Free plane curves with a linear Jacobian syzygy

TL;DR

This paper classifies plane free curves with a linear Jacobian syzygy, providing a complete description of their algebraic and geometric structure through Hilbert-Burch matrices.

Contribution

It offers a complete classification of generators of the Jacobian syzygy module for such curves, revealing two possible Hilbert-Burch matrix forms.

Findings

Two possible Hilbert-Burch matrix forms identified

Explicit equations of free curves derived

Geometric interpretation via integer points in triangles

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.