Colengths of fractional ideals and Tjurina number of a reducible plane curve

TL;DR

This paper refines formulas for the Tjurina number of reducible plane curves by analyzing colengths of fractional ideals, confirming a conjecture by Dimca, and connecting algebraic invariants with the Jacobian ideal.

Contribution

It introduces more manageable formulas for colengths of fractional ideals and applies them to derive a new Tjurina number formula, confirming Dimca's conjecture.

Findings

New formula for Tjurina number of reducible plane curves

Affirmative proof of Dimca's conjecture

Connection established between Kähler differentials and Jacobian ideal

Abstract

In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a conjecture proposed by A. Dimca in [7]. Our results are obtained by establishing more manageable formulas to compute the colengths of fractional ideals of the local ring associated with the algebroid (not necessarily a complete intersection) curve with several branches. We then apply these results to the Jacobian ideal of a plane curve over $\mathbb C$ to get a new formula for its Tjurina number and a proof of Dimca's conjecture. We end the paper by establishing a connection between the module of K\"ahler differentials on the curve modulo its torsion, seen as a fractional ideal, and its Jacobian ideal, explaining the relation between the present approach and that of [1].