Jet schemes and generating sequences of divisorial valuations in dimension two

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 ${\text{A}}^2.$ 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 ${\text{A}}^2,$ we construct an algebraic embedding ${\text{A}}^2\hookrightarrow {\text{A}}^N,N\ge 2$ such that $v$ is the trace of a monomial valuation on ${\text{A}}^N.$ We explain how results in this direction give a constructive approach to a conjecture of Teissier on resolution of singularities by one toric morphism.