Abstract
In this appendix, we discuss some of the ideas behind Hironaka’s characteristic polyhedron. In particular, we present pictures that are often hidden in the background.
The goal is to introduce the reader to the notion of Hironaka’s characteristic polyhedron. Existing references often work in the most general setting in order to develop and to prove new interesting results involving the characteristic polyhedron in the greatest possible generality. This has the drawback that one has to handle a heavy load of technicalities, in which a reader, who is inexperienced on the topic, might easily get lost in. While an introduction with all technical details may be found in Cossart et al. (Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, Lecture Notes in Mathematics, the present monograph, 2013, Chap. 8), we highlight the ideas behind the characteristic polyhedron and behind the notions appearing within its context.
Bernd Schober: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany e-mail: bernd.schober@uni-oldenburg.de
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Notes
- 1.
For a given scheme X which is singular at a point x ∈ X, this setting is obtained by embedding X ⊂ Z into a regular scheme Z and passing to the situation in the local ring \( \mathcal {O}_{Z,x} \) (if such an embedding exists), or if we pass to the completion of \( \mathcal {O}_{X,x} \) and then apply the Cohen structure theorem.
- 2.
In all references known to the author but [CJS], the ordering is (u, y) instead of (y, u). We decided to remain coherent with it, but we warn the reader to be careful, when looking into articles, where the characteristic polyhedron appears.
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Acknowledgements
I thank Vincent Cossart, Uwe Jannsen and Shuji Saito for offering me to write an appendix to their marvelous monograph [CJS] and for their comments on an earlier version of the appendix. Moreover, I thank Dan Abramovich and Bernard Teissier for their very helpful suggestions and comments.
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Cossart, V., Jannsen, U., Saito, S. (2020). Appendix. Hironaka’s Characteristic Polyhedron. Notes for Novices(B.Schober). In: Desingularization: Invariants and Strategy. Lecture Notes in Mathematics, vol 2270. Springer, Cham. https://doi.org/10.1007/978-3-030-52640-5_18
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