# The minimal Tjurina number of irreducible germs of plane curve singularities

## TL;DR

This paper establishes a formula for the minimal Tjurina number of irreducible plane curve singularities, confirming a conjecture and linking it to resolution multiplicities, advancing understanding of singularity invariants.

## Contribution

It provides a closed-form expression for the minimal Tjurina number based on resolution multiplicities, answering a question posed by Dimca and Greuel.

## Key findings

- Derived a formula for minimal Tjurina number in terms of resolution data
- Confirmed the relation between Milnor and Tjurina numbers for irreducible germs
- Built on previous work by Genzmer, Wall, and Mattei

## Abstract

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer, Wall and Mattei.