Jet schemes of quasi-ordinary surface singularities

TL;DR

This paper characterizes the structure of jet schemes of two-dimensional quasi-ordinary surface singularities, linking their components to the singularity's topological type and computing the log canonical threshold through specific divisors.

Contribution

It provides a complete description of jet scheme components for these singularities and introduces a weighted graph encoding their topological type, also identifying divisors computing the log canonical threshold.

Findings

Jet scheme components are classified and linked to the topological type.

A weighted graph encodes the irreducible components and their properties.

A divisor computing the log canonical threshold is identified, revealing non-monomial contributions.

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 $\mathbb{A}^2$).