Curves with Jacobian syzygies of the same degree

TL;DR

This paper investigates complex projective plane curves with Jacobian syzygies generated by their minimal degree component, revealing a surprisingly large class including certain line arrangements with specific singularities.

Contribution

It characterizes a broad class of curves with Jacobian syzygies generated in minimal degree, including smooth, maximal Tjurina, and specific line arrangements.

Findings

Examples include smooth and maximal Tjurina curves.

Line arrangements with only double and triple points are in this class under certain conditions.

The class of such curves is surprisingly large.

Abstract

In this notes we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However, this class of curves seems to be surprisingly large. In particular, any line arrangement $\mathcal A$ of $d=2k+1\geq 5 $ lines having only double and triple points is in this class if the number of triple points is $k$ and if they are all situated on a line $L \in \mathcal A$, see Theorem 6.2