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 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 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.

Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves

“…This fact implies that C 2 is a jumping line for E C 1 in the case (1). The claim in case (2) follows from [12,Corollary 4.6 and Example 4.8]. Indeed, these results imply that we have…”

“…Using for instance [12,Example 4.8] we see that in this case d L 0 1 (C 1 ) = r 1 . The same equality holds for all the other non free reduced plane curves C 1 satisfying 2r 1 < d 1 , see [12,Corollary 4.5]. This fact implies that C 2 is a jumping line for E C 1 in the case (1).…”

“…Recall now the construction of the Bourbaki ideal B(C 1 , ρ ′ 1 ) associated to the curve C 1 and to the minimal degree syzygy ρ ′ 1 given by (3.2), as described in [12,Section 5]. It follows that the Bourbaki ideal B(C 1 , ρ ′ 1 ) is contained in the principal ideal generated by D. This is a contradiction, since the Bourbaki ideal B(C 1 , ρ ′ 1 ) defines a subscheme which is either empty (when C 1 is a free curve), or zero-dimensional, see [12,Theorem 5.1].…”