Jet schemes of complex plane branches and equisingularity

TL;DR

This paper characterizes the structure of jet schemes of complex plane branches, linking their irreducible components and codimensions to the semigroup generators, thereby revealing their topological type.

Contribution

It provides explicit formulas for the irreducible components and codimensions of jet schemes of complex branches, connecting algebraic and topological properties.

Findings

Formulas for the number of irreducible components N(m)

Explicit expressions for their codimensions

Jet scheme structure encodes the topological type of the branch

Abstract

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.