Abstract: The famous Sylvester-Gallai theorem asserts that a line arrangement in the real projective plane has at least one double point unless all the lines pass one point. The Dirac-Motzkin conjecture, which was almost solved by Green-Tao, asserts in the same setup, the number of double points is at least the half of the number of lines. However, these results do not hold true for lines in the complex projective plane.

We study this double point problem for line arrangements in the complex projective plane by using algebra of line arrangements, called the logarithmic vector fields and its nice property called the freeness. By using it we prove the Anzis-Tohaneanu conjecture that is the Dirac-Motzkin conjecture for supersolvable complex line arrangements. We conjecture that every line arrangement which has no double points is free, and study this conjecture from the surjectivity of some restriction maps.