Arc valuations on smooth varieties

More, Yogesh

doi:10.1016/j.jalgebra.2009.02.002

Cited by 3 publications

(5 citation statements)

References 19 publications

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“…Let be an affine -variety. Being given a semivaluation on V trivial on k , one defines the closed set of as the Zariski closure of the set (which is nonempty by [Ish05, Proposition 2.11] in case is a divisorial valuation, and by [Mor09, Proposition 3.12] in general).…”

Section: Arc Schemes and Valuationsmentioning

confidence: 99%

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Bourqui

Cañón

2023

Forum of Mathematics, Sigma

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Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

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Section: Arc Schemes and Valuationsmentioning

confidence: 99%

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Bourqui

Cañón

2023

Forum of Mathematics, Sigma

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“…For a full treatment, including proofs, we refer the reader to [ELM04,Voj07,Ish08,dFEI08,Mor09,EM09].…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

“…In this section we review the theory of arc spaces. For a full treatment, including proofs, we refer the reader to [ELM04,Voj07,Ish08,dFEI08,Mor09,EM09].…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

“…It is straightforward to check that ord γv,ϕ = v. Trivial valuations can be written as v = ord γ , where γ is any constant arc for which γ(0) is the home of v. Among all arcs giving the same semi-valuation v, there is a distinguished one, characterized by being maximal with respect to specialization in the arc space. Namely, we consider the family Ish08,dFEI08,Mor09], and its generic point γ v verifies ord γv = v. Any other arc inducing v is a specialization of γ v . Following the terminology of [Mor09], we call C v the maximal arc set associated to v. If v is trivial, we have that…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

“…Namely, we consider the family Ish08,dFEI08,Mor09], and its generic point γ v verifies ord γv = v. Any other arc inducing v is a specialization of γ v . Following the terminology of [Mor09], we call C v the maximal arc set associated to v. If v is trivial, we have that…”

Section: Generalities On Arc Spacesmentioning

confidence: 99%

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The arc space of the Grassmannian

Docampo¹,

Nigro²

2017

Advances in Mathematics

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Abstract. We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.

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The Nash problem for torus actions of complexity one

Bourqui,

Langlois,

Mourtada

2024

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration.

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Arc valuations on smooth varieties

Cited by 3 publications

References 19 publications

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

The arc space of the Grassmannian

The Nash problem for torus actions of complexity one

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