Torical modification of Newton non-degenerate ideals

Aroca, Fuensanta; Gómez-Morales, Mirna; Shabbir, Khurram

doi:10.1007/s13398-012-0100-8

Cited by 16 publications

(23 citation statements)

References 15 publications

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“…When X is of that type, we determine again a natural family of irreducible components of X Sing m , m ≥ 1 whose associated divisorial valuations are monomial, hence defined by some vectors in N 3 . For all of the nonisolated forms of RTP-singularities except when X is of type B k−1,2l−1 , we show that these vectors give a regular subdivision Σ of the dual Newton fan of X and hence a nonsingular toric variety Z Σ ; since our singularities are Newton non-degenerate [26,2,1], this gives a birational toric morphism Z Σ −→ A 3 which is an embedded resolution of X ⊂ A 3 ; the irreducible components of the exceptional divisor correspond to the natural set of irreducible components of X Sing m . When X is of type B k−1,2l−1 , we again build a toric embedded resolution from the irreducible components of the jet schemes which does not factor through the toric map associated with the dual Newton fan (such resolutions of non-degenerate singularities also appear when one considers an embedded resolution in family; work in progress of Leyton-Alvarez, Mourtada and Spivakovsky).…”

Section: Introductionmentioning

confidence: 75%

“…The irreducibility of the inverse image results from the fact that C 6,3 is the product of an affine space and an A 1 -singularity and the jet schemes of such singularity are irreducible [20,18] (what applies here for A 1 is also true for any rational singularity). The components of π −1 m,6 (C 6,3 ) are not the essential components, they are associated with non-monomial valuations and they have the same weight vector, namely (2,1,3). They are encoded in Figure 3 (to the most right of the graph) by the dashed arrow which starts at the vertex weighted by the vector (2, 1, 3) and the equation x 2 2 + z 3 y 1 = 0.…”

Section: Rtp-singularities Of Type E 60mentioning

confidence: 99%

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The embedded Nash problem of birational models of rational triple singularities

Karadeniz¹,

Mourtada²,

Plénat³

et al. 2020

Preprint

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Introduction 1 Acknowledgements 3 2. RTP-singularities 3 3. Jet schemes 4 4. RTP-singularities of type E 6,0 6 4.1. Jet Schemes of E 6,0 7 4.2. Toric Embedded Resolution of E 6,011 5. RTP-singularities of type A k−1,l−1,m−1 12 5.1. Jet Schemes and toric Embedded resolution of A k−1,l−1,m−1 when k = l ≤ m 12 5.2. Jet Schemes and toric Embedded resolution of A k−1,l−1,m−1 when k ≥ l ≥ m 16 6. Jet Schemes and Toric Embedded Resolution of B k−1,m 17 7. Jet Schemes and Toric Embedded Resolution of C k−1,l+1 23 8. Jet Schemes and Toric Embedded Resolution of D k−1 24 9. Jet Schemes and Toric Embedded Resolution of E 7,0 26 10. Jet Schemes and Toric Embedded Resolution of E 0,7 28 11. Jet Schemes and Toric Embedded Resolution of F k−1 29 12. Jet Schemes and Toric Embedded Resolution of H n 31 References 34 2000 Mathematics Subject Classification. 58K20.This work is supported by the projects "PHC Bosphore no.42613UE", "TUBITAK no.118F320" and "Galatasaray University no.16.504.001BAP". The second named author is also partially supported by Project ANR-LISA-17-CE40-0023.

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Section: Introductionmentioning

confidence: 75%

Section: Rtp-singularities Of Type E 60mentioning

confidence: 99%

The embedded Nash problem of birational models of rational triple singularities

Karadeniz¹,

Mourtada²,

Plénat³

et al. 2020

Preprint

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“…These equations realize C as a complete intersection in (C g+1 , 0). Even more, this complete intersection is Newton non-degenerate in the sense of [AGS13] and [Tev07]. It was proven (resp.…”

Section: Space Monomial Curves With a Plane Semigroupmentioning

confidence: 86%

The monodromy conjecture for a space monomial curve with a plane semigroup

Martín-Morales¹,

Veys²,

Vos³

2021

Publicacions Matemàtiques

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This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded Q-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

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“…In this section we recall the construction and some properties of the Newton polyhedron of a polynomial and the corresponding toric variety associated to its normal fan. Most of the statements in this section may also be found in [2].…”

Section: Newton Polyhedramentioning

confidence: 93%

On Archimedean zeta functions and Newton polyhedra

Aroca

Gómez-Morales

León-Cardenal

2019

Journal of Mathematical Analysis and Applications

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Let f be a polynomial function over the complex numbers and let φ be a smooth function over C with compact support. When f is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to f and φ. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to f . More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, and necessary in the study of local zeta functions through resolution of singularities.Our results refine the corresponding results of Varchenko and generalize the results of Denef and Sargos in the real case, to the complex setting.

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Torical modification of Newton non-degenerate ideals

Abstract: Dedicated to professor Hironaka in his 80th birthday.

Cited by 16 publications

References 15 publications

The embedded Nash problem of birational models of rational triple singularities

The embedded Nash problem of birational models of rational triple singularities

The monodromy conjecture for a space monomial curve with a plane semigroup

On Archimedean zeta functions and Newton polyhedra

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