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 P 2 , we study the set of jumping lines for the rank two vector bundle ThCi on 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 ThCi 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.