Free plane curves with a linear Jacobian syzygy

2025/12/11 by Beorchia, Valentina, Gallet, Matteo, Logar, Alessandro
#13D02 #13P99 #14H45 #15A21 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.10928

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.

Citations

• Cohen--Macaulay ideals of codimension two and the geometry of plane points
• Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies
• A characterization of quasi-homogeneous singularities of free and nearly free plane curves
• Worst singularities of plane curves of given degree
• A bound for the Milnor number of plane curve singularities
• On the syzygies and Alexander polynomials of nodal hypersurfaces

Related

• The 0-th Fitting ideal of the Jacobian module of a plane curve
• Curves with Jacobian syzygies of the same degree
• Jacobian syzygies and plane curves with maximal global Tjurina numbers
• Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves
• On the Hilbert vector of the Jacobian module of a plane curve