Colengths of fractional ideals and Tjurina number of a reducible plane curve
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913503990775808 |
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| author | Hefez, Abramo Hernandes, Marcelo Escudeiro |
| author_facet | Hefez, Abramo Hernandes, Marcelo Escudeiro |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Colengths of fractional ideals and Tjurina number of a reducible plane curve Hefez, Abramo Hernandes, Marcelo Escudeiro Algebraic Geometry 13H10, 14H20 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]. |
| title | Colengths of fractional ideals and Tjurina number of a reducible plane curve |
| topic | Algebraic Geometry 13H10, 14H20 |
| url | https://arxiv.org/abs/2409.11153 |