On complex supersolvable line arrangements

Takuro Abe; Alexandru Dimca

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arXiv · 1907.12497

On complex supersolvable line arrangements

Takuro Abe·Alexandru Dimca

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Abstract

We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower bound for the number of double points $n_2$ in an $m$--homogeneous supersolvable line arrangement of $d$ lines is also considered. When $3 \leq m \leq 5$, or when $m \geq \frac{d}{2}$, or when there are at least two modular points, we show that $n_2 \geq \frac{d}{2}$, as conjectured by B. Anzis and S. O. Toh\u aneanu. This conjecture is shown to hold also for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements having a generic vertex.

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Takuro Abe, Alexandru Dimca. 2019-07-29. On complex supersolvable line arrangements. https://arxiv.org/abs/1907.12497

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