Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918300985851904 |
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| author | Dimca, Alexandru Sticlaru, Gabriel |
| author_facet | Dimca, Alexandru Sticlaru, Gabriel |
| contents | In this paper we compute an explicit closed formula for the total Tjurina number $τ(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This formula allows a completely new view point on the classical upper bounds for the total Tjurina number $τ(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall. This approach yields in particular a new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of a reduced plane curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08583 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves Dimca, Alexandru Sticlaru, Gabriel Algebraic Geometry Commutative Algebra In this paper we compute an explicit closed formula for the total Tjurina number $τ(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This formula allows a completely new view point on the classical upper bounds for the total Tjurina number $τ(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall. This approach yields in particular a new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of a reduced plane curve. |
| title | Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2601.08583 |