Addition-deletion results for the minimal degree of logarithmic derivations of arrangements

Abstract

We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane, and give a number of applications. First, we prove the existence of Tjurina maximal line arrangements in a lot of new situations. Then, starting with Ziegler's example of a pair of arrangements of $d=9$ lines with $n_3=6$ triple points in addition to some double points, having the same combinatorics, but distinct minimal degree of a logarithmic derivation, we construct new examples of such pairs, for any number $d\ge 9$ of lines, and any number $n_3\ge 6$ of triple points. Moreover, we show that such examples are not possible for line arrangements having only double and triple points, with $n_3\le 5$.

v2: We prove the existence Tjurina maximal line arrangements in many new cases, see Theorem 4.4 and Proposition 4.6

References in corpus (3)

• Average four-genus of two-bridge knots
• Jacobian syzygies and plane curves with maximal global Tjurina numbers
• Triangular arrangements on the projective plane