On the jumping lines of bundles of logarithmic vector fields along plane curves
A Dimca, G Sticlaru - 2020 - projecteuclid.org
A Dimca, G Sticlaru
2020•projecteuclid.orgFor 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 T⟨C⟩ 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 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�…
lines for the rank two vector bundle T⟨C⟩ 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 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�…
Abstract
For a reduced curve in the complex projective plane , we study the set of jumping lines for the rank two vector bundle on whose sections are the logarithmic vector fields along . We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of and with the Bourbaki ideal of the module of Jacobian syzygies of . In particular, when the vector bundle is unstable, a line is a jumping line if and only if it meets the -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.
Project Euclid