Minimal number of nodes in a complex line arrangement. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:35:54Z http://mathoverflow.net/feeds/question/72663 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72663/minimal-number-of-nodes-in-a-complex-line-arrangement Minimal number of nodes in a complex line arrangement. Fei YE 2011-08-11T08:05:27Z 2011-08-13T10:20:53Z <p>Let $\mathcal{A}$ be a collection of $n$ lines. Assume that $\mathcal{A}$ is not a pencil. It is known (see <a href="http://www.springerlink.com/content/320p742475v6q746/" rel="nofollow">http://www.springerlink.com/content/320p742475v6q746/</a>) that if all lines are in $\mathbb{RP}^2$, then there are at least $6n/13$ nodes. </p> <p>What is the minimal number of nodes, if all lines are in $\mathbb{CP}^2$?</p> http://mathoverflow.net/questions/72663/minimal-number-of-nodes-in-a-complex-line-arrangement/72827#72827 Answer by rita for Minimal number of nodes in a complex line arrangement. rita 2011-08-13T10:20:53Z 2011-08-13T10:20:53Z <p>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 $c_1^2>3c_2$.</p>