Arcs, valuations and the Nash map

Ishii, Shihoko

doi:10.1515/crll.2005.2005.588.71

Cited by 35 publications

(44 citation statements)

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“…On the other hand, bijectivity of the Nash mapping has been shown for many classes of surfaces (see [6], [10], [8], [9], [12], [13], [17], [19], [20], [21], [22], [24], [25], [26]). The techniques leading to the proof of each of these cases are different in nature, and the proofs are often complicated.…”

Section: Introductionmentioning

confidence: 99%

The Nash problem for surfaces

Bobadilla

Pereira

2012

Ann. Math.

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Section: Introductionmentioning

confidence: 99%

The Nash problem for surfaces

Bobadilla

Pereira

2012

Ann. Math.

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“…Since Z is a scheme of finite type over K [[s, t]], we need to work with special differentials. We follow the treatment given in [15,Appendix A].…”

Section: Canonical Divisorsmentioning

confidence: 99%

Terminal valuations and the Nash problem

Fernex

Docampo

2015

Invent. math.

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Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal valuation over X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of X. In dimension two, this result gives a new proof of the theorem of Fern\'andez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.Comment: v2: 21 pages, minor changes and corrections following the referees' reports. To appear in Invent. Mat

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“…The space of arcs in the torus acts on the arc space of a toric variety (see [15,16]). Suppose then that l := l(ν, s) > 0.…”

Section: Arcs and Jets On A Toric Singularitymentioning

confidence: 99%

“…Here H * ν denotes the set of arcs through (Z Λ , 0) which have generic point in the torus and a given order ν ∈ M * . The set H * ν is an orbit of the natural action of the arc space of the torus on the arc space of the toric variety Z Λ (see [15,16]). …”

Section: Introductionmentioning

confidence: 99%

Arithmetic motivic Poincaré series of toric varieties

Pablos

Pérez

2013

Algebra Number Theory

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Abstract. The arithmetic motivic Poincaré series of a variety V defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterlé series when V is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper we study this motivic series when V is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we deduce explicitly a finite set of candidate poles for this invariant.

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Arcs, valuations and the Nash map

Cited by 35 publications

References 18 publications

The Nash problem for surfaces

The Nash problem for surfaces

Terminal valuations and the Nash problem

Arithmetic motivic Poincaré series of toric varieties

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