Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

Blanco, Rocio

doi:10.1002/mana.201100108

Cited by 11 publications

(22 citation statements)

References 18 publications

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“…standard textbooks about toric geometry like [12]), there are combinatorial variants of both main algorithmic approaches to Hironaka resolution, i.e. one in the flavour of Bierstone and Milman [3] and one in the philosophy of Villamayor [4], [5].…”

Section: Combinatorial Algorithms For the Binomial Casementioning

confidence: 99%

“…This combinatorial game provides an ideal generated by hyperbolic equations that is locally monomial. This ideal needs to be rewritten as a monomial one to achieve a log-resolution, see [4] and [5] for details. 3 Another benefit of the purely combinatorial approach is the applicability of the algorithm in positive characteristic (in contrast to the general algorithms).…”

Section: Combinatorial Algorithms For the Binomial Casementioning

confidence: 99%

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Desingularization Algorithms: A Comparison from the Practical Point of View

Blanco¹,

Frühbis–Krüger²

2012

Harmony of Gröbner Bases and the Modern Industrial Society - The Second CREST-CSBM International Conference

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Over the last decade, implementations of several desingularization algorithms have appeared in various contexts. These differ as widely in their methods and in their practical efficiency as they differ in the situations in which they may be applied. The purpose of this article is to give a brief overview over a selection of these approaches and the applicability of the respective implementations in Singular.

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Section: Combinatorial Algorithms For the Binomial Casementioning

confidence: 99%

Section: Combinatorial Algorithms For the Binomial Casementioning

confidence: 99%

Desingularization Algorithms: A Comparison from the Practical Point of View

Blanco¹,

Frühbis–Krüger²

2012

Harmony of Gröbner Bases and the Modern Industrial Society - The Second CREST-CSBM International Conference

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“…The purpose of the present article is to show how central techniques of characteristic zero are interesting in positive or mixed characteristic and can be developed so that they can be applied to certain classes of singularities. Non-trivial examples are the resolution of varieties defined by binomial ideals ( [12], [14], [15]), or of generic determinantal varieties (section 5).…”

Section: Introductionmentioning

confidence: 99%

Partial Local Resolution by Characteristic Zero Methods

Schober

2018

Results Math

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We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution via blowing ups in regular centers. We illustrate this for generic determinantal varieties. The article is partially expository and is addressed to non-experts who aim to construct resolutions for other special classes of singularities in positive or mixed characteristic. A first approximation for the singularity of E = (J, b) at M is given by the tangent cone pair T M (E) = (In M (J, b), b) (Definition 1.23). It is a pair on the associated graded ring of R, gr M (R) := i≥0 M i /M i+1 (which is isomorphic to a polynomial ring over the residue field), and In M (J, b) is a homogeneous ideal generated by the b-initial forms of elements of J (Definition 1.16). From this, we obtain a cone C := Spec(gr M (R)/In M (J, b)).Associated to C there is the directrix Dir M (E) of E (at M ). This is the largest sub-vector space V of Spec(gr M (R)) leaving the cone C stable under translation, C + V = C (Defintion

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“…In section 1 we briefly recall some key definitions given in Part I [6]. The combinatorial algorithm 2.4 is constructed in section 2.…”

Section: Introductionmentioning

confidence: 99%

Desingularization of Binomial Varieties in Arbitrary Characteristic. Part Ii: Combinatorial Desingularization Algorithm

Blanco

2011

The Quarterly Journal of Mathematics

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In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial equations without any restriction, including monomials and p-th powers, where p is the characteristic of the base field.In particular, this algorithm works for toric ideals. However, toric geometry tools are not needed, the algorithm is constructed following the same point of view as Villamayor algorithm of resolution of singularities in characteristic zero.

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Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

Cited by 11 publications

References 18 publications

Desingularization Algorithms: A Comparison from the Practical Point of View

Desingularization Algorithms: A Comparison from the Practical Point of View

Partial Local Resolution by Characteristic Zero Methods

Desingularization of Binomial Varieties in Arbitrary Characteristic. Part Ii: Combinatorial Desingularization Algorithm

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