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

Alexandru Dimca; Gabriel Sticlaru

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arXiv · 1804.06349

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

Alexandru Dimca·Gabriel Sticlaru

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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.

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BibTeX RIS

Alexandru Dimca, Gabriel Sticlaru. 2018-11-22. On the jumping lines of bundles of logarithmic vector fields along plane curves. https://arxiv.org/abs/1804.06349

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