Toward Resolution of Singularities over a Field of Positive Characteristic—Part I. Foundation; the language of the idealistic ﬁltration

Kawanoue, Hiraku

doi:10.2977/prims/1201012043

Cited by 39 publications

(59 citation statements)

References 44 publications

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“…For example, a procedure is introduced in the work of Kawanoue and Matsuki [35,36], which also parallels this reduction. See also [34,50].…”

Section: Sections 2 and 3 Include An Overview Of Invariants That Havementioning

confidence: 99%

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

Benito

Villamayor

2011

Math. Ann.

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“…For example, a procedure is introduced in the work of Kawanoue and Matsuki [35,36], which also parallels this reduction. See also [34,50].…”

Section: Sections 2 and 3 Include An Overview Of Invariants That Havementioning

confidence: 99%

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

Benito

Villamayor

2011

Math. Ann.

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“…Rees algebras extend to a Diff-algebras, and this extension is naturally compatible with integral closure of algebras ( [31]). This interplay of Diff-algebras and integral closure is studied by Kawanoue in [22], and in [25], papers which present new ideas and technics in positive characteristic, and also provide an upper semi-continuous function with a different approach.…”

Section: Part 1 Introductionmentioning

confidence: 99%

Elimination with Applications to Singularities in Positive Characteristic

Villamayor¹

2008

Publ. Res. Inst. Math. Sci.

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“…For a brief summary of the entire series, including its goal and the overview of the Idealistic Filtration Program (called the IFP for short), we refer the reader to the introduction in Part I [Kaw07]. The outline of Part II is presented below.…”

Section: Overview Of the Seriesmentioning

confidence: 99%

Toward Resolution of Singularities over a Field of Positive Characteristic (The Idealistic Filtration Program)<br>Part II. Basic invariants associated to the idealistic ﬁltration and their properties

Kawanoue¹,

Matsuki²

2010

Publ. Res. Inst. Math. Sci.

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This is the second of a series of four papers entitled "Toward resolution of singularities over a field of positive characteristic (the Idealistic Filtration Program)". The goal is to present the IFP, and to ultimately construct an explicit algorithm guided by the program.In the classical setting in characteristic zero, resolution of singularities was carried out by induction on dimension. We take a so-called "hypersurface of maximal contact" to reduce the dimension by one. In the algorithm, we construct the strand of invariants "inv classic " of the following form:where the unit (w, s) consists of the weak order w and the number s of the "old" components in the boundary. Going from one unit to the next, the dimension of the object which we use to extract the information to compute the invariants drops by one, manifesting the induction on dimension. We run the algorithm with the center of blowup determined as the maximal locus of "inv classic ".In our new setting in positive characteristic, we no longer have a hypersurface of maximal contact. However, we try to carry out the induction on "invariant σ", which indicates the behavior of "a Leading Generator System". The notion of an LGS plays the role of a collective substitute for a hypersurface of maximal contact in the IFP. Accordingly, in our new algorithm, we construct the strand of invariants "inv" of the following form:where the unit (σ, e µ, s) consists of the above mentioned σ, followed by e µ and s, which correspond to w and s in the classical setting, respectively. Going from one unit to the next, the invariant σ of the LGS of the object, namely an idealistic filtration, strictly drops, manifesting the induction on the invariant σ. We run the new algorithm with the center of blowup determined as the maximal locus of "inv".The main purpose of this paper, Part II of the series, is to study the basic properties of the invariants that appear in the strand of invariants "inv", establishing the upper semicontinuity of the pair (σ, e µ) among others.

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Toward Resolution of Singularities over a Field of Positive Characteristic—Part I. Foundation; the language of the idealistic ﬁltration

Cited by 39 publications

References 44 publications

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

Elimination with Applications to Singularities in Positive Characteristic

Toward Resolution of Singularities over a Field of Positive Characteristic (The Idealistic Filtration Program)<br>Part II. Basic invariants associated to the idealistic ﬁltration and their properties

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