Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves

TL;DR

This paper investigates how the minimal degree of Jacobian syzygies of a union of two curves relates to their decomposition, providing new geometric criteria for jumping lines in the context of quasihomogeneous singularities.

Contribution

It introduces novel geometric criteria for jumping lines based on the decomposition of curves and the invariant mdr(f), especially in quasihomogeneous singularity cases.

Findings

Relations between mdr(f) and curve decomposition are established.

New criteria for identifying jumping lines for logarithmic vector fields.

Results apply to curves with quasihomogeneous singularities.

Abstract

Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know, for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve $C$.