Hironaka's characteristic polygon and effective resolution of surfaces

Details Hironaka's characteristic polygon and effective resolution of surfaces Hironaka's characteristic polygon and effective resolution of surfaces

2006-04-18

arxiv.org/abs/math/0604384v1

Comptes Rendus Math\'ematiques 344 (2007) 309-312

Mathematics

Algebraic Geometry

7 pages, 1 figure

Scientific paper

Hironaka's concept of 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, at least with the original definitions.

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