54 Result(s)

Chapter and Conference Paper

Additional Invariants in the Case e _x(X) = 2

In order to show key Theorem 6.40 in Chap. 6, we recall further invariants for singularities, which were defined by Hironaka. The definition works for any dimension, as long as the directrix is 2-dimensiona...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Proof in the Case \(e_x(X)=\overline {e}_x(X)=2\) , II: Separable Residue Extensions

In this chapter we prove Theorem 13.7 below, which implies Key Theorem 6.40 under the assumption that the residue fields of the initial points of X n \(\mathcal {X}_n\) are separably algebraic ov...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Non-existence of Maximal Contact in Dimension 2

Let Z be an excellent regular scheme and let X ⊂ Z be a closed subscheme.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Functoriality, Locally Noetherian Schemes, Algebraic Spaces and Stacks

In this chapter we reformulate the obtained functoriality for our resolution and apply this to obtain resolution of singularities for (two-dimensional) locally noetherian excellent schemes, algebraic spaces an...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Proof in the Case e _x(X) = es _x(X) = 2 , I: Some Key Lemmas

In this chapter we prepare some key lemmas for the proof of Theorem 6.40.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

An Alternative Proof of Theorem 6.17

We give another Proof of Theorem 6.17 which uses more classical tools in algebraic geometry.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Basic Invariants for Singularities

In this chapter we introduce some basic invariants for singularities.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

\(\mathcal {B}\) -Permissible Blow-Ups: The Embedded Case

Let Z be a regular scheme and let 𝔹 ⊂ Z \(\mathbb {B} \subset Z\) be a simple normal crossing divisor on Z. For each x ∈ Z, let 𝔹 ( x ) \(\mathbb {B}(x)\) be the subdivisor of 𝔹 $$ \mathbb {...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Main Theorems and Strategy for Their Proofs

We will treat the following two situations in a parallel way: (E)

(embedded case) X is an excellent noetherian scheme, i : X↪Z is a closed immersion into an excellent regular noetherian scheme Z, and ℬ $...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Characteristic Polyhedra of J ⊂ R

In this chapter we are always in Setup A (beginning of Chap. 7). We introduce a polyhedron Δ(J, u) which plays a crucial role in this monograph. It will provide us with useful invariants of singularities of Sp...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Introduction

Let X be an irreducible and reduced excellent noetherian scheme.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Permissible Blow-Ups

We discuss some fundamental results on the behavior of the ν ^∗-invariants, the e-invariants, and the Hilbert-Samuel functions under permissible blow-ups.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Termination of the Fundamental Sequences of \( \mathcal {B}\) -Permissible Blow-Ups, and the Case e _x(X) = 1

In this chapter we prove the Key Theorem 6.35 in Chap. 6, 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...

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

\( \mathcal {B}\) -Permissible Blow-Ups: The Non-embedded Case

Let X be a locally noetherian scheme. We start with the following definition.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

(u)-standard Bases

Let R be a regular noetherian local ring with maximal ideal 𝔪 \(\mathfrak {m}\) and residue field k = R ∕ 𝔪 \(k=R/\mathfrak {m}\) , and let J ⊆ 𝔪 \(J \subseteq \mathfrak {m}\) be an ideal.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Proof in the Case e _x(X) = e _x(X) = 2 , III: Inseparable Residue Extensions

In this chapter we complete the proof of key Theorem 6.40 (see Theorem 14.4 below).

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Transformation of Standard Bases Under Blow-Ups

In this chapter we will study the transformation of a standard base under permissible blow-ups, in particular with respect to near points in the blow-up.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)

Chapter and Conference Paper

Appendix. Hironaka’s Characteristic Polyhedron. Notes for Novices(B.Schober)

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.

Vincent Cossart, Uwe Jannsen, Shuji Saito in Desingularization: Invariants and Strategy (2020)