Jet schemes and generating sequences of divisorial valuations in dimension two

Mourtada, Hussein

doi:10.1307/mmj/1488510030

Cited by 18 publications

(15 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , x g ) of ν C , where [AM], [Mou2], [Spi] or [Tei1]. For i = 0 and i = 1, the condition ν C (x i ) =β i is equivalent to the assumptions that we put above on the variables x 0 and x 1 , respectively.…”

Section: Space Monomial Curves With Plane Semigroupsmentioning

confidence: 99%

The motivic Igusa zeta function of a space monomial curve with a plane semigroup

Mourtada

Veys

Vos

2021

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.

show abstract

Section: Space Monomial Curves With Plane Semigroupsmentioning

confidence: 99%

The motivic Igusa zeta function of a space monomial curve with a plane semigroup

Mourtada

Veys

Vos

2021

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…The difficulty reflects the fact that the structure of the semigroup of values S A (ν) = ν(A \ {0}) is closely related to some of the birational maps providing embedded local uniformizations of ν and can be extremely complicated. It is well understood in the case when A has dimension 1 (see [Zar06,GT00]) and for regular local rings of dimension 2 [Spi90,CV14,Mou17]. It is known for certain valuations dominating 2-dimensional quotient singularities [Dut20] and for certain valuations dominating 3-dimensional regular local rings [Kas16].…”

Section: Introductionmentioning

confidence: 99%

On the construction of valuations and generating sequences on hypersurface singularities

2021

Self Cite

View full text Add to dashboard Cite

Suppose that (K, ν) is a valued field, f (z) ∈ K[z] is a unitary and irreducible polynomial and (L, ω) is an extension of valued fields, where L = K[z]/(f (z)). Let A be a local domain with quotient field K dominated by the valuation ring of ν, and assume that f (z) is in A[z]. We describe the structure of the associated graded ring gr) for the filtration defined by ω as an extension of the associated graded ring of A for the filtration defined by ν. This gives an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr ω A[z]/(f (z)) generate it as a gr ν A-algebra as well as the relations between these images. We also work out the interactions of our method with phenomena such as the nontameness and the defect of an extension. For a valuation ν of rank 1 and a separable extension of valued fields (K, ν) ⊂ (L, ω) as above, our algorithm produces a generating sequence in a local birational extension A 1 of A dominated by ν if and only if there is no defect. In this case, gr ω A 1 [z]/(f (z)) is a finitely presented gr ν A 1 -module.

show abstract

“…-The theory of plane curves and jet schemes has been used to give a precise description of minimal generating sequences of valuations in dimension 2 ( [28], [2], [22], [13]). -Curvettes are being used in [27] whose results have later been generalised to non-algebraically closed fields by Cutkosky and Vinh in [11].…”

Section: Introductionmentioning

confidence: 99%