Free plane curves with a linear Jacobian syzygy
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912758459531264 |
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| author | Beorchia, Valentina Gallet, Matteo Logar, Alessandro |
| author_facet | Beorchia, Valentina Gallet, Matteo Logar, Alessandro |
| contents | The study of planar free curves is a very active area of research, but a structural study of such a class is missing. We give a complete classification of the possible generators of the Jacobian syzygy module of a plane free curve under the assumption that one of them is linear. Specifically, we prove that, up to similarities, there are two possible forms for the Hilbert-Burch matrix. Our strategy relies on a translation of the problem into the accurate study of the geometry of maximal segments of a suitable triangle with integer points. Following this description, we are able to determine precisely the equations of free curves and the associated Hilbert-Burch matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free plane curves with a linear Jacobian syzygy Beorchia, Valentina Gallet, Matteo Logar, Alessandro Commutative Algebra Algebraic Geometry 13D02, 13P99, 14H45, 15A21 The study of planar free curves is a very active area of research, but a structural study of such a class is missing. We give a complete classification of the possible generators of the Jacobian syzygy module of a plane free curve under the assumption that one of them is linear. Specifically, we prove that, up to similarities, there are two possible forms for the Hilbert-Burch matrix. Our strategy relies on a translation of the problem into the accurate study of the geometry of maximal segments of a suitable triangle with integer points. Following this description, we are able to determine precisely the equations of free curves and the associated Hilbert-Burch matrices. |
| title | Free plane curves with a linear Jacobian syzygy |
| topic | Commutative Algebra Algebraic Geometry 13D02, 13P99, 14H45, 15A21 |
| url | https://arxiv.org/abs/2512.10928 |