Jet schemes and generating sequences of divisorial valuations in dimension two

TL;DR

This paper introduces a novel jet scheme-based method to describe divisorial valuations in two dimensions, enabling the recovery of approximate roots and constructing embeddings that relate valuations to monomial valuations, advancing resolution of singularities.

Contribution

It provides a new jet scheme approach to minimal generating sequences of divisorial valuations and constructs embeddings linking valuations to monomial valuations, supporting Teissier's conjecture.

Findings

Recovered approximate roots from jet scheme equations.

Constructed embeddings where valuations are monomial.

Progress towards a constructive resolution of singularities.

Abstract

Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation $v$ centered at the origin of $\textbf{A}^2,$ we construct an algebraic embedding $\textbf{A}^2\hookrightarrow \textbf{A}^N,N\geq 2$ such that $v$ is the trace of a monomial valuation on $\textbf{A}^N.$ We explain how results in this direction give a constructive approach to a conjecture of Teissier on resolution of singularities by one toric morphism.