2024/09/17 by Hefez, Abramo, Hernandes, Marcelo Escudeiro · 2 citations
#13H10 #14H20 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.11153
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].