Jet schemes of quasi-ordinary surface singularities

Abstract

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 ${\mathbb{A}}^3$), 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 contributed 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 contributed by a monomial divisorial valuation (in suitable coordinates of ${\mathbb{A}}^2$).

References in corpus (2)

• Quasi-ordinary singularities: tree model, discriminant and irreducibility
• Geometric motivic Poincaré series of quasi-ordinary singularities

Cited by in corpus (1)

• On the embedded Nash problem