On the $b$-exponents of generic isolated plane curve singularities

TL;DR

This paper investigates the behavior of $b$-exponents in generic isolated plane curve singularities, confirming Yano's conjecture in certain cases and exploring the limitations of natural generalizations through examples.

Contribution

The authors prove Yano's conjecture for germs with two Puiseux pairs and distinct monodromy eigenvalues, and analyze the dependence of $b$-exponents on topology, highlighting cases where natural generalizations fail.

Findings

Yano's conjecture holds for two Puiseux pairs with distinct eigenvalues.

Natural generalizations of the conjecture do not always apply, as shown by counterexamples.

The structure of the resolution graph influences the behavior of $b$-exponents.

Abstract

In 1982, Tamaki Yano proposed a conjecture predicting how is the set of $b$-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In 1986, Pi.~Cassou-Nogu\`es proved the conjecture for the one Puiseux pair case. In a previous work the authors proved the conjecture for two Puiseux pairs germs whose complex algebraic monodromy has distinct eigenvalues. A natural problem induced by Yano's conjecture is, for a generic equisingular deformation of an isolated plane curve singularity germ to study how the set of $b$-exponents depends on the topology of the singularity. The natural generalization suggested by Yano's approach holds in suitable examples (for the case of isolated singularites which are Newton non-degenerated, commode and whose set of spectral numbers are all distincts). Morevover we show with an example that this natural generalization is not correct. We restrict to germs whose complex algebraic monodromy has distinct eigenvalues such that the embedded resolution graph has vertices of valency at most 3 and we discuss some examples with multiple eigenvalues.