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arXiv · 2512.10928

Free plane curves with a linear Jacobian syzygy

Valentina Beorchia·Matteo Gallet·Alessandro Logar
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Abstract

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.

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Valentina Beorchia, Matteo Gallet, Alessandro Logar. 2025-12-11. Free plane curves with a linear Jacobian syzygy. https://arxiv.org/abs/2512.10928

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