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JET SCHEMES OF QUASI-ORDINARY SURFACE SINGULARITIES

Part of: Birational geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  20 September 2019

HELENA COBO  and
HUSSEIN MOURTADA
HELENA COBO
Affiliation:
Departamento de Álgebra, Universidad de Sevilla, Spain email helenacobo@gmail.com
HUSSEIN MOURTADA
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7, Bâtiment Sophie Germain, 75013 Paris, France email hussein.mourtada@imj-prg.fr
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Abstract

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. 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 (equivalent to a divisorial valuation on $\mathbb{A}^{3}$), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$. This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$, where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$).

MSC classification

Primary: 14E18: Arcs and motivic integration 14J17: Singularities
Type
Article
Information
Nagoya Mathematical Journal , Volume 242 , June 2021 , pp. 77 - 164
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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