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

TL;DR

This paper investigates the jumping lines of a rank two vector bundle of logarithmic vector fields along plane curves, revealing their relation to algebraic properties like the Jacobian module and Bourbaki ideal, and revisiting classical results.

Contribution

It establishes new links between jumping lines and algebraic invariants of plane curves, extending classical results to the context of logarithmic vector bundles.

Findings

Jumping lines correspond to intersections with the Bourbaki ideal when the bundle is unstable.

The study connects geometric properties of jumping lines with algebraic invariants of the curve.

Classical theorems by Barth, Hartshorne, and Hulek are applied to this specific setting.

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\langle C \rangle $ 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\langle C \rangle $ 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.