Question

Let $\mathcal{A}$ be a collection of $n$ lines. Assume that $\mathcal{A}$ is not a pencil. It is known (see http://www.springerlink.com/content/320p742475v6q746/) that if all lines are in $\mathbb{RP}^2$, then there are at least $6n/13$ nodes.

What is the minimal number of nodes, if all lines are in $\mathbb{CP}^2$?

Answer 1

In a paper by Hirzebruch "Singularities of algebraic surfaces and characteristic numbers", Cont. Math. vol. 58, part I, 1986, there is an inequality for the number of double and triple points of an arrangement of $n$ complex lines. Denote by $t_k$ the number of points lying on precisely $k$ lines; if $t_n=t_{n-1}=t_{n-2}=0$, then $$t_2+3t_3/4\ge n+\sum_{r\ge 5}(2r-9)t_r.$$ This inequality is obtained from the Bogomolov-Miyaoka-Yau inequality for surfaces of general type: taking a suitable cover of the plane branched on an arrangement not satisfying the inequality above and desingularizing one would obtain a surface of general type with $c_1^2>3c_2$.