Free plane curves with a linear Jacobian syzygy

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.

💡 Research Summary

The paper addresses the classification of plane free curves whose Jacobian syzygy module contains a linear relation. A plane curve C = V(g) of degree n is called free when its Jacobian ideal J_g has projective dimension two, which is equivalent to the module Syz(J_g) being free of rank two. The authors focus on the situation where one of the two generators of Syz(J_g) is a triple of linear forms (A, B, C). By applying projective linear changes of coordinates, they reduce the possible linear triples to three normal forms, and then show that only two of these actually give rise to non‑trivial free curves: (i) A = a x, B = b y, C = c z with a, b, c not all equal and a ≠ 0, and (ii) A = y, B = z, C = 0.

In the first case, the Hilbert‑Burch matrix of the Jacobian ideal can be written as

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