Jet schemes of complex plane branches and equisingularity

Mourtada, Hussein

doi:10.5802/aif.2675

Cited by 19 publications

(42 citation statements)

References 13 publications

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“…z n − x b 2 ). Therefore comparing graphs associated to γ and γ ′ with γ = γ ′ , we deduce from Theorem 3.3 in [26], that the graphs must be different. Now, suppose it is true for g − 1 characteristic exponents, and we will prove it for g. From Proposition 4.13 we deduce that is sufficient to prove that the graphs associated to the sets {γ 1 , .…”

Section: And In General Formentioning

confidence: 96%

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Jet Schemes of Quasi-Ordinary Surface Singularities

Cobo¹,

Mourtada²

2019

Nagoya Math. J.

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In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity. We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on A 3 ), that computes the log canonical threshold of the singularity embedded in A 3 . This provides us with pairs X ⊂ A 3 whose log canonical thresholds are not contributed by monomial divisorial valuations. Note that for a pair C ⊂ A 2 , where C is a plane curve, the log canonical threshold is always contributed by a monomial divisorial valuation (in suitable coordinates of A 2 ). 1 , . . . , x 1 n d ]]). Moreover, some special exponents (called the characteristic exponents) which belong to the support of this series, are complete invariants of the topological type of the singularity (see [16]). In particular, they determine invariants which come from resolution of singularities, like the log canonical threshold or the Motivic zeta functions ([8], [3], [9]). They also give insights about the construction of a resolution of singularities ([6], [7], [18], [39]).Our aim is to construct some comparable complete invariants for all type of singularities. We search for such invariants in the jet schemes. For m ∈ N, the m-jet scheme, denoted by X m , is a scheme that parametrizes morphisms Spec C[t]/(t m+1 ) −→ X. Intuitively we can think of it as parametrizing arcs in an ambient space, which have a large contact, depending on m, with X. We know already that some invariants which come from resolution of singularities are encoded in jet schemes ([32], [13]).We want to extract from the jet schemes information about the singularity, which can be expressed in terms of invariants of resolutions of singularities. With this, our next goal is to construct a resolution of singularities by using invariants of jet schemes. For specific types of singularities, the knowledge of the irreducible components of the jet schemes X m of a singular variety X, together with some invariants of them, like dimension or embedding dimension, permits to determine deep invariants of the singularity of X: the topological type in the case of curves (see [26]), and the analytical type in the case of normal toric surfaces (see [27] and [28]). Moreover, in the case of irreducible plane curves, the minimal embedded resolution can be constructed from the jet schemes ([23]), and the same for rational double point singularities ([31]). Notice that, understanding the structure of jet schemes for particular singularities, remains a difficult problem. These structures have been studied in [40] and [12] for determinantal varieties, in [26] for plane curve singularities, [27] and [28] for normal toric surfaces, in [29] for rational double point surface singularities, and in [3...

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Section: And In General Formentioning

confidence: 96%

“…(ii.b) If j ′ (m, ν) > g 1 , then we are in the case g 2 = g 1 + 1 and j ′ (m, ν) = g 1 + 1. There exists an integer 1 ≤ r < k g1+1 (ν) such that (26) l g1+1 (ν) + re g1+1 ≤ m < l g1+1 (ν) + (r + 1)e g1+1 .…”

Section: And In General Formentioning

confidence: 99%

Jet Schemes of Quasi-Ordinary Surface Singularities

Cobo¹,

Mourtada²

2019

Nagoya Math. J.

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“…While we have that the global jet schemes of surfaces having such a singularity is irreducible [Mu], we prove that for m ∈ N, m big enough, the number N (m) of irreducble components of the schemes of m−jets centered at the singular point is independent of m. Note that this is not the case for instance for plane curves [Mo1]. This may not be the case also when we have rational singularities which are not locally complete intersection [Mo2].…”

Section: Introductionmentioning

confidence: 95%

“…But in general, there is no direct relation between the irreducbile components of jet schemes and those of the arc space. For instance, the function N (m), number of irreducible components of the schemes of m−jets centered in the singular point of a plane curve, goes to infinity when m goes to infinity [Mo1], while the space of arcs centered at the singular point is irreducible. Note that the Nash map is bijective for rational double point singularities [P1], [PS], [Pe], [Le], for surfaces with rational singularities [Re1], [Re2] and in general for surface singularities [deBPe].…”

Section: Introductionmentioning

confidence: 99%

Jet schemes of rational double point singularities

Mourtada¹,

Cobo²,

Lejeune-Jalabert³

2014

Valuation Theory in Interaction

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We prove that for m ∈ N, m big enough, the number of irreducble components of the schemes of m−jets centered at a point which is a double point singularity is independent of m and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an E 6 singularity to its "minimal" embedded resolutions of singularities.

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“…The subject of this note is the study of the jet schemes of toric surfaces. Beside being the simplest toric singularities, this class of singularities is interesting from two points of view: on one hand, these surfaces are examples of varieties having rational singularities, but which are not necessary local complete intersection, therefore we cannot characterize their rationality by [13] via their jet schemes; on the other hand, despite that these singularities are not complete intersections and therefore we do not have a definition of non-degeneration with respect to their Newton polyhedra in the sense of Kouchnirenko [9], they heuristically are non-degenerate because they are desingularized with one toric morphism, so from a jet-scheme theoretical point of view, their jet schemes should not give rise vanishing components [11] (i.e. projective systems of irreducible components whose limit in the arc space are included in the arc space of the singular locus); this follows from Remark 2.3.…”

Section: Introductionmentioning

confidence: 97%

Jet schemes of toric surfaces

Mourtada¹

2011

Comptes Rendus. Mathématique

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24 pages, 1 figure. Version préliminaire (avril 2011) d'un travail publié sous forme définitive (mai 2011).For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give rise to projective systems, to which we associate a weighted graph. We prove that the data of this graph is equivalent to the data of the analytical type of $S.$ Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for $m$ large enough, the set of components with index of speciality $1,$ is in 1-1 correspondance with the set of exceptional divisors that appear on the minimal resolution of $S.

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Jet schemes of complex plane branches and equisingularity

Cited by 19 publications

References 13 publications

Jet Schemes of Quasi-Ordinary Surface Singularities

Jet Schemes of Quasi-Ordinary Surface Singularities

Jet schemes of rational double point singularities

Jet schemes of toric surfaces

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