Titlebook: Desingularization: Invariants and Strategy; Application to Dimen Vincent Cossart,Uwe Jannsen,Shuji Saito Book 2020 The Editor(s) (if applic

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,Characteristic Polyhedra of ,???,,In this chapter we are always in Setup A (beginning of Chap. .). We introduce a polyhedron Δ(., .) which plays a crucial role in this monograph. It will provide us with useful invariants of singularities of Spec(.∕.) (see Chap. .). It also give us a natural way to transform a (.)-standard base of . into a standard base of . (see Corollary 8.26).

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,Termination of the Fundamental Sequences of ,-Permissible Blow-Ups, and the Case ,,(,)?=?1,In this chapter we prove the Key Theorem . in Chap. ., by deducing it from a stronger result, Theorem 10.2 below. Moreover we will give an explicit bound on the length of the fundamental sequence, by the .-invariant of the polyhedron at the beginning. First we introduce a basic setup.

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,Additional Invariants in the Case ,,(,)?=?2,In order to show key Theorem . in Chap. ., we recall further invariants for singularities, which were defined by Hironaka. The definition works for any dimension, as long as the directrix is 2-dimensional.

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,Proof in the Case ,,(,)?=?,,(,)?=?2 , III: Inseparable Residue Extensions,In this chapter we complete the proof of key Theorem . (see Theorem 14.4 below).