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arXiv · 1908.06885

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

Takuro Abe·Alexandru Dimca·Gabriel Sticlaru
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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 $d\geq 9$ of lines, and any number $n_3\geq 6$ of triple points. Moreover, we show that such examples are not possible for line arrangements having only double and triple points, with $n_3 \leq 5$.

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Takuro Abe, Alexandru Dimca, Gabriel Sticlaru. 2019-08-19. Addition-deletion results for the minimal degree of logarithmic derivations of arrangements. https://arxiv.org/abs/1908.06885

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