TY - JOUR

T1 - Double Points of Free Projective Line Arrangements

AU - Abe, Takuro

N1 - Funding Information:
This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B)
Publisher Copyright:
© The Author(s) 2020.

PY - 2022/2/1

Y1 - 2022/2/1

N2 - We prove the Anzis-Tohaneanu 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.

AB - We prove the Anzis-Tohaneanu 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.

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U2 - 10.1093/imrn/rnaa145

DO - 10.1093/imrn/rnaa145

M3 - Article

AN - SCOPUS:85106219752

SN - 1073-7928

VL - 2022

SP - 1811

EP - 1824

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 3

ER -