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arXiv · 2409.11153

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

Abramo Hefez·Marcelo Escudeiro Hernandes
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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 Kähler 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].

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Abramo Hefez, Marcelo Escudeiro Hernandes. 2024-09-17. Colengths of fractional ideals and Tjurina number of a reducible plane curve. https://arxiv.org/abs/2409.11153

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