
title = {Hironaka's characteristic polygon and effective resolution of surfaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {309--312},
publisher = {Elsevier},
volume = {344},
number = {5},
year = {2007},
doi = {10.1016/j.crma.2007.01.021},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.01.021/}
}
TY - JOUR AU - Piedra, Ramon AU - Tornero, José M. TI - Hironaka's characteristic polygon and effective resolution of surfaces JO - Comptes Rendus. Mathématique PY - 2007 SP - 309 EP - 312 VL - 344 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.01.021/ DO - 10.1016/j.crma.2007.01.021 LA - en ID - CRMATH_2007__344_5_309_0 ER -
%0 Journal Article %A Piedra, Ramon %A Tornero, José M. %T Hironaka's characteristic polygon and effective resolution of surfaces %J Comptes Rendus. Mathématique %D 2007 %P 309-312 %V 344 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.01.021/ %R 10.1016/j.crma.2007.01.021 %G en %F CRMATH_2007__344_5_309_0
Piedra, Ramon; Tornero, José M. Hironaka's characteristic polygon and effective resolution of surfaces. Comptes Rendus. Mathématique, Tome 344 (2007) no. 5, pp. 309-312. doi : 10.1016/j.crma.2007.01.021. http://www.numdam.org/articles/10.1016/j.crma.2007.01.021/
[1] Resolution of Singularities of Embedded Algebraic Surfaces, Academic Press, New York, 1966
[2] R. Astier, Uniformisation locale des surfaces d'Artin–Schreier en caractéristique positive, Ph.D. Thesis, Université de Versailles, 2002
[3] Desingularization in dimension 2 (Cossart, V.; Giraud, J.; Hermann, M., eds.), Resolution of Surface Singularities, Lecture Notes in Mathematics, vol. 1101, Springer, Berlin, 1984
[4] Excellent surfaces and their taut resolution (Hauser, H.; Lipman, J.; Oort, F.; Quirós, A., eds.), Resolution of Singularities, Progress in Mathematics, vol. 181, Birkhäuser, Basel, 2000
[5] Characteristic polyhedra of singularities, J. Math. Kyoto Univ., Volume 7 (1967), pp. 251-293
[6] Desingularization of excellent surfaces (Cossart, V.; Giraud, J.; Hermann, M., eds.), Resolution of Surface Singularities, Lecture Notes in Mathematics, vol. 1101, Springer, Berlin, 1984
[7] Risoluzione delle singolarità puntualli delle superficie algebriche, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., Volume 33 (1897), pp. 66-86
[8] On the Structure of Embedded Algebroid Surfaces, Proceedings of Symposia in Pure Mathematics, Part 2, vol. 40, American Mathematical Society, 1983
[9] A solution to Hironaka's polyhedra game (Artin, M.; Tate, J., eds.), Arithmetic and Geometry II, Progress in Mathematics, vol. 36, Birkhäuser, Boston, 1983
[10] Algebraic Surfaces, Springer, Berlin, 1935
[11] Reduction of singularities of algebraic three dimensional varieties, Ann. of Math., Volume 45 (1944), pp. 472-542
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