On the jumping lines of bundles of logarithmic vector fields along plane curves

Abstract

For a reduced curve $C:f=0$ in the complex projective plane ${\mathbb{P}}^2$, we study the set of jumping lines for the rank two vector bundle $T⟨C⟩$ on ${\mathbb{P}}^2$, whose sections are the logarithmic vector fields along $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T⟨C⟩$ is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourbaki ideal, a result going back to Schwarzenberger. Other classical general results by Barth, Hartshorne and Hulek resurface in the study of this special class of rank two vector bundles.

arXiv commentv6: several references added, material reorganized a little