Algebraic Geometry
Hironaka's characteristic polygon and effective resolution of surfaces
Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 309-312.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.021
Piedra, Ramon 1; Tornero, José M. 1

1 Departamento de Álgebra, Universidad de Sevilla, P.O. 1160, 41080 Sevilla, Spain
@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, Volume 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] Abhyankar, S.S. 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] Cossart, V. 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] Hauser, H. 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] Hironaka, H. Characteristic polyhedra of singularities, J. Math. Kyoto Univ., Volume 7 (1967), pp. 251-293

[6] Hironaka, H. 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] Levi, B. Risoluzione delle singolarità puntualli delle superficie algebriche, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., Volume 33 (1897), pp. 66-86

[8] Luengo, I. On the Structure of Embedded Algebroid Surfaces, Proceedings of Symposia in Pure Mathematics, Part 2, vol. 40, American Mathematical Society, 1983

[9] Spivakovsky, M. 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] Zariski, O. Algebraic Surfaces, Springer, Berlin, 1935

[11] Zariski, O. Reduction of singularities of algebraic three dimensional varieties, Ann. of Math., Volume 45 (1944), pp. 472-542

Cited by Sources: