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2020 On the jumping lines of bundles of logarithmic vector fields along plane curves
Alexandru Dimca, Gabriel Sticlaru
Publ. Mat. 64(2): 513-542 (2020). DOI: 10.5565/PUBLMAT6422006

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.

Citation

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Alexandru Dimca. Gabriel Sticlaru. "On the jumping lines of bundles of logarithmic vector fields along plane curves." Publ. Mat. 64 (2) 513 - 542, 2020. https://doi.org/10.5565/PUBLMAT6422006

Information

Received: 23 November 2018; Revised: 12 September 2019; Published: 2020
First available in Project Euclid: 3 July 2020

zbMATH: 07236052
MathSciNet: MR4119261
Digital Object Identifier: 10.5565/PUBLMAT6422006

Subjects:
Primary: 14J60
Secondary: 13D02 , 14B05 , 14H50 , 32S05 , 32S22

Keywords: Jacobian module , jumping line , logarithmic vector fields , plane curve , splitting type , stable bundle , vector bundle

Rights: Copyright © 2020 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.64 • No. 2 • 2020
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