On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity

Popescu-Pampu, Patrick

doi:10.1215/s0012-7094-04-12413-3

Cited by 15 publications

(20 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The singular locus of a quasi-ordinary singularity is determined, after Lipman, by its characteristic exponents (see [24] and [34]). Definition 3.14.…”

Section: Definitions and Notationsmentioning

confidence: 99%

Jet Schemes of Quasi-Ordinary Surface Singularities

Cobo¹,

Mourtada²

2019

Nagoya Math. J.

View full text Add to dashboard Cite

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...

show abstract

“…The singular locus of a quasi-ordinary singularity is determined, after Lipman, by its characteristic exponents (see [24] and [34]). Definition 3.14.…”

Section: Definitions and Notationsmentioning

confidence: 99%

Jet Schemes of Quasi-Ordinary Surface Singularities

Cobo¹,

Mourtada²

2019

Nagoya Math. J.

View full text Add to dashboard Cite

show abstract

“…Suppose that the reduced discriminant locus of the quasi-ordinary projection π has equation x 1 · · · x c = 0. The integer c is called in [34] the equisingular dimension of (S, 0), since (S, 0) may be viewed as a topologically equisingular deformation of a quasi-ordinary hypersurface germ of dimension c, but not smaller. See [2] and also Lipman's work [27].…”

Section: Quasi-ordinary Singularitiesmentioning

confidence: 99%

Quasi‐ordinary singularities, essential divisors and Poincaré series

Pérez¹,

Hernando²

2009

Journal of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. We define Poincaré series associated to a toric or analytically irreducible quasiordinary hypersurface singularity, (S, 0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multigraded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincaré series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincaré series associated to the set of divisorial valuations corresponding to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincaré series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.

show abstract

“…(See [12].) Make the Euclidean division of g by f (i) , by induction we get the f (i) -adic representation of g which is of the form g = c l i (f (i) ) l i .…”

Section: Proposition 33 If the Sequence Of Distinguished Exponents mentioning

confidence: 99%

“…This gives us the claimed adic representation. The uniqueness comes from the fact that the Y -degrees of the terms c l 0 ···l i (f (0) ) l 0 · · · (f (i) ) l i are pairwise distinct (see Lemma 7.2 of [12]). The only thing which remains to prove is the inequality 0 l i n i+1 − 1.…”

Section: Proposition 33 If the Sequence Of Distinguished Exponents mentioning

confidence: 99%

A construction for a class of valuations of the field k(X1,…,Xd,Y) with large value group

Moghaddam

2008

Journal of Algebra

View full text Add to dashboard Cite

Given any algebraically closed field k of characteristic zero and any totally ordered abelian group G of rational rank less than or equal to d, we construct a valuation of the field k(X 1 , . . . , X d , Y ) with value group G. In the case of rational rank equal to d this valuation is induced by a formal fractional power series parametrization of a transcendental hypersurface in affine (d + 1)-space which is naturally approximated by a sequence of quasi-ordinary hypersurfaces. The value semigroup ν(k[X, Y ] \ {0}) is the direct limit of the semigroups associated to these quasi-ordinary hypersurfaces.

show abstract

On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity

Cited by 15 publications

References 24 publications

Jet Schemes of Quasi-Ordinary Surface Singularities

Jet Schemes of Quasi-Ordinary Surface Singularities

Quasi‐ordinary singularities, essential divisors and Poincaré series

A construction for a class of valuations of the field k(X1,…,Xd,Y) with large value group

Contact Info

Product

Resources

About