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

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 determine 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 C1 and C2, 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.