Extremal properties of line arrangements in the complex projective plane

Abstract

The main goal of the paper is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erdös-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito.

On conic-line arrangements with nodes, tacnodes, and ordinary triple points

“…Example 4. 13 Consider now the case with m = 4, and note that there are many solutions to (4.1). Take d 1 = 1 and d 2 = 2, then we can find the following Diophantine solution, namely t = 2 and n 2 = 1.…”

A few introductory remarks on line arrangements

“…Melchior's result shows that a non-trivial real line arrangement must not only have positive t 2 but in fact there must be at least 3 double points. Interesting generalizations of Melchior's inequality in the realms of complex line arrangements have been obtained by Hirzebruch [27], see also [39] and the article by Piotr Pokora in this volume [36].…”

Conic configurations via dual of quartic curves

“…In that paper, that conic arrangement appears in the context of generalized Kummer surfaces: the desingularization of the double cover branched over Č is a K3 surface X on which the pull-back of the conics are union of (−2)-curves which have as well an interesting configuration. The freeness of the arrangement of curves C 12 is studied in [5], where we learned that this configuration has been also independently discovered in [2].…”