Abstract

We prove the Anzis–Tohăneanu conjecture, that is, the Dirac–Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point that can be regarded as the Sylvester–Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also, we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

中文翻译：

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双点自由投影线排列

摘要

我们证明了 Anzis-Tohăneanu 猜想，即 Dirac-Motzkin 猜想，用于在任意特征零场上的投影平面中的超可解线排列。此外，我们证明了线的划分自由排列至少包含一个双点，对于某些自由排列，可以将其视为 Sylvester-Gallai 定理。这是一般结果的推论，如果您将一条线添加到自由投影线排列，则该线必须包含至少一个双点。此外，我们证明了与超可解线排列和双点数有关的一些猜想和一个未解决的问题。