Maximal divisorial sets in arc spaces

Ishii, Shihoko

doi:10.2969/aspm/05010237

Cited by 23 publications

(24 citation statements)

References 11 publications

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“…We want to prove that if C α,β is an irreducible component of Cont ≥n (m ξ ) then r X,Ψ α,β > 1 (equivalently α = 0 and β = 0). Since X is a toric variety, by [30,Lemma 3.11] we have that (7.4.1)…”

Section: Fat Irreducible Components Of Contact Loci and Hironaka's Ordermentioning

confidence: 99%

“…Contac loci and (maximal) divisorial sets Definition 1.6. ( [22], [30]) Let a be a sheaf of ideals on X. Then one can define: In the following paragraphs we recall some results from [22], [21] and [30] regarding the expression of the subsets (1.6.1) and (1.6.2) in terms of irreducible components in the space of arcs of X and their connection with the notion of maximal divisorial sets from Definition 1.5.…”

Section: Divisorial Arcs and Maximal Divisorial Setsmentioning

confidence: 99%

“…This paper concerns the study of an invariant that is used in constructive resolution of singularities and how it can be read in the space of arcs of a given variety. More precisely, we explore how this invariant shows up when considering the so called contact loci with a singular closed point ξ, say Cont ≥n (m ξ ), i.e, the set of arcs that have order at least n at the maximal ideal m ξ of ξ for n ∈ N (see [22], [21] and [30] where the structure of these sets is studied).…”

Section: Introductionmentioning

confidence: 99%

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Contact loci and Hironaka's order

Bravo,

Encinas,

Pascual-Escudero

2018

Preprint

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We study contact loci sets of arcs and the behavior of Hironaka's order function defined in constructive Resolution of singularities. We show that this function can be read in terms of the irreducible components of the contact loci sets at a singular point of an algebraic variety. ContentsIntroduction 1 1. Arcs, valuations and contac loci 5 2. Nash multiplicity sequences, the persistance, and the Q-persistance 8 3. Rees algebras 11 4. Local presentations of the Multiplicity 13 5. Hironaka's order function, the persistance, and the Q-persistance 15 6. Generic values in contact loci sets 17 7. Fat irreducible components of contact loci and Hironaka's order 20 8. On the values of the Q-persistance 24 References 25

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Section: Fat Irreducible Components Of Contact Loci and Hironaka's Ordermentioning

confidence: 99%

Section: Divisorial Arcs and Maximal Divisorial Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Contact loci and Hironaka's order

Bravo,

Encinas,

Pascual-Escudero

2018

Preprint

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“…q−1 ) −1 (p q−1 ) and call it the maximal divisorial set corresponding to the divisorial valuation q • val E . This definition is the same as that in [4] and [13] in case char k = 0.…”

Section: The Arc Space and Jet Schemes Of A Varietymentioning

confidence: 99%

Singularities in arbitrary characteristic via jet schemes

Ishii,

Reguera

2015

Preprint

Self Cite

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This paper summarizes recent results concerning singularities with respect to the Mather-Jacobian log discrepancies over an algebraically closed field of arbitrary characteristic. The basic point is that the inversion of adjunction with respect to Mather-Jacobian discrepancies holds under arbitrary characteristic. Using this fact, we will reduce several geometric properties of the singularities to jet scheme problems and try to avoid discussions that are distinctive to characteristic 0.

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“…Ishii [Ishii2,Definition 2.8], we associate to a valuation v a subset C (v) ⊆ X ∞ in the following way. Definition 1.2.…”

Section: Valuations and Subsets Of The Arc Spacementioning

confidence: 99%

Arc valuations on smooth varieties

2009

Journal of Algebra

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Let X be a nonsingular variety (with dim X 2) over an algebraically closed field k of characteristic zero. Let α : Spec kJtK → X be an arc on X, and let v = ord α be the valuation given by the order of vanishing along α. We describe the maximal irreducible sub-both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. As a corollary, we get that v is determined by its sequence of centers. Also, when X is a surface, our construction also applies to any divisorial valuation v, and in this case C (v) coincides with the one introduced in [L. Ein, R. Lazarsfeld, M. Musta¸tǎ, Contact loci in arc spaces,

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Maximal divisorial sets in arc spaces

Cited by 23 publications

References 11 publications

Contact loci and Hironaka's order

Contact loci and Hironaka's order

Singularities in arbitrary characteristic via jet schemes

Arc valuations on smooth varieties

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