On the generalized Nash problem for smooth germs and adjacencies of curve singularities

TL;DR

This paper investigates the generalized Nash problem for arcs on smooth surface germs, establishing its combinatorial nature and exploring its connections with adjacency relations among plane curve singularities.

Contribution

It demonstrates that the generalized Nash problem can be reduced to combinatorial data and links it to various adjacency concepts in plane curve singularities.

Findings

The Nash problem is combinatorial in nature.

Connections between arc space inclusions and curve singularity adjacencies are established.

Provides a framework for analyzing singularities via arc space relations.

Abstract

In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities.