Quasihomogeneous isolated singularities in terms of syzygies and foliations
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| Format: | Preprint |
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2025
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| _version_ | 1866915504818487296 |
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| author | Hassanzadeh, Hamid Nejad, Abbas Nasrollah Simis, Aron |
| author_facet | Hassanzadeh, Hamid Nejad, Abbas Nasrollah Simis, Aron |
| contents | One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to the well-known result of Saito's a syzygy-theoretic characterization of a quasihomogeneous singularity affording an effective computational criterion. In the second theme, one explores the Milnor-Tjurina difference number from a commutative algebra viewpoint. Building on the Briancon-Skoda theorem and exponent, we extend previously known inequalities by Dimca and Greuel to arbitrary dimension and provide algebraic formulas involving the syzygy-theoretic part and reduction exponents. In the last theme one recovers and bring up to an algebraic light a result of Camacho and Movasati by establishing a couple of characterizations of quasihomogeneous isolated singularities in terms of the generators of the module of invariant vector fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasihomogeneous isolated singularities in terms of syzygies and foliations Hassanzadeh, Hamid Nejad, Abbas Nasrollah Simis, Aron Commutative Algebra Algebraic Geometry 13A30, 13D02, 13D07, 14B05 One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to the well-known result of Saito's a syzygy-theoretic characterization of a quasihomogeneous singularity affording an effective computational criterion. In the second theme, one explores the Milnor-Tjurina difference number from a commutative algebra viewpoint. Building on the Briancon-Skoda theorem and exponent, we extend previously known inequalities by Dimca and Greuel to arbitrary dimension and provide algebraic formulas involving the syzygy-theoretic part and reduction exponents. In the last theme one recovers and bring up to an algebraic light a result of Camacho and Movasati by establishing a couple of characterizations of quasihomogeneous isolated singularities in terms of the generators of the module of invariant vector fields. |
| title | Quasihomogeneous isolated singularities in terms of syzygies and foliations |
| topic | Commutative Algebra Algebraic Geometry 13A30, 13D02, 13D07, 14B05 |
| url | https://arxiv.org/abs/2509.16933 |