A higher-order tangent map and a conjecture on the higher Nash blowup of curves

Chávez-Martínez, Enrique; Duarte, Daniel; Flores, Arturo Giles

doi:10.1007/s00209-020-02579-5

Cited by 4 publications

(4 citation statements)

References 23 publications

(32 reference statements)

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“…There are generalizations of this result for n 1 in some cases [5,9,11]. On the other hand, it is well known that this result is not true if char.K/ > 0.…”

Section: Nash Blowups and Nobile's Theoremmentioning

confidence: 79%

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Nash blowups in prime characteristic

Duarte

Núñez-Betancourt

2021

Rev. Mat. Iberoam.

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We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's theorem for quotient varieties and hypersurfaces. We also prove a weaker version for F -pure varieties.Main Theorem (see Theorem 3.10). Let X be a normal irreducible variety. If Nash 1 .X / Š X, then X is a non-singular variety.

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“…There are generalizations of this result for n 1 in some cases [5,9,11]. On the other hand, it is well known that this result is not true if char.K/ > 0.…”

Section: Nash Blowups and Nobile's Theoremmentioning

confidence: 79%

“…This question has been settled recently for varieties over C: it has an affirmative answer for curves [30], but it is false in general [29]. Higher versions of Nobile's theorem have been proved for some families of varieties [5,9,11].…”

Section: Introductionmentioning

confidence: 99%

Nash blowups in prime characteristic

Duarte

Núñez-Betancourt

2021

Rev. Mat. Iberoam.

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“…To get this, in the first section of this paper we enunciate some knowing results about the combinatorial shape of the normalization of higher Nash modification in toric case. This results comes from [2,9] and allows us reduce the problem to find some subsets of vectors such that a matrix defined by this vectors has determinant no zero and are minimum for each essential divisor.…”

Section: Introductionmentioning

confidence: 99%

“…The higher Nash blowup of toric varieties was studied in [2]. Let us recall some tools from those papers that we will use.…”

Section: Introductionmentioning

confidence: 99%

Factorization of normalization of Nash blow-up of order n of An by the minimal resolution

Chávez-Martínez

2021

Preprint

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We show that the morphism of the normalization of n-th Nash blowup of the toric surface singularity A n can be factorized by the minimal resolution of A n . It was known that the normalization of the n-th Nash blowup of a toric surface and the minimal resolution is also a toric surface. Using this, we find the essential divisor of the minimal resolution in the subdivision that defines the n-th Nash blowup. As a consequence we obtain the factorization desired.

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Nash blowups of toric varieties in prime characteristic

Duarte¹,

Jeffries²,

Núñez-Betancourt³

2023

Collect. Math.

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We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.

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A higher-order tangent map and a conjecture on the higher Nash blowup of curves

Cited by 4 publications

References 23 publications

Nash blowups in prime characteristic

Nash blowups in prime characteristic

Factorization of normalization of Nash blow-up of order n of An by the minimal resolution

Nash blowups of toric varieties in prime characteristic

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